A roll of the dice

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SweetPea
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Re: A roll of the dice

Postby SweetPea » Wed Jan 30, 2013 12:22 am

John Jackson wrote:
SweetPea wrote:I wonder if examining the words used could clarify things.

What has become clear to me is that how the puzzle is worded is crucial.

The maths is simple - it's the conceptualization based on the framing of the problem (or how it's perceived) that causes the difficulty with conceptualizing it.

In this instance, it's the difference between knowing that one die is a 1 and one of the dice is a 1 that seems to be what's causing the confusion (including in me to begin with).

Very subtle, but very real, differences in the level of information we have makes a difference to how we (attempt to) solve it and what the correct answer actually is.

I'm not sure whether there's an easy way to explain the solution to the problem or whether it's just a case of you 'get it' or you don't.
I'm looking at what "outcomes" means wrt the result of a roll, compared to it's use when making or altering the square.
I think the overlap of the use of the term serves to increase problems in perceiving what is being described after the roll.
I think most everyone can get it if the answer is presented differently. It wouldn't be as much fun, though.
How do the Deniers get so lucky?
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Re: A roll of the dice

Postby Gord » Wed Jan 30, 2013 3:36 am

SweetPea wrote:The argument as I see it, is that the roll has already occurred, therefore there are not two possible die with a 1 showing - only 1 die , and it's already been "decided"; the roll already occurred.

In probability, it doesn't matter whether the event has occured yet or is going to occur; it only matters that the outcome remains unknown. Once the dice are rolled, they definitely have a specific outcome. In this particular problem, that outcome is known to one person, who then reveals partial information to us.

John Jackson wrote:The maths is simple - it's the conceptualization based on the framing of the problem (or how it's perceived) that causes the difficulty with conceptualizing it.

The difficulty comes in understanding what we know, vs. what we think we know. It's taken as a given that the dice roll will always have one and only one result; whatever that result is, once it has happened, the probability that it has happened will be 100%: For instance, if a 3 is rolled, then the probability that a 3 was rolled is 100%.

In this riddle, the information that we come to know is unusual. "At least one of the dice rolled a 1" informs us, paradoxically, that the result is both known and unknown -- in this case, two observers make us question what is known vs. what is unknown. But while the first observer can know the final, specific result, the second observer (which is a group including us, the ones trying to solve the problem) cannot.

With this seeming confusion over what is known or unknown, we turn to semantics. We understand intuitively that the first observer knows the results of both dice, and so we feel that one of the dice is "locked in" with a result of 1. This makes us feel that we can then eliminate one of the dice from the probability matrix, and that we can determine the chance of the remaining die result as if only one die had been rolled by itself. But the reality is counterintuitive; we do not know which die rolled the 1, and therefore cannot eliminate either die from the matrix.

"Matrix", of course, refers to my chart of the results of the two dice:

Image

But while the information given us cannot eliminate either die, it can eliminate specific results of each die. We know at least one 1 was rolled, and so those results containing no 1s can be eliminate from the matrix, leaving us with 11 remaining possibilities, of which only one contains two 1s.
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Re: A roll of the dice

Postby Gord » Wed Jan 30, 2013 3:39 am

Gord wrote:Image

P.S. I stole that image from the internet. Ignore the "T2", it's not germane to the discussion.

P.P.S. I actually expected to find a lot more examples of the 2 dice matrix for me to choose from, but thanks to the Matrix movies, the google result of my search was swamped with Keanu Reeves and leather trenchcoats! :P
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Re: A roll of the dice

Postby Donnageddon » Wed Jan 30, 2013 4:33 am

SweetPea wrote:
Donnageddon wrote:I am still waiting for MM to eat crow.

To say that, means you've actually understood nada.


Heck, SweetPea, as long as I don't understand you, I reckon I'm fine.
My name is not Donna.

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Re: A roll of the dice

Postby SweetPea » Wed Jan 30, 2013 4:42 am

Donnageddon wrote:
SweetPea wrote:
Donnageddon wrote:I am still waiting for MM to eat crow.

To say that, means you've actually understood nada.


Heck, SweetPea, as long as I don't understand you, I reckon I'm fine.
Then why are you waiting for MM to eat crow, Dearie?
How do the Deniers get so lucky?
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Re: A roll of the dice

Postby SweetPea » Wed Jan 30, 2013 5:22 am

Gord wrote:In probability, it doesn't matter whether the event has occured yet or is going to occur; it only matters that the outcome remains unknown. Once the dice are rolled, they definitely have a specific outcome. In this particular problem, that outcome is known to one person, who then reveals partial information to us.
...
The difficulty comes in understanding what we know, vs. what we think we know. It's taken as a given that the dice roll will always have one and only one result; whatever that result is, once it has happened, the probability that it has happened will be 100%: For instance, if a 3 is rolled, then the probability that a 3 was rolled is 100%.

