A roll of the dice

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Re: Right turn at Albuquerque
SweetPea wrote:How about a twist on things?
Suppose Gord rolls the dice behind blinds and only reveals to himself one die at a time. If he opens the first door ( Blue) and finds a 1, he reports finding a 1 and does not open the Red door.
Does that scenario help out?
No.
You're checking one die at a time. All we know is that at least one of the dice is a 1. They're quite different scenarios and will give different answers.
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Re: Right turn at Albuquerque
No, that Gord is checking them and it doesn't have to be separately, he could reveal all at one time. And he can roll them in the past too with no difference.John Jackson wrote:SweetPea wrote:How about a twist on things?
Suppose Gord rolls the dice behind blinds and only reveals to himself one die at a time. If he opens the first door ( Blue) and finds a 1, he reports finding a 1 and does not open the Red door.
Does that scenario help out?
No.
You're checking one die at a time. All we know is that at least one of the dice is a 1. They're quite different scenarios and will give different answers.
If the die behaves naturally ( allows us to be always expecting 1 out of 6 for any die thrown), it's easier to see what is going on
We can make Gord know and remain not knowing instead.
The only difference is change in perception when we look at it this way or that.
He could use both dice and look at them together, he could roll 60 pair of dice at once and look at all results together.
How do the Deniers get so lucky?
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 SweetPea
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Re: A roll of the dice
besides, look what weird answer it gives
The routine of having that Gord open the blue door first is only to show conceptual agreement between what is given and the way the 1 in 6 is played out each time, and from there we find out how many rolls our Gord is to be concerned with. .
I'm looking at it this way: out of lots and lot of rolls of a pair of dice, the results fall a certain way based on 1 in 6 die being a 1. That doesn't change no matter what, even if you roll ten thousand dice at once out of a bucket..
So the question after we know the figures for a set number of rolls, is how many rolls is our Gord concerned with of the total, to get the result we obtained for him. We can subtract the number of rolls that had no 1's from the total number of rolls, to find out. ..
The routine of having that Gord open the blue door first is only to show conceptual agreement between what is given and the way the 1 in 6 is played out each time, and from there we find out how many rolls our Gord is to be concerned with. .
I'm looking at it this way: out of lots and lot of rolls of a pair of dice, the results fall a certain way based on 1 in 6 die being a 1. That doesn't change no matter what, even if you roll ten thousand dice at once out of a bucket..
So the question after we know the figures for a set number of rolls, is how many rolls is our Gord concerned with of the total, to get the result we obtained for him. We can subtract the number of rolls that had no 1's from the total number of rolls, to find out. ..
Last edited by SweetPea on Mon Jan 28, 2013 3:12 pm, edited 1 time in total.
How do the Deniers get so lucky?
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Re: A roll of the dice
The next step.
The die that is known to be a 1 will not change its number spontaneously if the unknown die is revealed to be a 1. Do you agree that yhis is true as well?
The die that is known to be a 1 will not change its number spontaneously if the unknown die is revealed to be a 1. Do you agree that yhis is true as well?

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Re: A roll of the dice
Vanguard wrote:The die that is known to be a 1
We don't know that a particular die is a 1. We only know that at least one of them is.
"One die is a 1" and "one of the dice is a 1" are not the same thing.
We have to get over this hurdle before moving on: we do not know which die is showing the 1 at any time.
Agreed?
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Re: A roll of the dice
We can know ideally how many rolls do not have any 1's.John Jackson wrote:Vanguard wrote:The die that is known to be a 1
We don't know that a particular die is a 1. We only know that at least one of them is.
How do the Deniers get so lucky?
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Re: A roll of the dice
When Gord was erasing from the full square, did he know which roll produced the combinations he was erasing ?
If he didn't know which roll the combination came from, he could actually be erasing a roll which contained a 1 or a snake eyes!
Same deal, essentially, as you are worried about with Van
Gord knows the combinations happened ideally, but does not know in which roll they happened.
We know ideally for large numbers of reps, rather than by actual outcome of the particular experiment.
That's why we can say by the 1 in 6 rule, that of 120 die rolls, 20 turn out to be 1, ideally, no matter what.
