A roll of the dice

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Re: A roll of the dice
Gord wrote:Scott Mayers wrote:it is odd should the magician just say, "your wrong" or "you are an idiot".
you're
Precisely!! (Not that I was presenting the puzzle, though)
I eat without fear of certain Death from The Tree of Knowledge because with wisdom, we may one day break free from its mortal curse.

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Re: A roll of the dice
Gord wrote:Scott Mayers wrote:Simply for the question as asked above, you can ask whether 'trust' counts here.
That's true for every riddle ever told. Also for every test question ever asked on any exam, ever.
What's 3 time 9?
Twentyseven.
Wrong! I lied! The real question was: what's 9 times 8. You said twentyseven, so you fail maths class.
So yes, you can assume I may have been lying when I opened the question with the statement, "Here's one for your statistical analysis." But honestly, I don't see the point.
I'm obviously new to all this, but this thread is a crack up!
The general idea is to recall that probability is the number of ways an event can happen divided by the number of possible outcomes (sample space). The sample space is defined by the problem.
so here, the sample space is all possible outcomes in which two dice are thrown and at least one die shows a '1'. Gord showed there are 11 possible outcomes in which one die shows a '1', and of the 11 there's only one way to get snakeeyes. so 1/11.
apologies if I'm stating this in the same way as someone else (I admit I didn't read every response), but I think this is the simplest way to describe this.

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Re: A roll of the dice
Thanks, Tallboy. I think this is just one of those math problems that people think just feels wrong, and they won't accept the math because the answer seems to go against their intuition.
"Knowledge grows through infinite timelessness"  the random fictional Deepak Chopra quote site
"Imagine an ennobling of what could be"  the New Age BS Generator site
"You are also taking my words out of context."  Justin
"Nullius in verba"  The Royal Society ["take nobody's word for it"]
#ANDAMOVIE
Is Trump in jail yet?
"Imagine an ennobling of what could be"  the New Age BS Generator site
"You are also taking my words out of context."  Justin
"Nullius in verba"  The Royal Society ["take nobody's word for it"]
#ANDAMOVIE
Is Trump in jail yet?

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Re: A roll of the dice
Gord wrote:Thanks, Tallboy. I think this is just one of those math problems that people think just feels wrong, and they won't accept the math because the answer seems to go against their intuition.
Wording is everything though. If presented a 'puzzle', the error that many puzzle makers present (just as many math problem questions do when wording) is to assume a colloquial question to a "trick" problem which makes one require interpreting something 'cultural' about the semantics and grammar particular people use. In your 'puzzle', repeated here:
Gord wrote:Here's one for your statistical analysis:
I'm sitting at a table. You walk in and hand me a pair of dice. Each die has six sides, and you are certain that they are fair  in other words, they aren't loaded at all, and they would be acceptable to anyone who wanted to roll fairly.
You sit down across the table from me. Then I roll both dice behind a screen where you can't see them.
"Did you roll at least one 1?" you ask me.
I look at the dice. "Yes," I reply. "At least one of these two dice shows a 1 on top."
Now here's the question: What are the odds that I rolled two 1s?
Although you demand assuming 'fairness' of the die, it begs questioning why you would 'slip' in : "...Then I roll both dice behind a screen where you can't see them." If a magician were to even be demonstrating some 'trick', they would obviously be either mistaken to assume the audience should default to trust they assume you are remaining 'fair' even if you had some ability to determine the dice fair. To prevent this misunderstanding would require that you ask the audience to toss the die unless you as a magician were just joking as part of the intended performance.
I know you were likely thinking your question lacked any 'fault' and would be understood to treat your option to hide your roll 'fair' too. But then you must be EXPLICIT with this since the nature of the kind of question is intended to evoke close scrutiny. So it MATTERS if you ask a question that raises potential ambiguity to a serious question. It was this to which I find most probability/statistic questions worthy of serious criticism. You can't expect faith of precision merely on select interpretation of the one presenting the puzzle or question to excuse their own subtle use of words to be careless when you demand others of just the opposite with the actual puzzle in question.
So to prevent confusion should at least require us to do the rolling, not yourself.
So, instead, if you said,
You sit across the table from me. Then I ask YOU to roll both die. Then, what is the chance that any 'one' would show up on either die? [Assuming you are not a trickster yourself, that is!]
The extension to ask us to trust you AND also hide it (why is it necessary to hide it?) justly assures one should question your integrity. After all, a good magician could prove an outcome of 100% of ones on every throw.
I eat without fear of certain Death from The Tree of Knowledge because with wisdom, we may one day break free from its mortal curse.

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Re: A roll of the dice
The results must be hidden in order for the answer to be 1/11. If you can see the dice, then the answer is 1/6 because you know which one rolled the 1 that is reported in the question.
You are certain the dice are fair. I roll them where you cannot see them, and report that at least one die shows a 1. Any ambiguity is brought into the question by you, not by the question  as I said before, and which Tallboy quoted earlier, this is "true for every riddle ever told. Also for every test question ever asked on any exam, ever." If you want to insist on your introduced ambiguity on an exam, you are going to fail the test question.
You are certain the dice are fair. I roll them where you cannot see them, and report that at least one die shows a 1. Any ambiguity is brought into the question by you, not by the question  as I said before, and which Tallboy quoted earlier, this is "true for every riddle ever told. Also for every test question ever asked on any exam, ever." If you want to insist on your introduced ambiguity on an exam, you are going to fail the test question.
"Knowledge grows through infinite timelessness"  the random fictional Deepak Chopra quote site
"Imagine an ennobling of what could be"  the New Age BS Generator site
"You are also taking my words out of context."  Justin
"Nullius in verba"  The Royal Society ["take nobody's word for it"]
#ANDAMOVIE
Is Trump in jail yet?
"Imagine an ennobling of what could be"  the New Age BS Generator site
"You are also taking my words out of context."  Justin
"Nullius in verba"  The Royal Society ["take nobody's word for it"]
#ANDAMOVIE
Is Trump in jail yet?

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Re: A roll of the dice
Gord wrote:The results must be hidden in order for the answer to be 1/11. If you can see the dice, then the answer is 1/6 because you know which one rolled the 1 that is reported in the question.
You are certain the dice are fair. I roll them where you cannot see them, and report that at least one die shows a 1. Any ambiguity is brought into the question by you, not by the question  as I said before, and which Tallboy quoted earlier, this is "true for every riddle ever told. Also for every test question ever asked on any exam, ever." If you want to insist on your introduced ambiguity on an exam, you are going to fail the test question.
I've heard Scott's reasoning before, and there is something to be said for it in certain situations. Most probability problems of this type are designed as way to understand more complicated problems by way of analogy, so to speak. so coin flipping problems are often used to understand genetics, odd's of DNA tests, etc. (indeed, does anyone outside of the gambling establishment really care about the odds of getting snakeeyes given one of the two die show a 1??) but these are done initially for simple cases. simplifying assumptions are almost always independence (results from one dies are independent of the other), random results (fairness), and that each roll has the same probability (usually referred to as IID: identically and independently distributed). under these conditions, we can come up with some answers that are often pretty cool (counterintuitive). This experiment is such a case.
if we assume that the rolls are not independent (correlated) or unfair then we run into major problems, especially when trying to come up with a closedform solution (something you can write out) so we often turn to simulations in this age of computers (no so pre1980ish). So in real life we do often have to deal with this, and it seems that's what Scott is saying. but if we assume cheating (nonindepence) then we have a dissertation rather than a cute little problem.
So we assume things like random throws, fairness, etc. for the purposes of problems like this. and we have to know how to solve these types of problems first before going to more complex ones.
p.s. Scott's point earlier about random number generators... yes, they are pseudorandom number generators as they do repeat at some point. 20odd years ago when i used to do these sorts of things the patterns would repeat at about 2^16, which is WAY beyond what we need for a problem like this to converge (i would assume 100 simulations is more than enough but we could easily go to 1000 just to make sure). it's the real complicated problems (like if we include cheating) that require large numbers of simulations where we may have to worry about patterns. but i would guess that newer algorithms and higher speed computers have improved things substantially.
i hope i'm interpreting both your positions accurately.

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Re: A roll of the dice
Gord wrote:Thanks, Tallboy. I think this is just one of those math problems that people think just feels wrong, and they won't accept the math because the answer seems to go against their intuition.
yes. my brother (an engineer) called me a 'conman' after explaining the montehall 3door problem to him! he just wouldn't buy it no matter what i said. There was a psychologist/statistician at Stanford by the name of Amos Twersky who used to write extensively on people's misinterpretation of probability. i highly recommend his work. here's his NY Times obituary from 1996:
http://www.nytimes.com/1996/06/06/us/am ... .html?_r=0

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Re: A roll of the dice
Tallboy wrote:Gord wrote:Thanks, Tallboy. I think this is just one of those math problems that people think just feels wrong, and they won't accept the math because the answer seems to go against their intuition.
yes. my brother (an engineer) called me a 'conman' after explaining the montehall 3door problem to him! he just wouldn't buy it no matter what i said. There was a psychologist/statistician at Stanford by the name of Amos Twersky who used to write extensively on people's misinterpretation of probability. i highly recommend his work. here's his NY Times obituary from 1996. it has a couple of examples of the types of probability problems he worked on:
http://www.nytimes.com/1996/06/06/us/am ... .html?_r=0

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Re: A roll of the dice
Tallboy wrote:Gord wrote:Thanks, Tallboy. I think this is just one of those math problems that people think just feels wrong, and they won't accept the math because the answer seems to go against their intuition.
yes. my brother (an engineer) called me a 'conman' after explaining the montehall 3door problem to him! he just wouldn't buy it no matter what i said. There was a psychologist/statistician at Stanford by the name of Amos Twersky who used to write extensively on people's misinterpretation of probability. i highly recommend his work. here's his NY Times obituary from 1996:
http://www.nytimes.com/1996/06/06/us/am ... .html?_r=0
Whenever I doubt the math, I try to do the experiment. I did both the Monty Hall and the question I asked in this thread to see if the results came close to those predicted by the maths, and they did.
Hey! Google tells me that Monty Hall (born Maurice Halperin) turned 95 on August 25th! He was born in my home town of Winnipeg.
"Knowledge grows through infinite timelessness"  the random fictional Deepak Chopra quote site
"Imagine an ennobling of what could be"  the New Age BS Generator site
"You are also taking my words out of context."  Justin
"Nullius in verba"  The Royal Society ["take nobody's word for it"]
#ANDAMOVIE
Is Trump in jail yet?
"Imagine an ennobling of what could be"  the New Age BS Generator site
"You are also taking my words out of context."  Justin
"Nullius in verba"  The Royal Society ["take nobody's word for it"]
#ANDAMOVIE
Is Trump in jail yet?