In this riddle, the information that we come to know is unusual. "At least one of the dice rolled a 1" informs us, paradoxically, that the result is both known and unknown -- in this case, two observers make us question what is known vs. what is unknown. But while the first observer can know the final, specific result, the second observer (which is a group including us, the ones trying to solve the problem) cannot.

With this seeming confusion over what is known or unknown, we turn to semantics. We understand intuitively that the first observer knows the results of both dice, and so we feel that one of the dice is "locked in" with a result of 1. This makes us feel that
Is it not Pragmatics that we would turn to, then ?
Edit: probably not Pragmatics, but why Semantics in this case ?
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Re: A roll of the dice

Postby Donnageddon » Wed Jan 30, 2013 5:49 am

SweetPea wrote:
Donnageddon wrote:
SweetPea wrote:
Donnageddon wrote:I am still waiting for MM to eat crow.

To say that, means you've actually understood nada.


Heck, SweetPea, as long as I don't understand you, I reckon I'm fine.
Then why are you waiting for MM to eat crow, Dearie?


And that is a non sequitur, Sweetcheeks. Pretty much everything I've seen you post is either gibberish or a non sequitur.
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Re: A roll of the dice

Postby SweetPea » Wed Jan 30, 2013 5:55 am

Donnageddon wrote:
SweetPea wrote:
Donnageddon wrote:
SweetPea wrote:
Donnageddon wrote:I am still waiting for MM to eat crow.

To say that, means you've actually understood nada.


Heck, SweetPea, as long as I don't understand you, I reckon I'm fine.
Then why are you waiting for MM to eat crow, Dearie?


And that is a non sequitur, Sweetcheeks. Pretty much everything I've seen you post is either gibberish or a non sequitur.
What you offered
Heck, SweetPea, as long as I don't understand you, I reckon I'm fine.
was a non sequitur, yes. You've offered a couple in a row now.
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Re: A roll of the dice

Postby Gord » Wed Jan 30, 2013 7:23 am

SweetPea wrote:Is it not Pragmatics that we would turn to, then ?
Edit: probably not Pragmatics, but why Semantics in this case ?

I dunno, I don't get out much.
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Re: A roll of the dice

Postby Donnageddon » Wed Jan 30, 2013 7:39 am

It gets ugly when someone posts an anti-semantic comment.
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Re: A roll of the dice

Postby SweetPea » Wed Jan 30, 2013 7:48 am

Donnageddon wrote:It gets ugly when someone posts an anti-semantic comment.

beyond the pale!
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Re: A roll of the dice

Postby Vanguard » Wed Jan 30, 2013 10:28 pm

SweetPea has taken the Red Pill. Welcome down the rabbit hole. You should be proud.

Gord set several conditions for the problem in question.
1. We are only discussing the result of one specific dice roll.
2. The dice roll we are discussing has already occurred and so has a verifiable result.
3. One of the two dice rolled came up a 1.

There are two assumptions we can make.
1. The die that came up a 1 will not change to say another number until after the instance is over.
2. The die that came up a 1 is distinct and identifiable. (E.g. One could potentially point at it and say 'That die says 1'.)
It seems to me that only the second assumption is in dispute. You say we don't know which die says 1. I say since the dice have already been rolled, we do. Gord knows.

The fact that the die is verifiable is obfuscated by the condition that the die that showed a 1 was rolled behind a screen. But Gord can see the die in question and can see that there is a verifiable result. There is no potential for the die that Gord can see is a 1 to be anything but a 1. Therefore it is distinct. If he can identify the die that says 1, then for Gord, the question is what is the chance that the other die says 1 as well. 6 sides on the other die, one of them says 1, the odds are 1/6 for Gord.

So, if you are on the other side of the screen, do you exist in an alternate reality? The odds are 1/6 for Gord because he can see the die, but 1/11 for you because you cannot? The reality is the same for both individuals, screen or not. An answer of 1/11 denies that an objective reality exists.

The possibility for either die to come up a 1 without it being distinct and identifiable only exists before the die are rolled. Once they are rolled, we know which die came up a 1. The conditions change after the die are rolled when discussing the result of a specific roll because the question of what will thr dice show has already been answered.