If he didn't know which roll the combination came from, he could actually be erasing a roll which contained a 1 or a snake eyes!
Same deal, essentially, as you are worried about with Van
Gord knows the combinations happened ideally, but does not know in which roll they happened.
We know ideally for large numbers of reps, rather than by actual outcome of the particular experiment.
That's why we can say by the 1 in 6 rule, that of 120 die rolls, 20 turn out to be 1, ideally, no matter what.
Last edited by SweetPea on Mon Jan 28, 2013 4:46 pm, edited 1 time in total.
How do the Deniers get so lucky?
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Re: A roll of the dice
Ah. John, this is where you go wrong.
It doesn't matter if die A or die B is a 1. It only matters that we know one of them is a 1. "One die is a 1" and "one if the dice is a 1" is the exact same thing
Although the dice are rolled simultaneously, the die are independent. It doesn't matter what another die says, the odds of rolling a 1 on any d6 are 1/6. By stipulating that one of the die is known no matter which die it is, only the unknown die has the potential to be anything but a 1. There are six sides on the unknown die. One of those faces is a 1. What are the odds that the unknown die also came up as a 1? Please answer that question.
It doesn't matter if die A or die B is a 1. It only matters that we know one of them is a 1. "One die is a 1" and "one if the dice is a 1" is the exact same thing
Although the dice are rolled simultaneously, the die are independent. It doesn't matter what another die says, the odds of rolling a 1 on any d6 are 1/6. By stipulating that one of the die is known no matter which die it is, only the unknown die has the potential to be anything but a 1. There are six sides on the unknown die. One of those faces is a 1. What are the odds that the unknown die also came up as a 1? Please answer that question.
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Re: A roll of the dice
One 1 in 6 rolls is odds of 5 to 1 againstVanguard wrote:Ah. John, this is where you go wrong.
It doesn't matter if die A or die B is a 1. It only matters that we know one of them is a 1.
Although the dice are rolled simultaneously, the die are independent. It doesn't matter what another die says, the odds of rolling a 1 on any d6 are 1/6.
How do the Deniers get so lucky?
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 SweetPea
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Re: Right turn at Albuquerque
This way Gord gets to check all the rolls with both die together at the same time ( except for the 20 rolls which showed a 1 in the blue door),and so get a figure for how many red door 1's there are.in 120 reps and so get a figure for how many reps Gord is concerned with out of the 120, . .John Jackson wrote:SweetPea wrote:How about a twist on things?
Suppose Gord rolls the dice behind blinds and only reveals to himself one die at a time. If he opens the first door ( Blue) and finds a 1, he reports finding a 1 and does not open the Red door.
Does that scenario help out?
No.
You're checking one die at a time. All we know is that at least one of the dice is a 1. They're quite different scenarios and will give different answers.
Last edited by SweetPea on Mon Jan 28, 2013 5:25 pm, edited 1 time in total.
How do the Deniers get so lucky?
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Re: A roll of the dice
Once you know that one of the die is a 1, there are no longer 11 possible outcomes. You lose the 5 outcomes that you know didn't happen. The die that shows a 1 does not show 2, 3, 4, 5 or 6. It doesnt matter if it is die A or die B. The die that says 1 will not say another number. The other 5 sides on the die that says 1 are not possible. The only possibilities are the 6 sides of the unknown die. As there are only 6 possible outcomes left, the denominator must be a 6.
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Re: A roll of the dice
Suppose we are The Gods and Gord is like a robot in that he can't quit until he's supposed to. As Gods we can send the combinations for each roll down in any order, so long as in the end of time, everything we do balances.
If Gord sees approx the right figures as he does ideally, by the end of a lot of rolls, he's satisfied. The number of rolls is known in advance by us so he can't quit on his own accord early when the lasagna comes by..
If we do it right Gord can't tell that magic happened.
Edited to fix
So we could send down all the combinations that have no 1s . Gord checks them all, both at once.
Then we send down all the combinations that have 1s behind the red door only. Gord checks them all both at once
That leaves only the combinations that have 1 in the blue door. We start to send them.