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Re: A roll of the dice
Gord wrote:The results must be hidden in order for the answer to be 1/11. If you can see the dice, then the answer is 1/6 because you know which one rolled the 1 that is reported in the question.
You are certain the dice are fair. I roll them where you cannot see them, and report that at least one die shows a 1. Any ambiguity is brought into the question by you, not by the question  as I said before, and which Tallboy quoted earlier, this is "true for every riddle ever told. Also for every test question ever asked on any exam, ever." If you want to insist on your introduced ambiguity on an exam, you are going to fail the test question.
I'm confused at your interpretation of 1/6. It is likely some language problem and partly what bothers me about how some use statistics and probability. I was pointing to the fact that when puzzles are presented by many using statistics, their mathematical background may not be a problem, but their lack of equal skill using logic in language IS. I find this a lot in half the textbooks on math especially with respect to 'problem' questions in that the language of the authors presenting the question are deluded into thinking they've appropriately asked the question correctly.
I have run into this in test questions by teachers with similar problems and end up having to write CONDITIONAL multiple solutions just to make sure they don't treat one answer 'wrong' when it is THEY who are making the error as 'teachers'. And then the students who fall for the trap by responding to one way are deemed as the idiots when this is NOT the case. It just ends up making many people turn away from math and science altogether, even IF they had real competence.
And to the Monty Hall problem, the same thing occurs with the way the question is posed AND to the lack of the puzzle poser to NOT see that they lack the same credibility to KNOW their own lack of Language logic as they do with their mathematical logic. I also see how this problem relates to confusion in Quantum Mechanics Interpretation. The Monty Hall problem is an excellent example of the problems concerning the 'trinity' relations of (1, 2, and 3) OR to (0, 1, and Infinity) as numbers being used in problems. For instance, the use of the 100door example is severely flawed BECAUSE the limited options using only 3 doors differs.
I don't want to go further into this as I had before unless we discuss the interpretation problems in QM, something I doubt many here are sufficiently familiar with.
I eat without fear of certain Death from The Tree of Knowledge because with wisdom, we may one day break free from its mortal curse.

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Re: A roll of the dice
Tallboy wrote:Gord wrote:The results must be hidden in order for the answer to be 1/11. If you can see the dice, then the answer is 1/6 because you know which one rolled the 1 that is reported in the question.
You are certain the dice are fair. I roll them where you cannot see them, and report that at least one die shows a 1. Any ambiguity is brought into the question by you, not by the question  as I said before, and which Tallboy quoted earlier, this is "true for every riddle ever told. Also for every test question ever asked on any exam, ever." If you want to insist on your introduced ambiguity on an exam, you are going to fail the test question.
I've heard Scott's reasoning before, and there is something to be said for it in certain situations. Most probability problems of this type are designed as way to understand more complicated problems by way of analogy, so to speak. so coin flipping problems are often used to understand genetics, odd's of DNA tests, etc. (indeed, does anyone outside of the gambling establishment really care about the odds of getting snakeeyes given one of the two die show a 1??) but these are done initially for simple cases. simplifying assumptions are almost always independence (results from one dies are independent of the other), random results (fairness), and that each roll has the same probability (usually referred to as IID: identically and independently distributed). under these conditions, we can come up with some answers that are often pretty cool (counterintuitive). This experiment is such a case.
if we assume that the rolls are not independent (correlated) or unfair then we run into major problems, especially when trying to come up with a closedform solution (something you can write out) so we often turn to simulations in this age of computers (no so pre1980ish). So in real life we do often have to deal with this, and it seems that's what Scott is saying. but if we assume cheating (nonindepence) then we have a dissertation rather than a cute little problem.
So we assume things like random throws, fairness, etc. for the purposes of problems like this. and we have to know how to solve these types of problems first before going to more complex ones.
p.s. Scott's point earlier about random number generators... yes, they are pseudorandom number generators as they do repeat at some point. 20odd years ago when i used to do these sorts of things the patterns would repeat at about 2^16, which is WAY beyond what we need for a problem like this to converge (i would assume 100 simulations is more than enough but we could easily go to 1000 just to make sure). it's the real complicated problems (like if we include cheating) that require large numbers of simulations where we may have to worry about patterns. but i would guess that newer algorithms and higher speed computers have improved things substantially.
i hope i'm interpreting both your positions accurately.
True 'randomness' is NOT 'fair'. When 'fair', it loses being relatively indeterminate and why using large repeated samples are used. It is NOT logically rational to use an 'empirical' test to determine whether the Monty Hall problem is or is not true, for instance, something I get frustrated with some who believe this.
I eat without fear of certain Death from The Tree of Knowledge because with wisdom, we may one day break free from its mortal curse.

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Re: A roll of the dice
Scott Mayers wrote:Gord wrote:The results must be hidden in order for the answer to be 1/11. If you can see the dice, then the answer is 1/6 because you know which one rolled the 1 that is reported in the question.
You are certain the dice are fair. I roll them where you cannot see them, and report that at least one die shows a 1. Any ambiguity is brought into the question by you, not by the question  as I said before, and which Tallboy quoted earlier, this is "true for every riddle ever told. Also for every test question ever asked on any exam, ever." If you want to insist on your introduced ambiguity on an exam, you are going to fail the test question.
I'm confused at your interpretation of 1/6.
The odds of any die rolling a 1 is one in six. It's only when you know one of them rolled a 1 but you don't know which one it was that the chances of the second die also having rolled a 1 becomes one in eleven.
"Knowledge grows through infinite timelessness"  the random fictional Deepak Chopra quote site
"Imagine an ennobling of what could be"  the New Age BS Generator site
"You are also taking my words out of context."  Justin
"Nullius in verba"  The Royal Society ["take nobody's word for it"]
#ANDAMOVIE
Is Trump in jail yet?
"Imagine an ennobling of what could be"  the New Age BS Generator site
"You are also taking my words out of context."  Justin
"Nullius in verba"  The Royal Society ["take nobody's word for it"]
#ANDAMOVIE
Is Trump in jail yet?

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Re: A roll of the dice
Gord wrote:Tallboy wrote:Gord wrote:Thanks, Tallboy. I think this is just one of those math problems that people think just feels wrong, and they won't accept the math because the answer seems to go against their intuition.
yes. my brother (an engineer) called me a 'conman' after explaining the montehall 3door problem to him! he just wouldn't buy it no matter what i said. There was a psychologist/statistician at Stanford by the name of Amos Twersky who used to write extensively on people's misinterpretation of probability. i highly recommend his work. here's his NY Times obituary from 1996:
http://www.nytimes.com/1996/06/06/us/am ... .html?_r=0
Whenever I doubt the math, I try to do the experiment. I did both the Monty Hall and the question I asked in this thread to see if the results came close to those predicted by the maths, and they did.
that is a good idea. whenever doing complex probability problems (or one's i didn't understand well) i would run a
simulation to verify results.
Gord wrote:Hey! Google tells me that Monty Hall (born Maurice Halperin) turned 95 on August 25th! He was born in my home town of Winnipeg.
cool! i'm glad he's still with us and i didn't know he was Canadian. btw, the monty hall problem actually dates back to the 1800's. it was posed by joseph bertrand (box problem) then later by martin gardner (3 prisoner problem). it's a common problem in undergraduate senior probability course. whenever i hear mathematicians giving the wrong answer, it reminds me that mathematicians often know little of of specialties outside their area. Number theorists, topologists, and probabilists (sp?) often know little of each other's fields. there is no probability theorists or statisticians that i know of that would get this problem wrong... as i said it's a standard problem in an undergraduate intro to probability course, and the formal answer tests student's knowledge on how to use Bayes Theorem.

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Re: A roll of the dice
Scott Mayers wrote:Tallboy wrote:Gord wrote:The results must be hidden in order for the answer to be 1/11. If you can see the dice, then the answer is 1/6 because you know which one rolled the 1 that is reported in the question.
You are certain the dice are fair. I roll them where you cannot see them, and report that at least one die shows a 1. Any ambiguity is brought into the question by you, not by the question  as I said before, and which Tallboy quoted earlier, this is "true for every riddle ever told. Also for every test question ever asked on any exam, ever." If you want to insist on your introduced ambiguity on an exam, you are going to fail the test question.
I've heard Scott's reasoning before, and there is something to be said for it in certain situations. Most probability problems of this type are designed as way to understand more complicated problems by way of analogy, so to speak. so coin flipping problems are often used to understand genetics, odd's of DNA tests, etc. (indeed, does anyone outside of the gambling establishment really care about the odds of getting snakeeyes given one of the two die show a 1??) but these are done initially for simple cases. simplifying assumptions are almost always independence (results from one dies are independent of the other), random results (fairness), and that each roll has the same probability (usually referred to as IID: identically and independently distributed). under these conditions, we can come up with some answers that are often pretty cool (counterintuitive). This experiment is such a case.
if we assume that the rolls are not independent (correlated) or unfair then we run into major problems, especially when trying to come up with a closedform solution (something you can write out) so we often turn to simulations in this age of computers (no so pre1980ish). So in real life we do often have to deal with this, and it seems that's what Scott is saying. but if we assume cheating (nonindepence) then we have a dissertation rather than a cute little problem.
So we assume things like random throws, fairness, etc. for the purposes of problems like this. and we have to know how to solve these types of problems first before going to more complex ones.
p.s. Scott's point earlier about random number generators... yes, they are pseudorandom number generators as they do repeat at some point. 20odd years ago when i used to do these sorts of things the patterns would repeat at about 2^16, which is WAY beyond what we need for a problem like this to converge (i would assume 100 simulations is more than enough but we could easily go to 1000 just to make sure). it's the real complicated problems (like if we include cheating) that require large numbers of simulations where we may have to worry about patterns. but i would guess that newer algorithms and higher speed computers have improved things substantially.
i hope i'm interpreting both your positions accurately.
True 'randomness' is NOT 'fair'. When 'fair', it loses being relatively indeterminate and why using large repeated samples are used. It is NOT logically rational to use an 'empirical' test to determine whether the Monty Hall problem is or is not true, for instance, something I get frustrated with some who believe this.
i'm not sure i understand what you're saying, specifically "When 'fair', it loses being relatively indeterminate." can you expand on this? also, what is your answer to this question? apologies if you wrote this above and i missed it.
true randomness is not 'necessarily' fair. 'fairness' is a process... masking (closing your eyes, being behind a curtain, having observers insure 'fairness', etc.). but if the results are independent and truly random, that would indicate that the results of each roll are unpredictable (rolls are independent of previous rolls or any systematic 'rule'). so if truly random why would this not be an indication of fairness? and when i say 'fairness' i mean these random results many not have arisen from a 'fair' process, but the results are still indistinguishable from those arising from a fair process and therefore give us the correct answer.
and why is it not logically rational to use an empirical test? all the simulation is doing is playing the game 1000's of times. results will converge to the true answer (as shown by Gord and others) and a simple problem like this does not suffer from pseudorandom number generators repeating as one would not need many samples for convergence. i'm interested in understanding your position but i'm not sure i'm getting it.
btw, as i stated before, problems such as these are stylized examples used to illustrate a point. you must assumes certain things (fairness, randomness, etc.) and the result follows. once we understand these simple problems we are better equipped to understand more complicated ones. but the way the problem is posed, pretend it's an exam problem and answer assuming the setup posed.