It's a difficult concept I guess, but its a funny error to see people make on a skeptic forum.

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Re: A roll of the dice

Postby SweetPea » Thu Jan 31, 2013 12:08 am

So, if you are on the other side of the screen, do you exist in an alternate reality? The odds are 1/6 for Gord because he can see the die, but 1/11 for you because you cannot? The reality is the same for both individuals, screen or not. An answer of 1/11 denies that an objective reality exists.
Gord knows but we do not. Therefore we use probability calculations.

Suppose Gord did not know. Yes, the dice have already been thrown. An objective reality exists
Can you calculate the probabilities even though it has already been rolled ? Same thing.
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Re: A roll of the dice

Postby Vanguard » Thu Jan 31, 2013 12:28 am

So you agree the odds for Gord are 1/6? Do you think the odds are different on the same dice roll for different people?

If Gord did not know that one of the dice was a 1, then the odds would be 1/36.

Would Gord have asked the question had neither of the dice been a 1?
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Re: A roll of the dice

Postby SweetPea » Thu Jan 31, 2013 12:39 am

Vanguard wrote:If Gord did not know thst one of the dice was a 1, then the odds would be 1/36.

Would Gord have asked the question had he not confirmed the 1?
That doesn't make any difference. Forget about Gord and think about it as if a robot that can only detect a 1 and can do nothing else. It says there is a 1.

We do the calculation if we do not know but want to know the probability.
One in 36 if we don't know anything, and if we know one of the die is a 1, then we calculate.
1 in 11 ways to get snake eyes if we know that one die is a 1.
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Re: A roll of the dice

Postby John Jackson » Thu Jan 31, 2013 1:02 am

Vanguard wrote:The possibility for either die to come up a 1 without it being distinct and identifiable only exists before the die are rolled. Once they are rolled, we know which die came up a 1.

No we don't. We only know that one of them came up as a 1.

If you can't see the dice, how do you know which one came up as a 1?

Vanguard wrote:It's a difficult concept I guess, but its a funny error to see people make on a skeptic forum.

It has been suggested that you're 'trolling' here but I don't think you are. You simply haven't conceptualized the problem properly.

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Re: A roll of the dice

Postby SweetPea » Fri Feb 01, 2013 5:23 pm

Vanguard,
Try this.One die is blue and one is red

There are 36 ways to get results when you roll the pair of dice.
There are 11 ways.to get a result that has at least one of the die showing a 1.
One of those 11 ways is the result "snake eyes".

I'm adjusting a bit to get different views.
Is the above a coherent view ?
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Re: A roll of the dice

Postby SweetPea » Fri Feb 01, 2013 6:12 pm

Trying it in reverse:

The result "snake eyes" is one of 11 ways of getting a result with at least one of the die showing a 1
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Re: A roll of the dice

Postby Gord » Fri Feb 01, 2013 8:30 pm

SweetPea wrote:Is the above a coherent view ?

Yes. That alone worries me. :?
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Re: A roll of the dice

Postby SweetPea » Fri Feb 01, 2013 9:47 pm

Gord wrote:
SweetPea wrote:Is the above a coherent view ?

Yes. That alone worries me. :?

Is it a view that could be more revealing than your explanation ?
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Re: A roll of the dice

Postby Vanguard » Fri Feb 01, 2013 9:50 pm

Yes, SweetPea, that is correct. That doesn't have bearing on this question.

One of the dice is a 1. Will that die change to a different number if the second die is revealed to be a 1?

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Re: A roll of the dice

Postby SweetPea » Fri Feb 01, 2013 9:56 pm

Vanguard wrote:Yes, SweetPea, that is correct. That doesn't have bearing on this question.

One of the dice is a 1. Will that die change to a different number if the second die is revealed to be a 1?

No, whichever die it is will not change to be something else.
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Re: A roll of the dice

Postby Vanguard » Fri Feb 01, 2013 10:09 pm

So if I roll both dice again, is there a chance that the die that says a 1 will say another number if the other die says a 1?

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Re: A roll of the dice

Postby SweetPea » Fri Feb 01, 2013 10:13 pm

Vanguard wrote:So if I roll both dice again, is there a chance that the die that says a 1 will say another number if the other die says a 1?
Yes, it can say another number (regardless of the other die's number).
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Re: A roll of the dice

Postby Vanguard » Fri Feb 01, 2013 10:37 pm

So the simulation as ran introduces a condition that does not exist in the original problem?