Gord is relieved that things finally worked out naturally and normally and perfectly, when it gets to 120 reps  because when things were looking out of whack, he had calculated the minimum number of combos with 1 in the blue door it could take in a row until the ideal was reached  and it did ! All the last combos having 1 in the blue door.
If Gord sees approx the right figures as he does ideally, by the end of a lot of rolls, he's satisfied. The number of rolls is known in advance by us so he can't quit on his own accord early when the lasagna comes by..
If we do it right Gord can't tell that magic happened.
Edited to fix
So we could send down all the combinations that have no 1s . Gord checks them all, both at once.
Then we send down all the combinations that have 1s behind the red door only. Gord checks them all both at once
That leaves only the combinations that have 1 in the blue door. We start to send them.
Gord is relieved that things finally worked out naturally and normally and perfectly, when it gets to 120 reps  because when things were looking out of whack, he had calculated the minimum number of combos with 1 in the blue door it could take in a row until the ideal was reached  and it did ! All the last combos having 1 in the blue door.
Last edited by SweetPea on Mon Jan 28, 2013 6:35 pm, edited 5 times in total.
How do the Deniers get so lucky?
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Re: A roll of the dice
Vanguard wrote:Ah. John, this is where you go wrong.
No, it's where you go wrong.
Vanguard wrote:"One die is a 1" and "one if the dice is a 1" is the exact same thing
It isn't the same. This is the crucial insight you need. The amount of information in both statements is different.
Vanguard wrote:There are six sides on the unknown die. One of those faces is a 1. What are the odds that the unknown die also came up as a 1? Please answer that question.
The answer to your question is 1 in 6. But we don't know which die is the 'unknown' one. You're still treating the problem as if we know that one die is a 1 and then all we have to do is work out the probability that the 'unknown' die is also a 1.
But that is not what we're attempting to do.
We know that one of the dice is a 1.
This means that there are 11 possible combinations for how at least one die is a 1. Only one of them is the [1,1] combination.
All I can say, again, is that you need to do an experiment (as the problem is stated) and be surprised by the result.

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Re: A roll of the dice
Your perspective changes once you know that one of the die definately is a 1. You are confusing the number of possibilities from before you recieve the information and after you recieve it.
When you don't know what either dice say, the odds are 11/36 that you will get at least one 1. So, the dice have been rolled already and one die shows a 1. What are the chances that this specific dice roll produced at least one 1? Is it a certainty or is it 6/11?
When you don't know what either dice say, the odds are 11/36 that you will get at least one 1. So, the dice have been rolled already and one die shows a 1. What are the chances that this specific dice roll produced at least one 1? Is it a certainty or is it 6/11?
Last edited by Vanguard on Mon Jan 28, 2013 7:01 pm, edited 2 times in total.
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Re: A roll of the dice
I disagree, I think he's treating the problem not as if he knows, but rather as if it doesn't matter if he knows.John Jackson wrote:The answer to your question is 1 in 6. But we don't know which die is the 'unknown' one. You're still treating the problem as if we know that one die is a 1 and then all we have to do is work out the probability that the 'unknown' die is also a 1.
How do the Deniers get so lucky?
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Re: A roll of the dice
Sweapea, I've read through your posts, but I can't tell if you agree with my answer or not.
Are you arguing that the answer is 1 in 11, or 1 in 6?
Are you arguing that the answer is 1 in 11, or 1 in 6?
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Re: A roll of the dice
Yes, it doesn't matter which die, only that one of the dice is a 1.
There are some parallels that can be drawn from the Monty Hall Problem.
#1. If you pick the door with the car first, it doesn't matter which goat is shown, only that a goat is shown, in terms of the probability. Just like the dice in Gord's problem. It doesn't matter which die is known, only that one is known.
#2. The Little Green Lady example. If a person were to pick a door after the goat was revealed, they would have a 1/2 chance of winning, not 2/3. This is because the number of possible outcomes is completely different for the original player and the new one who came in the middle of the scenario. By revealing that there is a 1 on the die, Gord has already revealed a goat. The fact thay one die is known to be a 1 already completely alters the number of possible outcomes.
So, I ask again. What is the chance that one of the die is a 1 in Gord's problem? Is it a certainty or is it 6/11?
There are some parallels that can be drawn from the Monty Hall Problem.