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Re: A roll of the dice
Tallboy wrote:Hey! Google tells me that Monty Hall (born Maurice Halperin) turned 95 on August 25th! He was born in my home town of Winnipeg.
cool! i'm glad he's still with us and i didn't know he was Canadian. btw, the monty hall problem actually dates back to the 1800's. it was posed by joseph bertrand (box problem) then later by martin gardner (3 prisoner problem). it's a common problem in undergraduate senior probability course. whenever i hear mathematicians giving the wrong answer, it reminds me that mathematicians often know little of of specialties outside their area. Number theorists, topologists, and probabilists (sp?) often know little of each other's fields. there is no probability theorists or statisticians that i know of that would get this problem wrong... as i said it's a standard problem in an undergraduate intro to probability course, and the formal answer tests student's knowledge on how to use Bayes Theorem.[/quote]
I agree with the problem that special studies lack common ground. We no longer use a 'broad' foundational education beginning in philosophy and logic. The way its done now is about practicality only and so favors those with stronger clerical aptitudes, not logical ones.
The Monty Hall problem has absolutely NOTHING to do with Bayes Theorem. Or did I read that wrong?
I eat without fear of certain Death from The Tree of Knowledge because with wisdom, we may one day break free from its mortal curse.

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Re: A roll of the dice
Tallboy wrote:Scott Mayers wrote:True 'randomness' is NOT 'fair'. When 'fair', it loses being relatively indeterminate and why using large repeated samples are used. It is NOT logically rational to use an 'empirical' test to determine whether the Monty Hall problem is or is not true, for instance, something I get frustrated with some who believe this.
i'm not sure i understand what you're saying, specifically "When 'fair', it loses being relatively indeterminate." can you expand on this? also, what is your answer to this question? apologies if you wrote this above and i missed it.
I should have reedited my wording before publishing as it looks awkwardly said.
What is 'fair' means that the distribution of odds of a set of possibilities are EQUAL in distribution and that the actual occurrences OBEY outcomes that favor this. If 1/2 of the time X occurs, and this is measured to be the case, it suggests there is no actual 'randomness'. In other words, true randomness is NEVER 'fair'. If you could PREDICT an outcome in principle, it is 'determinate'; when you cannot, it is 'indeterminate'. Thus the likelihood of some outcome to occur is sincerely RANDOM, if it cannot be determined IN PRINCIPLE (is thus 'indeterminate). A probability IS 'determinate' only if it is actually 'fair' but loses its capacity to refer to something 'natural'.
This was what Einstein was referring to about questioning whether nature (God) throws dice with respect to Quantum Mechanics. While it is acceptable to treat probability as a PRACTICAL math to aid in determining truth, some interpret the nature of knowing a probability AND that it is IN PRINCIPLE a function of reality is the question here.
For example, if you toss a coin ONCE, what is the probability of it being heads (not tails)? It is either 100% or 0% because reality does not permit us to simultaneously have that EXACT same toss being done more than once. If you specifically assert that the probability is measured over a million times and demonstrates the probability of the SET of events is 50%, this is due to the approach of being able to repeat the event over and over which defeats the "random" reality. What you CAN say is that since the coin you toss in the single event has a probability 'space' treating the outcomes, heads to mean nontails, that 1 of EITHER a head or nonhead out of 2 possibilities will occur. But you cannot use this fact to assume that 1 out of 2 'times' [see the language here?] you get heads because you are assuring something 'fair' about tossing as being 'random' at the same time. This is just another way of saying that something is both determinate AND indeterminate at the same time.
This is what was troublesome to the Copenhagen Interpretation of QM that Einstein and others were trying to point out. If you ACCEPT contradiction to BE a normal function of reality [an 'in Principle' factor], this could be 'true' [like having multiple worlds occurring side by side], but you cannot assume that WHAT you observe IS in fact a REAL paradox/contradiction. So, for the double slit experiment, the assumption that the interference pattern is 'proof' of something IN PRINCIPLE is demonstrating nature as being paradoxical has to be questioned for the same reason you can't expect that tossing a coin will demonstrate a single flip that shows 1/2 of a head on the same toss with 1/2 of a tail. Its either 100% head and 0% tail or vice versa.
So the concern I am raising is in support of Einstein's position and how it relates to the math of probability in general. Many of the 'puzzles' presented can ALSO prove to be 'true' merely for those BELIEVING it. A good example is the Secretary Puzzle in which the math actually 'discovered' is used in reality to determine valued of certain futures. But when everyone believes in its value, even based on some error of interpretation, the 'experiments' to prove its validity will tend to also PROVE the math being used even if in error by that interpretation.
I eat without fear of certain Death from The Tree of Knowledge because with wisdom, we may one day break free from its mortal curse.

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Re: A roll of the dice
Scott Mayers wrote:I agree with the problem that special studies lack common ground. We no longer use a 'broad' foundational education beginning in philosophy and logic. The way its done now is about practicality only and so favors those with stronger clerical aptitudes, not logical ones.
not sure what you mean. can you give an example?
Scott Mayers wrote:The Monty Hall problem has absolutely NOTHING to do with Bayes Theorem. Or did I read that wrong?
the formal solution, indeed, the one a student would be expected to show on a formal probability exam, would use bayes theorem to solve:
http://angrystatistician.blogspot.com/2 ... hall.html

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Re: A roll of the dice
Tallboy wrote:Scott Mayers wrote:I agree with the problem that special studies lack common ground. We no longer use a 'broad' foundational education beginning in philosophy and logic. The way its done now is about practicality only and so favors those with stronger clerical aptitudes, not logical ones.
not sure what you mean. can you give an example?
Education today emphasizes teaching clerical factors up front, such as rote facts rather than 'proof' from the beginning. They utilize an 'abstract' topdown methodology rather than a foundational one. An example would be how one is required to first learn a higher order language prior to understanding the logic BENEATH this level. In pretwentieth century formal training, the emphasis was to BEGIN with questioning things up front....enticing people to question reality using LOGIC early on. We reverse this. Only when one is challenging their "Phd", [Philosophic degree] are they permitted to use "free" thought, for instance. Prior to that level, the emphasis is to require essays or papers to PROVE things based on OTHER people's insights. It focuses on granting 'credit' to others. So even, for instance, if you had a good logical capacity to 'prove' something without requiring other references, you are discouraged from this UNTIL you have reached the Mastery level.
Clerical factors thus favor those ENTERING things like science as those who have a GOOD memory and the capacity to 'follow' rather than think up front.
In general, we do not use the Euclidean approach where we assume simple things that we all agree on and then prove step by step what we hope they understand. Instead, we DEMAND that we have faith in the authorities of "science" as an institute PRIOR to being privileged to the PROOF.
Scott Mayers wrote:The Monty Hall problem has absolutely NOTHING to do with Bayes Theorem. Or did I read that wrong?
the formal solution, indeed, the one a student would be expected to show on a formal probability exam, would use bayes theorem to solve:
http://angrystatistician.blogspot.com/2 ... hall.html
That's going BEYOND a necessary way to prove the intended solutions. The WAY the puzzle is presented is purposely meant to appear OBSCURE by being elliptical (leaving OUT necessary premises). It is like a magician using some 'trick' only once because if people saw it again close up, it would no longer BE as profound or magical as it first appears.
Or, it is like if I wanted to Prove someone exists in the next room, instead of simply asking the person in doubt to go see for themselves directly, that you set up mirrors everywhere and then PROVE the logic of mirrors first. Sure, it can be done but is not only unnecessary, it presents more 'doubt' if you care to question the details.
I learned how to play chess initially by certain friends who had played since they were one month old. But upon learning the rules, I was simply too incompetent and yet they would never let you 'win' so that you can appreciate the fun of the game. These friends were NOT any more intellectually astute, ...they were clerically skilled. And while the game may be something interesting, I don't play it, even if it might also have interesting logical things to learn from it.
I eat without fear of certain Death from The Tree of Knowledge because with wisdom, we may one day break free from its mortal curse.

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Re: A roll of the dice
Scott Mayers wrote:That's going BEYOND a necessary way to prove the intended solutions. The WAY the puzzle is presented is purposely meant to appear OBSCURE by being elliptical (leaving OUT necessary premises). It is like a magician using some 'trick' only once because if people saw it again close up, it would no longer BE as profound or magical as it first appears.
first off, i think i know what you mean regarding the first part of this (deterministic vs nondeterministic systems, Einstein's comment on 'God doesn't roll dice,' etc. ), and i will respond to this tomorrow when i have more time. but i'll start with the paragraph above...
not sure what you mean by 'going BEYOND a necessary way to prove the intended solution.' this is a VERY simple application of Bayes theorem. it's primarily used to illustrate Bayes theorem in class, so students learn how to apply it. when they can apply it in simple cases such as these, they can then more easily use it on complex problems, and ultimately understand Bayesian analysis. that is the point of coming up with problems such as these.
i see nothing obscure or left out in the die problem. two fair die are thrown (fair meaning each number on each die will have equal prob of occurring) and you're told that one of the two die shows a '1'. we're then asked what the prob of the other die showing a '1' is. there are many forms of this problem... assume the probability of having a boy is equal to the probability of having a girl which is = 1/2. a couple has 2 kids and you're told one of them is a boy. what the chance the other is a boy.
i'm not sure what 'trick' is being employed here. and yes, once you figure the problem out isn't not so profound anymore. that's when you've reached an understanding of how conditional probability works. that's the point. the way the problem is stated, the person throwing the dice is not lying, cheating or tricking so adding these possibilities is going beyond the scope of the problem.

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Re: A roll of the dice
Scott Mayers wrote:
Education today emphasizes teaching clerical factors up front, such as rote facts rather than 'proof' from the beginning. They utilize an 'abstract' topdown methodology rather than a foundational one. An example would be how one is required to first learn a higher order language prior to understanding the logic BENEATH this level. In pretwentieth century formal training, the emphasis was to BEGIN with questioning things up front....enticing people to question reality using LOGIC early on. We reverse this. Only when one is challenging their "Phd", [Philosophic degree] are they permitted to use "free" thought, for instance. Prior to that level, the emphasis is to require essays or papers to PROVE things based on OTHER people's insights. It focuses on granting 'credit' to others. So even, for instance, if you had a good logical capacity to 'prove' something without requiring other references, you are discouraged from this UNTIL you have reached the Mastery level.
in a phd program (not sure why you have phd in quotes) you are required to know the current research in the field inside out before expounding on it. otherwise you are speaking from ignorance, as many of the comments such a person would make have been addressed before. one can challenge ideas any time they want, but you need to be up to speed on the current advances in the field beforehand.
Scott Mayers wrote:In general, we do not use the Euclidean approach where we assume simple things that we all agree on and then prove step by step what we hope they understand. Instead, we DEMAND that we have faith in the authorities of "science" as an institute PRIOR to being privileged to the PROOF.
no one demands this in math. you can read the proof yourself and if you disagree you can state why. nothing is hidden.