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Re: A roll of the dice

Postby SweetPea » Fri Feb 01, 2013 11:11 pm

Vanguard wrote:So the simulation as ran introduces a condition that does not exist in the original problem?
please explain
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Re: A roll of the dice

Postby Vanguard » Sat Feb 02, 2013 12:50 am

Well, if you design an experiment that hasn't controlled for all the variables in the hypothesis except for the variable in question, do you have a valid experiment?

The question is asked after the die roll has happened and a 1 is identified. The question, simplified, is "What is the probability that the other die not identified as a 1 is also a 1?"

This is a diffetent question from "If I rolled both dice and got a 1, what is the chance that I got snake-eyes?

These two questions have different answers. In one the dice roll has already occurred and the die that says 1 can only say 1. In the other, the dice roll has not occurred so either die could come up as a 1.

To set up an experiment for the first, you simulate the specific dice roll in question by placing one die on the table saying 1 and roll the unknown die. As you agreed, there is no chance that the die that says a 1 can become anything else so you don't roll it.

To set up an experiment for the second, you roll 2 die several times and count the frequency of snake-eyes and only one 1 rolls.

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Re: A roll of the dice

Postby SweetPea » Sat Feb 02, 2013 3:00 am

Vanguard wrote:Well, if you design an experiment that hasn't controlled for all the variables in the hypothesis except for the variable in question, do you have a valid experiment?

The question is asked after the die roll has happened and a 1 is identified. The question, simplified, is "What is the probability that the other die not identified as a 1 is also a 1?"

This is a diffetent question from "If I rolled both dice and got a 1, what is the chance that I got snake-eyes?

These two questions have different answers. In one the dice roll has already occurred and the die that says 1 can only say 1. In the other, the dice roll has not occurred so either die could come up as a 1.

To set up an experiment for the first, you simulate the specific dice roll in question by placing one die on the table saying 1 and roll the unknown die. As you agreed, there is no chance that the die that says a 1 can become anything else so you don't roll it.
Say the one die you arbitrarily might pick. is the blue die.
What are the chances that the blue die would be a die showing 1, in rolls which have results meeting the conditions { at least one die showing 1)?
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Re: A roll of the dice

Postby Vanguard » Sat Feb 02, 2013 3:17 am

50%. You can control for that in our experement by doing half the rolls with the blue die set as the 1 and the other half with the red die set as the 1. You will find the result is the same whichever die you set to be the 1 however.

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Re: A roll of the dice

Postby SweetPea » Sat Feb 02, 2013 3:18 am

Vanguard wrote:50%.You can control for that in our experement by doing half the rolls with the blue die set as the 1 and the other half with the red die set as the 1. You will find the result is the same whichever die you set to be the 1 however.
All you are controlling for there is for loading or weighting type differences in the die. Right? That's a quality control or fairness control.

It's already assumed or given that the dice used are statistically indistinguishable from fair or perfect.
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Re: A roll of the dice

Postby SweetPea » Sat Feb 02, 2013 4:20 am

Vanguard wrote:
What are the chances that the blue die would be a die showing 1, in rolls which have results meeting the conditions { at least one die showing 1)?
50%. .
It's 6 out of 12 - not 6 out of 11 ?

Gord? Coherent or not coherent?
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Re: A roll of the dice

Postby Vanguard » Sat Feb 02, 2013 4:28 am

What? No. Jeez. Really?

I was showing you that the color of the die doesn't matter..

If you roll both a red and a blue die, what are the odds that the red die will be a 1? What are the odds that the blue die will be a 1?

If the red die is a 1, what are the odds that the blue die is a 1? If the blue die is a 1, what are the odds that the red die is a 1?

If the red die is a 1, what are the odds that the red die isn't a 1? If the blue die is a 1, what are the odds that the blue die isn't a 1.

You really don't get it?

The question is "I rolled two dice. One of them is a 1. What are the odds that the other die is a 1 too?"

Your answer is "One time out of the number of possibilities on the die that doesn't say 1 plus all the possibilities on the die that does say one, even the ones that we already know didn't happen because I can't see the dice. But if you showed one to me the odds are 1/6."

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Re: A roll of the dice

Postby Vanguard » Sat Feb 02, 2013 4:33 am

If you roll 2d6 and tell me that one die is a 1, I know that 30/36 of the results didn't happen. That leaves 1/6 that did.

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Re: A roll of the dice

Postby SweetPea » Sat Feb 02, 2013 4:34 am

Vanguard wrote:What? No. Jeez. Really?

I was showing you that the color of the die doesn't matter..
Doesn't matter for what, though?