#1. If you pick the door with the car first, it doesn't matter which goat is shown, only that a goat is shown, in terms of the probability. Just like the dice in Gord's problem. It doesn't matter which die is known, only that one is known.
#2. The Little Green Lady example. If a person were to pick a door after the goat was revealed, they would have a 1/2 chance of winning, not 2/3. This is because the number of possible outcomes is completely different for the original player and the new one who came in the middle of the scenario. By revealing that there is a 1 on the die, Gord has already revealed a goat. The fact thay one die is known to be a 1 already completely alters the number of possible outcomes.
So, I ask again. What is the chance that one of the die is a 1 in Gord's problem? Is it a certainty or is it 6/11?

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Re: A roll of the dice
Vanguard wrote:Your perspective changes once you know that one of the die definately is a 1.
Yes, from not having a clue to knowing that at least one die is a 1.
Vanguard wrote:You are confusing the number of possibilities from before you recieve the information and after you recieve it.
Before we receive any information we know nothing about the state of either die. After we receive the information all we know is that at least one of the dice is a 1.
Vanguard wrote:When you don't know what either dice say, the odds are 11/36 that you will get at least one 1.
Correct.
In the problem as it is stated, we don't know which of the two dice is a 1.
Vanguard wrote:So, the dice have been rolled already and one die shows a 1.
No. One of the dice shows a 1.
Vanguard wrote:What are the chances that this specific dice roll produced at least one 1? Is it a certainty or is it 6/11?
Neither. As the information that we have is that one of the dice is a 1, we can only say that either one die or the other is a 1 therefore there are 11/36 combinations that match our knowledge and so it's 1 in 11 that a [1,1] occurred given the information we have.

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Re: A roll of the dice
Really? It's not certain that this dice roll produced at least one 1? Do you want to take another crack at that answer?
If it isn' certain or 6/11, I'm intrigued to know what you think the probability of there being at least one die that says 1 in the specific roll in question.
In your thinking, if I were to take the die that said a 1 out from behind the blind and set it down right in front of you and you looked right at it, would that alter the probability in any way?
If it isn' certain or 6/11, I'm intrigued to know what you think the probability of there being at least one die that says 1 in the specific roll in question.
In your thinking, if I were to take the die that said a 1 out from behind the blind and set it down right in front of you and you looked right at it, would that alter the probability in any way?
Last edited by Vanguard on Tue Jan 29, 2013 12:38 am, edited 1 time in total.
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Re: A roll of the dice
John Jackson, I think I love you. And I only say that to 3.141592654 people.
There are 11/36 ways to roll at least one one. The red die can roll a one 6 of those 11 ways, and the blue die can roll a one 6 of those 11 ways. In only 1 of those 11 ways can both the red and the blue dice roll ones.
There are 11/36 ways to roll at least one one. The red die can roll a one 6 of those 11 ways, and the blue die can roll a one 6 of those 11 ways. In only 1 of those 11 ways can both the red and the blue dice roll ones.
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Re: A roll of the dice
Yes. And if the Red die is a 1, then there is a 1/6 possibility that the Blue due is a 1. Alternatively, if the Blue die is a 1, then there is a 1/6 chance that the Red die is a 1.
These two scenarios are mutually exclusive. They cannot exist in the same universe. Hence there are 6 possible outcomes, not 11.
So there you go.
These two scenarios are mutually exclusive. They cannot exist in the same universe. Hence there are 6 possible outcomes, not 11.
So there you go.
Last edited by Vanguard on Tue Jan 29, 2013 1:02 am, edited 1 time in total.
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Re: A roll of the dice
I get approximately 1 in 11.1 but I'm too tired to check why because of your stupid dice. Hope you had a nice snooze.Gord wrote:Sweapea, I've read through your posts, but I can't tell if you agree with my answer or not.
Are you arguing that the answer is 1 in 11, or 1 in 6?
How do the Deniers get so lucky?
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Re: A roll of the dice
Wow. Nobody here but me knows math?
All you did was take the chance of getting a 1 on 2d6 (11/36) and the chance of getting snakeyes (1/36) and jam them together like this (1/11). Nowhere in all of probability or statistics will that method give you a probability of anything. Learn math.