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Re: A roll of the dice
Tallboy wrote:Scott Mayers wrote:That's going BEYOND a necessary way to prove the intended solutions. The WAY the puzzle is presented is purposely meant to appear OBSCURE by being elliptical (leaving OUT necessary premises). It is like a magician using some 'trick' only once because if people saw it again close up, it would no longer BE as profound or magical as it first appears.
first off, i think i know what you mean regarding the first part of this (deterministic vs nondeterministic systems, Einstein's comment on 'God doesn't roll dice,' etc. ), and i will respond to this tomorrow when i have more time. but i'll start with the paragraph above...
not sure what you mean by 'going BEYOND a necessary way to prove the intended solution.' this is a VERY simple application of Bayes theorem. it's primarily used to illustrate Bayes theorem in class, so students learn how to apply it. when they can apply it in simple cases such as these, they can then more easily use it on complex problems, and ultimately understand Bayesian analysis. that is the point of coming up with problems such as these.
i see nothing obscure or left out in the die problem. two fair die are thrown (fair meaning each number on each die will have equal prob of occurring) and you're told that one of the two die shows a '1'. we're then asked what the prob of the other die showing a '1' is. there are many forms of this problem... assume the probability of having a boy is equal to the probability of having a girl which is = 1/2. a couple has 2 kids and you're told one of them is a boy. what the chance the other is a boy.
i'm not sure what 'trick' is being employed here. and yes, once you figure the problem out isn't not so profound anymore. that's when you've reached an understanding of how conditional probability works. that's the point. the way the problem is stated, the person throwing the dice is not lying, cheating or tricking so adding these possibilities is going beyond the scope of the problem.
It remains AMBIGUOUS because you don't know (a) if he is telling the truth THAT one die is rolled, (b) or, inclusively, whether if assuming he IS telling the truth, that the stat he is asking for is about both die or ONLY the remaining die. In REALITY, when we NEED to use statistics or probability, we DO have the right to ask more questions and be sure we understand the problem. But if being TESTED for some math/logic question merely based solely on some specific set of words, is like having to determine whether OJ Simpson actually killed his wife or not. The evidence suggests he did based on various interpretations but still left sufficient problems simply BECAUSE of the ambiguity of all the details.
So I often find such questions insufficient to determine whether one is or is NOT good at the math and so 'cheats' those required to be tested where the problem goes beyond merely the math because it actually involves NONmathematical concepts, and usually, 'cultural' factors, like the WAY one uses their words and to what they leave out, intentionally or not.
That's why I like using the magician example. It is like a magician performing some trick ONCE and then demanding the audience to determine HOW the trick was done. And if one guesses one real POSSIBLE solution, the magician asserts it is NOT the case and THEN demonstrates how he did it specifically. How is the audience 'at fault'? AND, how does figuring out the ACTUAL way he did the trick make one better or worse at problem solving if either they lack sufficient information or can see multiple ways the trick can be rationally done?
And to this statement, "we're then asked what the prob of the other die showing a '1' is." is actually YOUR interpretation since this is not explicit in the Gord's presented question.
Gord wrote:You sit down across the table from me. Then I roll both dice behind a screen where you can't see them.
"Did you roll at least one 1?" you ask me.
I look at the dice. "Yes," I reply. "At least one of these two dice shows a 1 on top."
Now here's the question: What are the odds that I rolled two 1s?
The probability for the SECOND die only? Ambiguous since it would either be 1/6 if the 'elliptical' assumption is to base it on Gord's perception since he is HIDING it BUT KNOWS which die he has tossed. Just because the 'player' is unable to see the die shouldn't require him to determine whether this is asking for himself, the player, to assert some specific truth biased from 'seeing' through Gord's eyes or his own given that it is hidden. If it is the 11/36, this should just be asked in principle of what could be tossed WITHOUT hiding: "What are the odds that I roll two 1s?" But hiding it means you have to determine whether the die is arbitrary to the solution or not. So it could be either 1/6 or 11/36.
So the only correct answer is to do it conditionally:
Two solutions:
If you are asking what any other die is given one specific die is thrown, the odds are 1/6 for that remaining die.
If you are asking what the TOTAL odds are given one die certainly IS a 1, but not to judge which specific die is tossed, it is 11/36.
The second solution doesn't require hiding. So the 'hiding' appears to present the ambiguity with some elliptical assumption by Gord. If he meant the 2nd solution, you have to assume that you cannot assume 'trusting' him whether any one specific die is tossed or not. So it presents itself as a magicians performance.
Magician wants to prove that he can read your mind. So he asks you to silently pick some card in your mind and write it down. He mixes the deck and places one of the fiftytwo in a concealed bag. Then asks you what the card you were thinking was. You say, "Queen of Spades". The magician looks into the bag and simply says 'yes', that's it, without revealing it. [this was something similarly done recently in some performance for "America's got talent" as a joke part of his performance, but I couldn't remember which one it was to find the damn link. I looked for 45 minutes too!]
I eat without fear of certain Death from The Tree of Knowledge because with wisdom, we may one day break free from its mortal curse.

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Re: A roll of the dice
Tallboy wrote:
in a phd program (not sure why you have phd in quotes) you are required to know the current research in the field inside out before expounding on it. otherwise you are speaking from ignorance, as many of the comments such a person would make have been addressed before. one can challenge ideas any time they want, but you need to be up to speed on the current advances in the field beforehand.
The point of emphasizing PHD was that the traditional way to learn is to BEGIN with philosophy and formal logic AND to question things using philosophy, your OWN capacity to think, and not what some OTHER people have already said. To assume one FIRST catch up prior to even determining whether it is, is like what many demand of the skeptic as a PREREQUISITE to even permit logical analysis of ones religion: you must first READ the whole Bible. The trap is that even if you did, you'd still risk just one demanding you STILL just don't understand and demand you read some other extended work.
So, to actually appropriately learn requires beginning with questions and demand a step by step type of learning. Today's educations depends more on a topdown abstraction in that you may be given some 'hints' of the trustworthiness of the authorities, but you must trust most of the science as dictated first, and only THEN, are you allowed to even be 'qualified' to ask questions.
Scott Mayers wrote:In general, we do not use the Euclidean approach where we assume simple things that we all agree on and then prove step by step what we hope they understand. Instead, we DEMAND that we have faith in the authorities of "science" as an institute PRIOR to being privileged to the PROOF.
no one demands this in math. you can read the proof yourself and if you disagree you can state why. nothing is hidden.[/quote]
I mentioned the faith of 'science' not simply of math here. There is a distinction. But we also do not demand even one take Euclidean geometry in high school now. And where it exists, it is optional. So people are now taught to have faith in the Pythagorean Theorem and the emphasis is on USING it, not learning how or why it is or is not true. (just AS science is done too) Thus this places significance on those types of people who are more good at blind obedience and good memory skills [clerical] (like the social butterflies good at remembering names and who have good etiquette, and not good selfdriven thinkers).
So you get people who eventually DO get to that degree and what KIND of thinkers are they by the time they've just spent all those years learning? Do you not think that the VERY investment to these types NOT also make them more likely to justify their approach regardless of its validity or worth? Then the cycle continues as these become the next teachers expecting blind obedience and respect of their students.
I eat without fear of certain Death from The Tree of Knowledge because with wisdom, we may one day break free from its mortal curse.

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Re: A roll of the dice
Scott Mayers wrote:
It remains AMBIGUOUS because you don't know (a) if he is telling the truth THAT one die is rolled, (b) or, inclusively, whether if assuming he IS telling the truth, that the stat he is asking for is about both die or ONLY the remaining die. In REALITY, when we NEED to use statistics or probability, we DO have the right to ask more questions and be sure we understand the problem. But if being TESTED for some math/logic question merely based solely on some specific set of words, is like having to determine whether OJ Simpson actually killed his wife or not. The evidence suggests he did based on various interpretations but still left sufficient problems simply BECAUSE of the ambiguity of all the details.
i can simplify it for you... he's telling the truth and he's asking what the probability of rolling snakeeyes GIVEN you know that at least one of the two dice shows a 1. This information is sufficient to answer the problem. it may help to know that no one really cares about the probability here as much as whether you can solve this problem under simplifying assumptions. it's showing you know how to use the rules of probability (bayes thm) or some other method. It's a stylized example.
Scott Mayers wrote:So I often find such questions insufficient to determine whether one is or is NOT good at the math and so 'cheats' those required to be tested where the problem goes beyond merely the math because it actually involves NONmathematical concepts, and usually, 'cultural' factors, like the WAY one uses their words and to what they leave out, intentionally or not.
the question is asking a simple direct question. there's no nonmathematical concepts or cultural factors. two fair dice are rolled, everyone is telling the truth. you're told at least one is a '1' and asked what's the prob of snakeeyes. it really can't be any more straightforward.
Scott Mayers wrote:That's why I like using the magician example. It is like a magician performing some trick ONCE and then demanding the audience to determine HOW the trick was done. And if one guesses one real POSSIBLE solution, the magician asserts it is NOT the case and THEN demonstrates how he did it specifically. How is the audience 'at fault'? AND, how does figuring out the ACTUAL way he did the trick make one better or worse at problem solving if either they lack sufficient information or can see multiple ways the trick can be rationally done?
there's no trick here. it's a very simple problem. there are many ways to get to 1/11 and they're all acceptable. advanced prob students are required to learn advanced techniques and one way to learn them is to practice on simple problems.
Scott Mayers wrote:And to this statement, "we're then asked what the prob of the other die showing a '1' is." is actually YOUR interpretation since this is not explicit in the Gord's presented question.Gord wrote:You sit down across the table from me. Then I roll both dice behind a screen where you can't see them.
"Did you roll at least one 1?" you ask me.
I look at the dice. "Yes," I reply. "At least one of these two dice shows a 1 on top."
Now here's the question: What are the odds that I rolled two 1s?
i don't see a difference in the way i wrote it from what Gord wrote.
Scott Mayers wrote:The probability for the SECOND die only?
no. the conditional probability that the second die is a 1 given at least one of the two die is a 1. this is testing your ability understand sample spaces.
Scott Mayers wrote:Ambiguous since it would either be 1/6 if the 'elliptical' assumption is to base it on Gord's perception since he is HIDING it BUT KNOWS which die he has tossed. Just because the 'player' is unable to see the die shouldn't require him to determine whether this is asking for himself, the player, to assert some specific truth biased from 'seeing' through Gord's eyes or his own given that it is hidden.
not sure what you mean here. the 'hiding' thing is a heuristic. it's told this way to make it more understandable. seems it didn't work . use my formulation, it's the same thing: two die are rolled and at least one shows a '1'. what's the probability the other die is also a '1'.
Scott Mayers wrote:If it is the 11/36, this should just be asked in principle of what could be tossed WITHOUT hiding: "What are the odds that I roll two 1s?" But hiding it means you have to determine whether the die is arbitrary to the solution or not. So it could be either 1/6 or 11/36.
it's 1/11. and i reformulated the problem above without the 'hiding' part.
Scott Mayers wrote:So the only correct answer is to do it conditionally:
Two solutions:
If you are asking what any other die is given one specific die is thrown, the odds are 1/6 for that remaining die.
If you are asking what the TOTAL odds are given one die certainly IS a 1, but not to judge which specific die is tossed, it is 11/36.
the first solution is not what the problem says. the 'hiding' heuristic is to let you know that you don't know which die is initially showing a '1'. what you're saying here is what is the probability that you roll a 1 on the second die given you rolled a 1 on the first. and you still have to make the assumption the dice are independent to come up with 1/6.
[/quote]Scott Mayers wrote:The second solution doesn't require hiding. So the 'hiding' appears to present the ambiguity with some elliptical assumption by Gord. If he meant the 2nd solution, you have to assume that you cannot assume 'trusting' him whether any one specific die is tossed or not. So it presents itself as a magicians performance.
yes, this is the correct formulation. the assumption is he's telling the truth so you don't have to worry about it... no need to introduce any ambiguity.
so now that we're on the same page. please explain why you think the answer is 11/36?