If you roll both a red and a blue die, what are the odds that the red die will be a 1? What are the odds that the blue die will be a 1?
The odds are 5 to 1 against. For either die. Or 1 in 6 chances.

If the red die is a 1, what are the odds that the blue die is a 1?
1 in 6. Now include the odds that the red die is a die showing a 1.
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Re: A roll of the dice

Postby SweetPea » Sat Feb 02, 2013 4:36 am

Vanguard wrote:If you roll 2d6 and tell me that one die is a 1, I know that 30/36 of the results didn't happen. That leaves 1/6 that did.

But you don't know which 30 didn't happen.
What are the odds for either set of 30 that might be the ones that didn't happen, to be the 30 that didn't happen ?
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Re: A roll of the dice

Postby Vanguard » Sat Feb 02, 2013 4:52 am

Doesn't matter which 30. It's 30 if one die is a 1, and a different 30 if the other die is a 1. But it's 30 either way, no less.

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Re: A roll of the dice

Postby SweetPea » Sat Feb 02, 2013 5:05 am

Vanguard wrote:Doesn't matter which 30. It's 30 if one die is a 1, and a different 30 if the other die is a 1.
Yes a different set of 30. What are the odds that the 30 you picked, are the ones which didn't actually happen?
But it's 30 either way
So there are two ways. You might have picked right or you might have picked wrong. What are your odds?
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Re: A roll of the dice

Postby Gord » Sat Feb 02, 2013 11:07 am

SweetPea wrote:What are the chances that the blue die would be a die showing 1, in rolls which have results meeting the conditions { at least one die showing 1)?

It's 6 out of 12 - not 6 out of 11 ?

Gord? Coherent or not coherent?

It's 6 out of 11.

Remember, chance only applies when you don't know the result. When at least one of the two dice shows a 1, there are 11 possibilities. For six of those possibilities, the blue die is a 1, and for six of them the red die is a 1.

SweetPea wrote:But you don't know which 30 didn't happen.
What are the odds for either set of 30 that might be the ones that didn't happen, to be the 30 that didn't happen ?

If you don't know which die shows the 1, it's not 30/36 results that didn't happen, it's 26/36. (That's assuming, of course, that saying "one die is a 1" means both dice aren't 1s. If we stick with the original question, "at least one die is a 1", then there are 25/36 results that didn't happen.)
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SweetPea
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Re: A roll of the dice

Postby SweetPea » Sat Feb 02, 2013 5:05 pm

Gord wrote:
SweetPea wrote:What are the chances that the blue die would be a die showing 1, in rolls which have results meeting the conditions { at least one die showing 1)?

It's 6 out of 12 - not 6 out of 11 ?

Gord? Coherent or not coherent?

It's 6 out of 11.

Remember, chance only applies when you don't know the result. When at least one of the two dice shows a 1, there are 11 possibilities. For six of those possibilities, the blue die is a 1, and for six of them the red die is a 1.
Yes. That's why I questioned the 50 % answer from Vanguard.
Just because the probabilities are the same doesn't mean both are 50 %. Isn't that right? Still coherent?

Gord wrote:
SweetPea wrote:But you don't know which 30 didn't happen.
What are the odds for either set of 30 that might be the ones that didn't happen, to be the 30 that didn't happen ?

If you don't know which die shows the 1, it's not 30/36 results that didn't happen, it's 26/36. (That's assuming, of course, that saying "one die is a 1" means both dice aren't 1s. If we stick with the original question, "at least one die is a 1", then there are 25/36 results that didn't happen.)
Yes.

Vanguard wrote:
What are the chances that the blue die would be a die showing 1, in rolls which have results meeting the conditions { at least one die showing 1)?

50%.
.
So that's why I followed up with
It's 6 out of 12 - not 6 out of 11 ?

Gord? Coherent or not coherent?
Just checking.
After a time, we can expect you'll become accustomed to the emergent property of consilience arising from my posts!
How do the Deniers get so lucky?
viewtopic.php?f=16&t=24129

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Re: A roll of the dice

Postby SweetPea » Sat Feb 02, 2013 10:35 pm

Vanguard, do you see how both die have 6 out of 11 chances ? You count up the squares with at least one 1 in them and you see 11 of those squares.
There are 6 squares with results showing 1 on the blue die. There are 6 squares with results showing 1 on the red die, out of the 11 squares with at least one 1.

If you agree, then you'll follow up to notice that:
neither die has a 50 % probability!
How do the Deniers get so lucky?
viewtopic.php?f=16&t=24129


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