My faith in humanity has been downgraded.
All you did was take the chance of getting a 1 on 2d6 (11/36) and the chance of getting snakeyes (1/36) and jam them together like this (1/11). Nowhere in all of probability or statistics will that method give you a probability of anything. Learn math.
My faith in humanity has been downgraded.
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Re: A roll of the dice
It is fun watching people struggle against the obvious.
An infallible, incapable of lying robot rolls two completely legitimate, and completely the same dice behind a screen.
You know there is 36 possible combinations for the dice roll.
The robot tells you that one of the die is a 1.
You know there are 11 combinations where one of the die will be a 1 (you have no idea which die is a 1).
Out of these 11 combinations only 1 is for both die to be a 1.
Now, what are the chances that both die are 1?
The answer is 1 in 11.
Any other answer is just stoopid.
An infallible, incapable of lying robot rolls two completely legitimate, and completely the same dice behind a screen.
You know there is 36 possible combinations for the dice roll.
The robot tells you that one of the die is a 1.
You know there are 11 combinations where one of the die will be a 1 (you have no idea which die is a 1).
Out of these 11 combinations only 1 is for both die to be a 1.
Now, what are the chances that both die are 1?
The answer is 1 in 11.
Any other answer is just stoopid.
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Re: A roll of the dice
Donnageddon wrote:It is fun watching people struggle against the obvious.
I'm telling you, he's just trolling now. Nobody could be that amazingly dense and still type legibly.
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Re: A roll of the dice
SweetPea wrote:Gord wrote:Sweapea, I've read through your posts, but I can't tell if you agree with my answer or not.
Are you arguing that the answer is 1 in 11, or 1 in 6?
I get approximately 1 in 11.1 but I'm too tired to check why because of your stupid dice. Hope you had a nice snooze.
What?
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Re: A roll of the dice
Gord wrote:Donnageddon wrote:It is fun watching people struggle against the obvious.
I'm telling you, he's just trolling now. Nobody could be that amazingly dense and still type legibly.
I am still waiting for MM to eat crow.
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Re: A roll of the dice
This is the conceptual problem. I see it as arising from a difference in assumptions possibly due to language/terms being used.Vanguard wrote:Yes. And if the Red die is a 1, then there is a 1/6 possibility that the Blue due is a 1. Alternatively, if the Blue die is a 1, then there is a 1/6 chance that the Red die is a 1.
These two scenarios are mutually exclusive. They cannot exist in the same universe. Hence there are 6 possible outcomes, not 11.
So there you go.
The antagonistic views are
A/ These outcomes are possible outcomes
B/ These outcomes are outcomes that already occurred, "virtually".
It's not because of stupidity that our minds turn different ways on these things. The MHP solution generated lots of opposition from good minds.
How do the Deniers get so lucky?
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Re: A roll of the dice
I've found a major error in my proof now that I've rechecked.Gord wrote:SweetPea wrote:Gord wrote:Sweapea, I've read through your posts, but I can't tell if you agree with my answer or not.
Are you arguing that the answer is 1 in 11, or 1 in 6?
I get approximately 1 in 11.1 but I'm too tired to check why because of your stupid dice. Hope you had a nice snooze.
What?
How do the Deniers get so lucky?
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Re: A roll of the dice
SweetPea wrote:I've found a major error in my proof now that I've rechecked.Gord wrote:SweetPea wrote:Gord wrote:Sweapea, I've read through your posts, but I can't tell if you agree with my answer or not.
Are you arguing that the answer is 1 in 11, or 1 in 6?
I get approximately 1 in 11.1 but I'm too tired to check why because of your stupid dice. Hope you had a nice snooze.
What?
I understand approximately 1/3 of your posts. Rarely an entire one; usually just a part of each.
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Re: A roll of the dice
Gord wrote:SweetPea wrote:I've found a major error in my proof now that I've rechecked.Gord wrote:SweetPea wrote:Gord wrote:Sweapea, I've read through your posts, but I can't tell if you agree with my answer or not.
Are you arguing that the answer is 1 in 11, or 1 in 6?
I get approximately 1 in 11.1 but I'm too tired to check why because of your stupid dice. Hope you had a nice snooze.