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Re: A roll of the dice
Scott Mayers wrote:The point of emphasizing PHD was that the traditional way to learn is to BEGIN with philosophy and formal logic AND to question things using philosophy, your OWN capacity to think, and not what some OTHER people have already said. To assume one FIRST catch up prior to even determining whether it is, is like what many demand of the skeptic as a PREREQUISITE to even permit logical analysis of ones religion: you must first READ the whole Bible. The trap is that even if you did, you'd still risk just one demanding you STILL just don't understand and demand you read some other extended work.
i'm not sure why you think the 'traditional way to learn' is to begin with philosophy and logic. math programs i'm familiar with teach you HOW to do problems first (with a bit of philosophy) and the 'why' comes later. like writing, you learn the alphabet first, then grammar, then you write and question traditional structure. sticking with probability, you really don't know a lot of why's till you finish your phd. you learn the most when you start teaching. but you need to know the field first so that you don't ask questions that have already been answered. you yourself seem to agree with this... you were discussing Einstein and QM in a previous post and you wrote:
"I don't want to go further into this as I had before unless we discuss the interpretation problems in QM, something I doubt many here are sufficiently familiar with."
so it seems you agree that you need to know the subject matter before you're able to discuss it rationally.
Scott Mayers wrote:So, to actually appropriately learn requires beginning with questions and demand a step by step type of learning. Today's educations depends more on a topdown abstraction in that you may be given some 'hints' of the trustworthiness of the authorities, but you must trust most of the science as dictated first, and only THEN, are you allowed to even be 'qualified' to ask questions.
neither is 'appropriate'... they are just two different methods. when you teach children to read/write you don't start with the theory of linguistics or language. they won't get it. but they will get how to make letters and put words together. when we teach calculus we first teach students how to graph functions, then differentiate, integrate. we don't start with the theory behind it. indeed, the theory behind it is presented in a 'real analysis' course, and this is often one of the most difficult areas of mathematics. first you teach grammar, then writing. first you teach calculus, then probability (or real analysis, or topology, or...). it is often advantageous to learn how before why.
Scott Mayers wrote:In general, we do not use the Euclidean approach where we assume simple things that we all agree on and then prove step by step what we hope they understand. Instead, we DEMAND that we have faith in the authorities of "science" as an institute PRIOR to being privileged to the PROOF.
there is no 'demand' for anything and there is no 'faith' involved. if you want to study QM, you learn what Einstein and Planck suggested first; those that pioneered the field. you are free to disagree but you won't be taken seriously if you don't know the different arguments.
Scott Mayers wrote:
I mentioned the faith of 'science' not simply of math here. There is a distinction. But we also do not demand even one take Euclidean geometry in high school now. And where it exists, it is optional. So people are now taught to have faith in the Pythagorean Theorem and the emphasis is on USING it, not learning how or why it is or is not true. (just AS science is done too) Thus this places significance on those types of people who are more good at blind obedience and good memory skills [clerical] (like the social butterflies good at remembering names and who have good etiquette, and not good selfdriven thinkers).
it pains me to say this but most people couldn't care less about the Pythagorean thm. if they ever need to use it, they hire someone to do it for them or they employ it without understanding it. but this happens all the time. my parents spoke at least 4 languages, but they never understood rules of grammar. it isn't required. you can speak with good grammar by memorization and repetition/mimicking alone. and their speaking abilities were sufficient for their purposes.
Scott Mayers wrote:So you get people who eventually DO get to that degree and what KIND of thinkers are they by the time they've just spent all those years learning? Do you not think that the VERY investment to these types NOT also make them more likely to justify their approach regardless of its validity or worth? Then the cycle continues as these become the next teachers expecting blind obedience and respect of their students.
when i was a first year grad student in my first class, i asked a lot of questions. i was all over the place and i constantly took up class time with them. my prof finally said 'read chapter 2 first and get back to me with your questions later.' so i went home and read chapter 2, over and over. i didn't quite get it so i went to the library and got a few more books that discussed the same topic and spent the weekend reading those. when i came back to class my prof asked me if i had any questions and i smiled and said 'no.' he smiled too and walked away.
the point is that these are often difficult concepts. that's why people spend 4+ years getting their phd. you need to spend the time reading and understanding your subject on your own before asking every question under the sun. and when i did ask questions they were far more intelligent.
the phd is designed to give the student a thorough background in his subject area. most of the great work i've seen were done by mathematicians AFTER they received their phd's, which is how it was meant to be. i can still hear my advisor admonishing me 'don't try to win a nobel prize with your dissertation. get it done then go out and make your mark in the world.'

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Re: A roll of the dice
Tallboy wrote:Scott Mayers wrote:
It remains AMBIGUOUS because you don't know (a) if he is telling the truth THAT one die is rolled, (b) or, inclusively, whether if assuming he IS telling the truth, that the stat he is asking for is about both die or ONLY the remaining die. In REALITY, when we NEED to use statistics or probability, we DO have the right to ask more questions and be sure we understand the problem. But if being TESTED for some math/logic question merely based solely on some specific set of words, is like having to determine whether OJ Simpson actually killed his wife or not. The evidence suggests he did based on various interpretations but still left sufficient problems simply BECAUSE of the ambiguity of all the details.
i can simplify it for you... he's telling the truth and he's asking what the probability of rolling snakeeyes GIVEN you know that at least one of the two dice shows a 1. This information is sufficient to answer the problem. it may help to know that no one really cares about the probability here as much as whether you can solve this problem under simplifying assumptions. it's showing you know how to use the rules of probability (bayes thm) or some other method. It's a stylized example.
The information is CONDITIONALLY capable of 'solving' due to its logical ambiguity as I've explained. You no doubt simply are in sync with the WAY Gord uses language and why you may be unable to interpret the words in only ONE unique way. I don't NEED reinterpreting and think MUCH different than most. I see details more closely than most could or would bother to care doing.
Scott Mayers wrote:So I often find such questions insufficient to determine whether one is or is NOT good at the math and so 'cheats' those required to be tested where the problem goes beyond merely the math because it actually involves NONmathematical concepts, and usually, 'cultural' factors, like the WAY one uses their words and to what they leave out, intentionally or not.
the question is asking a simple direct question. there's no nonmathematical concepts or cultural factors. two fair dice are rolled, everyone is telling the truth. you're told at least one is a '1' and asked what's the prob of snakeeyes. it really can't be any more straightforward.
Here is an example of HOW I look at things: If you've ever seen an I.Q. or similar 'aptitude' tests, many devise them with certain thinking that the authors believe have specific answers. As a simplified part we all go through even in our elementary school years is to have three pictures of which we might be asked to circle two which are 'alike' or 'the same'; or, which is 'different'.
What I see that others do not is that ALL solutions have validity if the child is seriously doing the task (not having a wandering mind and answering randomly). And what many don't notice is that contrary to many, the apparent I.Q. being measured (or being 'taught') is ABOUT CULTURE, because ALL answers in any given three pictures are 'logically correct' under the condition that you understand what or how that person thinks. Yet the testing by many is believed to have a specific unique answer.
For example, if given (1) a circle, (2) a square, and (3) curved continuous line, like a short segment of a sine wave, which two do you think are "alike"? ANY two are 'correct' but actually can DETERMINE the WAY one thinks. Certainly, if the testers mark someone 'wrong' on this example, they are actually hardheaded and biased to some cultural assumption about the form of expected behavior when communicating. And when teachers are using this for children, they are DEFINING the cultural means to interpret things.
[NOTE: I made a picture of the above but for some reason I see no 'attachment' possibility any more.]
I was going to give you the potential reasons here but then thought I should test you or others first to see if you can 'see' what is common between any two,....or even all three, examples.
Scott Mayers wrote:That's why I like using the magician example. It is like a magician performing some trick ONCE and then demanding the audience to determine HOW the trick was done. And if one guesses one real POSSIBLE solution, the magician asserts it is NOT the case and THEN demonstrates how he did it specifically. How is the audience 'at fault'? AND, how does figuring out the ACTUAL way he did the trick make one better or worse at problem solving if either they lack sufficient information or can see multiple ways the trick can be rationally done?
there's no trick here. it's a very simple problem. there are many ways to get to 1/11 and they're all acceptable. advanced prob students are required to learn advanced techniques and one way to learn them is to practice on simple problems.
You can learn the techniques by SIMPLY defining them from scratch in a formal way. As to PROBLEM QUESTIONS [ie. puzzles], language is important. The problem with them is that they can be purposely defined to RULE out students who ARE potentially capable whom a teacher(s) simply just doesn't like. Take the example of the three imagetype comparison test as above. Since any answers(as in ALL similar 'tests') ALWAYS have a 'correct' result, you can BIAS against the one being testing to discriminate against them by saying any one of them is 'wrong' by showing that any of the others are 'correct'. I gave this example because it DOES have multiple solutions. But the same goes for many other problems like the one Gord presented. [and the way he did].
Scott Mayers wrote:And to this statement, "we're then asked what the prob of the other die showing a '1' is." is actually YOUR interpretation since this is not explicit in the Gord's presented question.Gord wrote:You sit down across the table from me. Then I roll both dice behind a screen where you can't see them.
"Did you roll at least one 1?" you ask me.
I look at the dice. "Yes," I reply. "At least one of these two dice shows a 1 on top."
Now here's the question: What are the odds that I rolled two 1s?
i don't see a difference in the way i wrote it from what Gord wrote.
I am responding to the initial OP. If you are interpreting that you wrote it differently, I'd have to treat your own wording as a 'new' puzzle because you are REinterpreting the same question using YOUR words. The Monty Hall Wiki does multiple reinterpretations for each challenge by other mathematicians and treats them distinct (New). This is appropriate other than to READ into the initial question certain assumptions NOT presented.
Scott Mayers wrote:The probability for the SECOND die only?
no. the conditional probability that the second die is a 1 given at least one of the two die is a 1. this is testing your ability understand sample spaces.Scott Mayers wrote:Ambiguous since it would either be 1/6 if the 'elliptical' assumption is to base it on Gord's perception since he is HIDING it BUT KNOWS which die he has tossed. Just because the 'player' is unable to see the die shouldn't require him to determine whether this is asking for himself, the player, to assert some specific truth biased from 'seeing' through Gord's eyes or his own given that it is hidden.
not sure what you mean here. the 'hiding' thing is a heuristic. it's told this way to make it more understandable. seems it didn't work . use my formulation, it's the same thing: two die are rolled and at least one shows a '1'. what's the probability the other die is also a '1'.
On the former, I understand that interpretation too. That the first die was already tossed, if the question (conditional) is asking the TOTAL probability of ANY die being tossed to be a '1', then YES, the probability is 100% certain (for trusting Gord, of course).
For the latter,
This is where I think 'we' (people, in general) err when we think we presented a specific puzzle in some assumed 'unique' way that is NOT.
I found this channel a few days ago to which I highly like and may help clarify PERCEPTUAL problems with using probability/statistics by HOW they are presented: Unmasking the Hidden Paradox in Data. He has others there too that likely apply. There he uses the example of HOW one uses a statistical presentation (a preferred selected result) by authors who are just as likely to not see their own error (assuming they are not actually BEING deceptive themselves.)
Scott Mayers wrote:If it is the 11/36, this should just be asked in principle of what could be tossed WITHOUT hiding: "What are the odds that I roll two 1s?" But hiding it means you have to determine whether the die is arbitrary to the solution or not. So it could be either 1/6 or 11/36.
it's 1/11. and i reformulated the problem above without the 'hiding' part.Scott Mayers wrote:So the only correct answer is to do it conditionally:
Two solutions:
If you are asking what any other die is given one specific die is thrown, the odds are 1/6 for that remaining die.
If you are asking what the TOTAL odds are given one die certainly IS a 1, but not to judge which specific die is tossed, it is 11/36.
the first solution is not what the problem says. the 'hiding' heuristic is to let you know that you don't know which die is initially showing a '1'. what you're saying here is what is the probability that you roll a 1 on the second die given you rolled a 1 on the first. and you still have to make the assumption the dice are independent to come up with 1/6.Scott Mayers wrote:The second solution doesn't require hiding. So the 'hiding' appears to present the ambiguity with some elliptical assumption by Gord. If he meant the 2nd solution, you have to assume that you cannot assume 'trusting' him whether any one specific die is tossed or not. So it presents itself as a magicians performance.
yes, this is the correct formulation. the assumption is he's telling the truth so you don't have to worry about it... no need to introduce any ambiguity.
so now that we're on the same page. please explain why you think the answer is 11/36?[/quote]
If both die are simply tossed, the probability will be 11/36 for ANY of them to at least be a '1':
All probabilities for ANY arrangements are 6*6=36. So for the first arbitrary die, it has 1/6 (= 6/36) of a chance to be a '1' as well as the second. There is one possibility though that is 'repeat' and so you must subtract it:
1/6 +1/6  1/36 = (6 + 6  1)/36 = 11/36
I eat without fear of certain Death from The Tree of Knowledge because with wisdom, we may one day break free from its mortal curse.

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Re: A roll of the dice
Tallboy wrote:Scott Mayers wrote:The point of emphasizing PHD was that the traditional way to learn is to BEGIN with philosophy and formal logic AND to question things using philosophy, your OWN capacity to think, and not what some OTHER people have already said. To assume one FIRST catch up prior to even determining whether it is, is like what many demand of the skeptic as a PREREQUISITE to even permit logical analysis of ones religion: you must first READ the whole Bible. The trap is that even if you did, you'd still risk just one demanding you STILL just don't understand and demand you read some other extended work.
i'm not sure why you think the 'traditional way to learn' is to begin with philosophy and logic. math programs i'm familiar with teach you HOW to do problems first (with a bit of philosophy) and the 'why' comes later. like writing, you learn the alphabet first, then grammar, then you write and question traditional structure. sticking with probability, you really don't know a lot of why's till you finish your phd. you learn the most when you start teaching. but you need to know the field first so that you don't ask questions that have already been answered. you yourself seem to agree with this... you were discussing Einstein and QM in a previous post and you wrote:
"I don't want to go further into this as I had before unless we discuss the interpretation problems in QM, something I doubt many here are sufficiently familiar with."
so it seems you agree that you need to know the subject matter before you're able to discuss it rationally.
Much of what I've learned has been on my own. After high school, I wasn't satisfied with HOW I learned and so began from scratch: From World History up to the Greeks, then to Philosophy, which begins with Logic, then to learning the traditional way using a BROAD stepbystep approach by the WAY things were discovered historically. It is FOUNDATIONAL. If I didn't understand something in one area, I'd stop and digress to determine what I was missing in another area, hopping back and forth to get a REAL understanding.
The basic grammar and prep for intellectual study is only necessary at a very elementary level. The present teaching is based on trying to efficiently get students through enough education to be 'practical' for the job market, and mostly FOR jobs that are 'clerical'. Even most scientists in practice are merely working UNDER the authority of others who actually do the thinking. They thus prefer now to attend to those who could "labor", so to speak.
You wouldn't know this if you were raised through the education system since the early to mid 1900s. I only discovered this because I was selfdriven and had to seek HOW to optimally understand. If you simply collect very old texts, you'd see a very significant difference in HOW they did this. That's why the era of the great scientists ended at about the 1960s.
But NO, I can relate to a child to teach much of what they refuse to teach until one today has their Master's degree. Note though that it is still relatively 'practical' WHEN you learn bottom up and broad because you sincerely don't rely on memorizing trivia that is simply meaningless data until you rationally derive a REASON WHY you need to know something other than to today's 'motives'. Most go to University simply because they are ingrained with a motive to get a good job. The Europeans, I believe, may still use some of this style. But for the most part, such method is limited to the more fortunate who can afford very good schools that do this now.
Scott Mayers wrote:So, to actually appropriately learn requires beginning with questions and demand a step by step type of learning. Today's educations depends more on a topdown abstraction in that you may be given some 'hints' of the trustworthiness of the authorities, but you must trust most of the science as dictated first, and only THEN, are you allowed to even be 'qualified' to ask questions.
neither is 'appropriate'... they are just two different methods. when you teach children to read/write you don't start with the theory of linguistics or language. they won't get it. but they will get how to make letters and put words together. when we teach calculus we first teach students how to graph functions, then differentiate, integrate. we don't start with the theory behind it. indeed, the theory behind it is presented in a 'real analysis' course, and this is often one of the most difficult areas of mathematics. first you teach grammar, then writing. first you teach calculus, then probability (or real analysis, or topology, or...). it is often advantageous to learn how before why.
I was teaching LOGIC to a fiveyear old. She was able to understand BUT I still remember her coming to me one day saying, "my head hurts from thinking!" I was only testing to see if I could teach her and it CAN be done. But mostly this requires a good parent to do this since it is very involved.
If you remembered your early life, you probably remembered asking questions like WHY is "knight" spelled that way. You are told things just like many are taught religion, by authoritarian figures that simply say, "just because". But you can teach a child how to understand HOW language is 'arbitrary' by allowing THEM to create or recreate symbols. For math, you teach them binary first but allow THEM to think it through step by step to understand FIRST, not memorize. The memory comes automatic. This is completely opposite to those who emphasize spelling as some virtuous intellectual pursuit.
So I completely disagree to the QUALITY of difference. The bottomupbroad approach is the way one actually BECOMES intellectual intrinsically. The topdown approach only works by the numbers. But society doesn't actually WANT too many brilliant thinkers but need many obedient and skilled employees. So while you CAN also learn this way, if FAVORS the naturally more fortunate who are accidentally good at clerical skills UP FRONT. This is like prescreening only for TALL people for Basketball. Sure some shorter people could become a promising player, but it also discriminates against others who might be as 'good'. For science and other higher education, the demand favors those who initially have higher memory for 'trivia' and why there is a false stereotype that those who have skill in trivia are somehow more wise.
Scott Mayers wrote:In general, we do not use the Euclidean approach where we assume simple things that we all agree on and then prove step by step what we hope they understand. Instead, we DEMAND that we have faith in the authorities of "science" as an institute PRIOR to being privileged to the PROOF.
there is no 'demand' for anything and there is no 'faith' involved. if you want to study QM, you learn what Einstein and Planck suggested first; those that pioneered the field. you are free to disagree but you won't be taken seriously if you don't know the different arguments.
The way much of it is taught DOES require 'faith'. Most 'skeptics' I know become skeptical only by a kind of 'popularity' appeal without actually "KNOWING" what or why religious thinking is at fault. It's just "silly" thinking (without an actual argument to 'why').
I already agree to a bottomup approach that includes understanding what Einstein or Planck did, for example. But we NOW do 'abstracting' which overviews a subject but only teases some of the thinking until you get to critical questions. Then this is glossed over and you must 'trust' these true up front. Take a grade 11 Chemistry class, for example. They teach you how and what has been DETERMINED through quantum mechanics but LACK the proofs that build a case for 'why' things are as they are. For instance, you learn THAT there are orbitals and the way they are 'summarized' as knowledge but do not 'evolve' ones mind to reproduce this wisdom on their own. And this happens in repeated stages with only a bit more added at each stage, to keep up with the other different classes simultaneously.
Scott Mayers wrote:
I mentioned the faith of 'science' not simply of math here. There is a distinction. But we also do not demand even one take Euclidean geometry in high school now. And where it exists, it is optional. So people are now taught to have faith in the Pythagorean Theorem and the emphasis is on USING it, not learning how or why it is or is not true. (just AS science is done too) Thus this places significance on those types of people who are more good at blind obedience and good memory skills [clerical] (like the social butterflies good at remembering names and who have good etiquette, and not good selfdriven thinkers).
it pains me to say this but most people couldn't care less about the Pythagorean thm. if they ever need to use it, they hire someone to do it for them or they employ it without understanding it. but this happens all the time. my parents spoke at least 4 languages, but they never understood rules of grammar. it isn't required. you can speak with good grammar by memorization and repetition/mimicking alone. and their speaking abilities were sufficient for their purposes.
Yes, many don't see the significance as you are appearing to here. It is HOW things are proven step by step that make learning the Pythagorean theorem valuable to learn the traditional way. But you flip LOGIC, with PRACTICE as the way to go. Where you think 'grammar' is secondary, I think 'spelling' is secondary. That is, you think that LOGIC should be simply 'learned' by varying experiences by focusing only on etiquette or culture. The more socially astute prefer learning by practice but many do not realize that this is often based on their inherent factors: genetics, (for things like good memory, like a pleasant LOOK or BEHAVIOR, that appeals to most), or environmental fortunes (wealth, travel, variety due to options available).
I remember how I used to be starving so often in school and so fatigued that it was hard to actually compete with having the energy in class growing up. It is for these cases that focusing on 'clerical' skills suits those already defaulted to some inherent factors. If you grew up with no 'allowance' and further discovered that if you tried to work that you were only penalized for it because you NOW must pay rent and food with it, MOTIVATIONAL approaches such as, "so you can get a job", isn't so motivating for those less fortunate.
So the emphasis should BE to UNDERSTAND things with reasoning that makes sense up front. If, for instance, you learn from Euclidean geometry that a "tangent" is a word from Greek that meant 'touching' and that it initially described a line that "just touches" a circle, not an abstract memorized 'function'; Or, that the Earth can be determined as being round by creating a shadow using a stick at two different places at the same time, like how the Erastesthenes (the name not significant to memorize, just his story of discovery). Or that "log" and "logic" are terms derived from "to Look (up)" as in a "log" that was an anal record of data...or that 'anal' came from the meaning we get as "annual" (a yearly record), not that thing that looks like this: *.
These methods are STILL 'practical' but make THINKING from evolving discovery as the thing being practiced, not the 'practice' of HOW to be sure to credit who, what, and where, to be sure that we respect copyright authors. Why is it important to KNOW the name of some scientist rather than the efforts. To me, it is not the logic (like Grammar) that needs to be trivialized and expected to be 'accidentally' inferred, but the cultural expectations of demanding we SPELL things correctly, or that we need to memorize our times tables. It is the clerical factors that come by accident and without force.
Scott Mayers wrote:So you get people who eventually DO get to that degree and what KIND of thinkers are they by the time they've just spent all those years learning? Do you not think that the VERY investment to these types NOT also make them more likely to justify their approach regardless of its validity or worth? Then the cycle continues as these become the next teachers expecting blind obedience and respect of their students.
when i was a first year grad student in my first class, i asked a lot of questions. i was all over the place and i constantly took up class time with them. my prof finally said 'read chapter 2 first and get back to me with your questions later.' so i went home and read chapter 2, over and over. i didn't quite get it so i went to the library and got a few more books that discussed the same topic and spent the weekend reading those. when i came back to class my prof asked me if i had any questions and i smiled and said 'no.' he smiled too and walked away.
the point is that these are often difficult concepts. that's why people spend 4+ years getting their phd. you need to spend the time reading and understanding your subject on your own before asking every question under the sun. and when i did ask questions they were far more intelligent.
the phd is designed to give the student a thorough background in his subject area. most of the great work i've seen were done by mathematicians AFTER they received their phd's, which is how it was meant to be. i can still hear my advisor admonishing me 'don't try to win a nobel prize with your dissertation. get it done then go out and make your mark in the world.'
You're missing the point. The 'time' one spends in anything makes them better in some aspect. I don't dismiss depth, as you should see from how much effort I put into my long posts. It is that the way you are taught reverses the actual WAY one becomes sufficiently intelligent. You NEED relevant motives that directly relate to understanding. I don't know what your intended inference of your experience with your professor implied. For me in classes, I NEVER wasted time on note taking AND believed in being sure to understand by ASKING lots of questions regardless of the teachers who would get annoyed. I aced my sciences and maths, and was at the top of my classes for them (when I returned BACK to school later).
The BIGGEST reason I don't like the present way of teaching/learning is because I personally have discovered really cool things beyond the capacity of anyone of a formal degree based on using the bottomup approach personally. It doesn't make me of the same 'qualifications' in the WAYS expected institutionally. But even MY approach IS very invested with at LEAST a similar 'degree' in time. But I have a strong foundation broadly that enables me to connect each part of nearly ALL subject areas that the present education lacks but makes them more 'specialized'.
I'm guessing this post is way off the topic now but relates, in the 'broad' sense, that I'm most familiar with. Bringing it back, I think we need to pay attention to DETAILS with respect to meaning and understanding based on logic and less on the 'how' well we perform at clerical aspects. I'm good at Calculus, for instance, but STILL have problems recalling what 8 x 7 is most of the time. I used to make many trivial errors when doing accounting (often by some mere pennies) but understand the logic of it.
I eat without fear of certain Death from The Tree of Knowledge because with wisdom, we may one day break free from its mortal curse.

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Scott Mayers wrote:The information is CONDITIONALLY capable of 'solving' due to its logical ambiguity as I've explained. You no doubt simply are in sync with the WAY Gord uses language and why you may be unable to interpret the words in only ONE unique way. I don't NEED reinterpreting and think MUCH different than most. I see details more closely than most could or would bother to care doing.
it's fine if you want to do conditioning. It's clear to me what the problem is not only from reading Gord's example, but also realizing that taking cheating into account immensely complicates the answer (perhaps even unanswerable, at least in a test setting) and asking what the prob of getting a 1 on the second die given you have one on the first is only testing your knowledge of independent events (answer is 1/6 of course). the problem itself says that both die are rolled and at least one die show a 1. So we're not rolling one at a time. indeed, if I told you I rolled one die and got a 1 and then asked you what the prob of getting a 1 on another roll you would use the independence assumption and just say 1/6. this problem is saying there is at least one '1' on two dice rolled.
Scott Mayers wrote:Here is an example of HOW I look at things: If you've ever seen an I.Q. or similar 'aptitude' tests, many devise them with certain thinking that the authors believe have specific answers. As a simplified part we all go through even in our elementary school years is to have three pictures of which we might be asked to circle two which are 'alike' or 'the same'; or, which is 'different'.
What I see that others do not is that ALL solutions have validity if the child is seriously doing the task (not having a wandering mind and answering randomly). And what many don't notice is that contrary to many, the apparent I.Q. being measured (or being 'taught') is ABOUT CULTURE, because ALL answers in any given three pictures are 'logically correct' under the condition that you understand what or how that person thinks. Yet the testing by many is believed to have a specific unique answer.
For example, if given (1) a circle, (2) a square, and (3) curved continuous line, like a short segment of a sine wave, which two do you think are "alike"? ANY two are 'correct' but actually can DETERMINE the WAY one thinks. Certainly, if the testers mark someone 'wrong' on this example, they are actually hardheaded and biased to some cultural assumption about the form of expected behavior when communicating. And when teachers are using this for children, they are DEFINING the cultural means to interpret things.
I don't think anyone is arguing that IQ tests don't have culturally biased questions. this analogy doesn't fit this specific problem. if an IQ tests asks what 2+2 is, the answer is 4. there is no cultural bias worth mentioning that would give another answer. the dice question is in this category. it's straightforward. no ambiguity or cultural issue.
to be clear, there have been culturally biased questions on IQ tests. you have to show that a particular problem is culturally biased. in this case is the dice problem we've been discussing. just because people have written ambiguous, culturallybiased questions in the past doesn't mean this one is.
Scott Mayers wrote:And to this statement, "we're then asked what the prob of the other die showing a '1' is." is actually YOUR interpretation since this is not explicit in the Gord's presented question.
I find it explicit in Gord's presentation and I can't see any other way to interpret it. you haven't shown me a reasonable alternative interpretation yet. I disagree that including possibilities that the person is lying is reasonable or can be inferred from the problem. and the heuristic that he's doing it behind a screen is to show that only the person rolling the dice knows the answer (you may assume he's being truthful). if it helps you, you can remove the screen and formulate the problem the way I did... two dice are rolled and at least one show a '1'. whats the prob that the other is a '1' as well. it's the same problem. if you don't think so, please show me how it can be reasonably interpreted another way.l
Scott Mayers wrote:I am responding to the initial OP. If you are interpreting that you wrote it differently, I'd have to treat your own wording as a 'new' puzzle because you are REinterpreting the same question using YOUR words. The Monty Hall Wiki does multiple reinterpretations for each challenge by other mathematicians and treats them distinct (New). This is appropriate other than to READ into the initial question certain assumptions NOT presented.
I wrote the problem without using the heuristic "behind a screen" and there being a conversation between the two people. it's not a new puzzle. I'm just rewriting it removing the heuristic. if you think they're different, please show why.
Scott Mayers wrote:Ambiguous since it would either be 1/6 if the 'elliptical' assumption is to base it on Gord's perception since he is HIDING it BUT KNOWS which die he has tossed.
the guy rolling the dice behind the curtain is rolling both dice at the same time, like at a craps table in vegas. there is no 'first' or 'second' die. that's why he says he rolls the dice and says 'at least one die shows a 1. this tells you that we don't know which die, or in other words, he rolled both at the same time like in craps. they could both be 1's, and that is in fact the probability the question is after.
Scott Mayers wrote:Just because the 'player' is unable to see the die shouldn't require him to determine whether this is asking for himself, the player, to assert some specific truth biased from 'seeing' through Gord's eyes or his own given that it is hidden.
there is no specific truth to assert. please state what that would be here.
Scott Mayers wrote:On the former, I understand that interpretation too. That the first die was already tossed, if the question (conditional) is asking the TOTAL probability of ANY die being tossed to be a '1', then YES, the probability is 100% certain (for trusting Gord, of course).
right, but that's not the question. if I tell you I rolled two die and at least one of them shows a 1, then the probability that one of them shows a 1 is 100%. not a very interesting problem. and again, there is no 'first die.' both are rolled at the same time.
Scott Mayers wrote:For the latter,
This is where I think 'we' (people, in general) err when we think we presented a specific puzzle in some assumed 'unique' way that is NOT.
I found this channel a few days ago to which I highly like and may help clarify PERCEPTUAL problems with using probability/statistics by HOW they are presented: Unmasking the Hidden Paradox in Data. He has others there too that likely apply. There he uses the example of HOW one uses a statistical presentation (a preferred selected result) by authors who are just as likely to not see their own error (assuming they are not actually BEING deceptive themselves.)
the video is an example of Simpson's Paradox. the Berkeley admissions example is the standard example given in class. it's very cool but has nothing to do with the dice problem we're discussing. laypeople may be fooled by this but a seasoned statistician would not. this is a very wellknown paradox and something statisticians look for when appropriate.
Scott Mayers wrote:so now that we're on the same page. please explain why you think the answer is 11/36
If both die are simply tossed, the probability will be 11/36 for ANY of them to at least be a '1':
All probabilities for ANY arrangements are 6*6=36. So for the first arbitrary die, it has 1/6 (= 6/36) of a chance to be a '1' as well as the second. There is one possibility though that is 'repeat' and so you must subtract it:
1/6 +1/6  1/36 = (6 + 6  1)/36 = 11/36
yes, that's right and you have the formulation correct to get to this point so you understand the setup. but that's not what the problem is asking. the problem is not asking what the prob of getting at least one '1' when two dice are rolled. it's asking for the conditional prob that given at least one die show's a '1', what is the prob that both show a 1. so you have one more step to do. of the 11 ways to have at least one die show a '1', there is only one way for both dice to show a '1', so 1/11. the fact that we are told in advance that at least one die shows a '1' means you can throw away all the 25 possibilities in which neither die show a '1'. we know they are not possible in this problem as the problem tells us that at least one die shows a '1' to begin with.

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Re: A roll of the dice
Scott Mayers wrote:
If both die are simply tossed, the probability will be 11/36 for ANY of them to at least be a '1':
All probabilities for ANY arrangements are 6*6=36. So for the first arbitrary die, it has 1/6 (= 6/36) of a chance to be a '1' as well as the second. There is one possibility though that is 'repeat' and so you must subtract it:
1/6 +1/6  1/36 = (6 + 6  1)/36 = 11/36
just to point something out... this is the 'addition rule' in probability. the possibility of a 'repeat' is that you count snakeeyes twice here, which is why you have to remove one snakeeye (1/36 chance). something Gord showed in his sample space matrix when answering the problem.
if you want to answer this problem using probability rules, you would need to use conditional prob rules. This is how we want stat students to answer the problem as the point of the problem is to give stat students practice in applying conditional prob rules.
Indeed, the question itself may admonish the student to use the rule of conditional prob so that they're forced to show their grasp of these concepts. the answer would go something like this:
Two dice are rolled
Let A = snakeeyes
Let B = at least one die shows a '1'
We want the Pr(AB) ... (read, the probability of A given B. the '' means 'given')
Pr(AB) = Pr(A and B) / Pr(B) (conditional probability rule)
= Pr(A)Pr(BA) / Pr(B) (algebraically the same as the first line)
Note:
Pr(A) = 1/36
Pr(BA) = 1 (i.e., given you have snakeeyes the prob that at least one die shows a '1' is 1 (100%).
Pr(B) = 11/36 (as Scott showed)
so Pr(AB) = Pr(A)Pr(BA) / Pr(B) = (1/36)(1)/(11/36) = 1/11.

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Re: A roll of the dice
Gord wrote:I have GOT to stop asking this question online. I just tried it on another forum, and got reamed out by several people who refused to accept my answer. Now they think I'm an idiot!
that's what makes it fun!?

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Re: A roll of the dice
Tallboy wrote:Scott Mayers wrote:And to this statement, "we're then asked what the prob of the other die showing a '1' is." is actually YOUR interpretation since this is not explicit in the Gord's presented question.
I find it explicit in Gord's presentation and I can't see any other way to interpret it.
Please, call me "the Gord". Now that I read it, I really really like it! Makes me sound ominous and powerful, like it belongs on the sides of several buildings.
Tallboy wrote:Gord wrote:I have GOT to stop asking this question online. I just tried it on another forum, and got reamed out by several people who refused to accept my answer. Now they think I'm an idiot!
that's what makes it fun!?
I had more fun posting it on a maths site.
"Knowledge grows through infinite timelessness"  the random fictional Deepak Chopra quote site
"Imagine an ennobling of what could be"  the New Age BS Generator site
"You are also taking my words out of context."  Justin
"Nullius in verba"  The Royal Society ["take nobody's word for it"]
#ANDAMOVIE
Is Trump in jail yet?
"Imagine an ennobling of what could be"  the New Age BS Generator site
"You are also taking my words out of context."  Justin
"Nullius in verba"  The Royal Society ["take nobody's word for it"]
#ANDAMOVIE
Is Trump in jail yet?

 Obnoxious Weed
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Re: A roll of the dice
Gord wrote:Tallboy wrote:Gord wrote:I have GOT to stop asking this question online. I just tried it on another forum, and got reamed out by several people who refused to accept my answer. Now they think I'm an idiot!
that's what makes it fun!?
I had more fun posting it on a maths site.
Like this one: http://mymathforum.com/appliedmath/228 ... iddle.html
I registered there under the name "Soup Dominion" to ask after I'd tried this riddle on another nonmaths website, with different wording to the way I presented it here:
Bill and Ted are sitting at a table. Bill rolls two dice (normal dice, unweighted, six sides each, numbered 1 to 6) behind a screen where Ted can't see what he's rolled. One of the dice is red and the other is blue.
"Did at least one of those dice come up as a 1?" Ted asks.
"Yes," Bill replies truthfully. "At least one of the dice rolled a 1."
Now that you have all that information, what are the chances that Bill rolled two 1s?
Everyone on the maths website who bothered to post agreed with me. Almost everyone on the nonmaths website who posted (even the professor of physics and the professor of computer science!) disagreed with me.
"Knowledge grows through infinite timelessness"  the random fictional Deepak Chopra quote site
"Imagine an ennobling of what could be"  the New Age BS Generator site
"You are also taking my words out of context."  Justin
"Nullius in verba"  The Royal Society ["take nobody's word for it"]
#ANDAMOVIE
Is Trump in jail yet?
"Imagine an ennobling of what could be"  the New Age BS Generator site
"You are also taking my words out of context."  Justin
"Nullius in verba"  The Royal Society ["take nobody's word for it"]
#ANDAMOVIE
Is Trump in jail yet?

 Real Skeptic
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 Joined: Fri Jun 13, 2008 3:31 am
Re: A roll of the dice
Gord wrote:Please, call me "the Gord". Now that I read it, I really really like it! Makes me sound ominous and powerful, like it belongs on the sides of several buildings.
If I ever catch you entering a forum thread like "The Fonz" from Happy Days, with both thumbs in the air, I will personally drive up to Canada and drop Justin Trudeau's ego on all your toes sequentially.

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Re: A roll of the dice
Gord wrote:Tallboy wrote:Scott Mayers wrote:And to this statement, "we're then asked what the prob of the other die showing a '1' is." is actually YOUR interpretation since this is not explicit in the Gord's presented question.
I find it explicit in Gord's presentation and I can't see any other way to interpret it.
Please, call me "the Gord". Now that I read it, I really really like it! Makes me sound ominous and powerful, like it belongs on the sides of several buildings.
or perhaps el gordo, or el gordodito, if you're not into that brevity thing... man
Tallboy wrote:Gord wrote:I have GOT to stop asking this question online. I just tried it on another forum, and got reamed out by several people who refused to accept my answer. Now they think I'm an idiot!
that's what makes it fun!?
I had more fun posting it on a maths site. [/quote][/quote]
i'll check that out. dunningkruger was strong with this thread!

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Re: A roll of the dice
Matthew Ellard wrote:Gord wrote:Please, call me "the Gord". Now that I read it, I really really like it! Makes me sound ominous and powerful, like it belongs on the sides of several buildings.
If I ever catch you entering a forum thread like "The Fonz" from Happy Days, with both thumbs in the air, I will personally drive up to Canada and drop Justin Trudeau's ego on all your toes sequentially.
lmao!!

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Re: A roll of the dice
Gord wrote:Everyone on the maths website who bothered to post agreed with me. Almost everyone on the nonmaths website who posted (even the professor of physics and the professor of computer science!) disagreed with me.
do you have a link for the nonmaths website? i thought i heard it all but some of these are quite creative...
i've seen mathematicians who were not stat/prob people get this wrong. well, they got the montyhall problem wrong. tbh i've never seen physics, cs or mathematicians get a simple dice problem like this wrong. it's quite illuminating, although i'm not sure of what lol

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Re: A roll of the dice
I eat without fear of certain Death from The Tree of Knowledge because with wisdom, we may one day break free from its mortal curse.

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Re: A roll of the dice
Tallboy wrote:Hi Scott i did NOT mean you regarding dunningkruger. i was referring to people who, in addition to having the wrong answer, were condescending/arrogant in their replies to Gord (see the first few pages before you joined the conversation). it's the condescending tone that i'm referring to, and Gord's mentioned that in other blogs he was actually called an 'idiot.' it was to this comment and the people that made it that ii made the DK reference about. your responses have always been polite.
although i don't think you are thinking about the problem correctly, you are making very valid points on the misuse of stats and your thoughts on teaching 'bottomup' (i have no problem with your approach on this and it was OT so i let it alone). as you can prob tell, i have a strong interest in the misuse of stats/prob, and it is a great learning experience for me to see flaws in my or others arguments. reading what you and others say, i think i have a far better understanding of this problem than i had before!
I didn't interpret this as about me necessarily but HAVE had this accused of me before. I believe I mentioned it before regarding specifically the behavior you are discussing. I feel that we could reduce much online problems for even those who initially may insult because we are all naturally emotional and may not intend it. But some are also 'rational' in their heads but have a harder time communicating it (yet?) and so granting they'll possibly get emotionally frustrated can make them also become the very things they are being accused of (being 'weird' in some way). So that link and movie is just a an aside that just accidentally suits the very justification for the research that Dunning Kruger opted to tackle. It originated literally from a court case of some apparent criminal who tried to rob a bank (or some other crime) who thought he really WAS invisible and so raised question to HOW, if he was serious, such a STRONG belief in something can be caused. "Griff, the Invisible" doesn't credit Dunning Kruger but accidentally entertains how and why some people DO turn to such apparently weird beliefs. ...Sometimes even based on real interpretations of even science itself. [The girl in it turned out attracted to Griff for her own extended belief that she could likely eventually walk through walls based on quantum mechanics interpretation of "superposition" and to the nature of matter to be mostly made of up mere space.
By the way, I opened a thread, First Principles in logic and science... to extend our own discussion on this if you or others are interested. It crosses many of the digressions I keep going into about logic and science which might be unclear to some.
I eat without fear of certain Death from The Tree of Knowledge because with wisdom, we may one day break free from its mortal curse.

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Re: A roll of the dice
Scott Mayers wrote:Tallboy,
I have nothing more to say on this. I personally DON'T normally care about supposed "puzzles" not based in reallife examples but get improperly used as justification in reallife areas. If it's just entertainment, that's fine.
these puzzles are designed to teach basic prob to students. we need this basis to be able to attack more complex problems, like the ones you mention that include lying and cheating. in vegas, casino's actually play games with dice, cards, etc. so these are reallife problems for them.
Scott Mayers wrote:Statistics and probability are abused more often BY those most embraced in it because they are actually deluded by the fact that the relative 'truth' of them DEPEND on treating people as fixed probabilities themselves. So we MUST be more concerned about the details ESPECIALLY in this particular area of math....it is used most specifically by CON ARTISTS who depend on the people to behave predictably upon trusting statistics and probability as though it were a more 'advanced' form of thinking than other math.
it is! (just kidding). yes, there are people that lie with stats all the time (have you been watching the US presidential debates?). The point being that if you have a strong understanding of stats and biasness (both taught in stats courses) then you will be less likely to be taken in by conmen. so the way to protect yourself from being conned with stats, is to learn stats!
Scott Mayers wrote:Las Vegas is based and dependent on this. The stock market is based and dependent upon this.
casino owners make lot's of $$ using probability correctly and banking on the public not using it correctly (the gamblers fallacy, emotion, etc.). it relies on the publics' misunderstanding of probability (I find this somewhat reprehensible, btw). the stock market/hedge fund guys use prob/stats in the same way to predict stock prices. you can tell them that their probabilities are logically unsound, or that they don't deal with the real world problems, and they will smile back and wave to you as they drive off in their Ferrari's on their way to their $10M vacation house in the Hamptons.
stats is not deterministic. all stat models include an error term, so prediction is not perfect. it's a tool to predict future events when the reallife situation is too complex for deterministic systems. when Einstein said 'God doesn't roll dice with the universe' he's right. if we knew exactly how the universe worked we wouldn't need stats. but some things are unknowable. perhaps some day in the future we could predict how people will vote from a blood test. but until then, we take polls that have a margin of error. and these polls, if done correctly (representative random sample, etc.) are pretty darn accurate.

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Re: A roll of the dice
Tallboy wrote:Gord wrote:Everyone on the maths website who bothered to post agreed with me. Almost everyone on the nonmaths website who posted (even the professor of physics and the professor of computer science!) disagreed with me.
do you have a link for the nonmaths website?
No, it's long gone. The original forum disappeared over 12 years ago, then the second version was created, then it disappeared and a third came along, then it was gone and the fourth and final version was up until someone important died (the husband of the woman hosting the final version; the original version of the forum was run by the guy who died, from a progress brain disease). They're all gone now. I'm 90% certain the riddle was posted on the fourth incarnation.
"Knowledge grows through infinite timelessness"  the random fictional Deepak Chopra quote site
"Imagine an ennobling of what could be"  the New Age BS Generator site
"You are also taking my words out of context."  Justin
"Nullius in verba"  The Royal Society ["take nobody's word for it"]
#ANDAMOVIE
Is Trump in jail yet?
"Imagine an ennobling of what could be"  the New Age BS Generator site
"You are also taking my words out of context."  Justin
"Nullius in verba"  The Royal Society ["take nobody's word for it"]
#ANDAMOVIE
Is Trump in jail yet?