What?
I understand approximately 1/3 of your posts. Rarely an entire one; usually just a part of each.
Well, I stayed late trying to show a proof ( eventually reaching a weird result).
When you asked about my answer wrt "1 out of 6 times or 1 out of 11" , I could tell you the result i got  but not why it turned out weird ( 1 in 11.1 )
Now that I am refreshed I found a major error once I looked it over.
How do the Deniers get so lucky?
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 Gord
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Re: A roll of the dice
1/11.1 isn't that weird. It's very close to the average result of 1/11. There's always going to be a margin of error; when I did the die rolling experiment, my results were closer to 1/10 than 1/11.
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 SweetPea
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Re: A roll of the dice
but it was a theoretical result, a virtual result. And a major error in method is the reason, not minor variability. The closeness to 11 was the "temptingly" weird part. That was a fluke.Gord wrote:1/11.1 isn't that weird. It's very close to the average result of 1/11. There's always going to be a margin of error; when I did the die rolling experiment, my results were closer to 1/10 than 1/11.
How do the Deniers get so lucky?
viewtopic.php?f=16&t=24129
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 Gord
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Re: A roll of the dice
Oh.
Then I have no idea how you got 1/11.1. So again, I didn't understand 2/3 of your post.
Then I have no idea how you got 1/11.1. So again, I didn't understand 2/3 of your post.
"Knowledge grows through infinite timelessness"  the random fictional Deepak Chopra quote site
"You are also taking my words out of context."  Justin
"Nullius in verba"  The Royal Society ["take nobody's word for it"]
#ANDAMOVIE
"You are also taking my words out of context."  Justin
"Nullius in verba"  The Royal Society ["take nobody's word for it"]
#ANDAMOVIE

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Re: A roll of the dice
Gord wrote:John Jackson, I think I love you. And I only say that to 3.141592654 people.
1 in 6, surely!
 Gord
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Re: A roll of the dice
John Jackson wrote:Gord wrote:John Jackson, I think I love you. And I only say that to 3.141592654 people.
1 in 6, surely!
I like pi.
Mmmmm, piiiiiiiii....
"Knowledge grows through infinite timelessness"  the random fictional Deepak Chopra quote site
"You are also taking my words out of context."  Justin
"Nullius in verba"  The Royal Society ["take nobody's word for it"]
#ANDAMOVIE
"You are also taking my words out of context."  Justin
"Nullius in verba"  The Royal Society ["take nobody's word for it"]
#ANDAMOVIE
 SweetPea
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Re: A roll of the dice
I wonder if examining the words used could clarify things.
When we say the number of "possible outcomes" of a roll, is 36, everyone agrees, but when it's about the particular case ( in the puzzle), there is a fundamental disagreement.
A difference in opinion appears when an outcome or result is proposed to have already occurred.
In the puzzle, extra information is given, and so some of the possible "outcomes" previously considered, are now going to be eliminated. What is to be eliminated, is disagreed over.
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.
I tried phrasing the problem in a different way. Instead of calling them "possible outcomes", I call them "ways the dice might behave" or something like that. Ways that outcomes could come about. Like "routes" to get there.
Then when the outcome of the roll is a 1 showing and an unknown, as in this puzzle, I do not wrongly assume that it makes no difference which die has the 1, because I recognize that the number we are discussing is only the ways that such a described combo could come into being.
When we say the number of "possible outcomes" of a roll, is 36, everyone agrees, but when it's about the particular case ( in the puzzle), there is a fundamental disagreement.
A difference in opinion appears when an outcome or result is proposed to have already occurred.
In the puzzle, extra information is given, and so some of the possible "outcomes" previously considered, are now going to be eliminated. What is to be eliminated, is disagreed over.
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.
I tried phrasing the problem in a different way. Instead of calling them "possible outcomes", I call them "ways the dice might behave" or something like that. Ways that outcomes could come about. Like "routes" to get there.
Then when the outcome of the roll is a 1 showing and an unknown, as in this puzzle, I do not wrongly assume that it makes no difference which die has the 1, because I recognize that the number we are discussing is only the ways that such a described combo could come into being.
How do the Deniers get so lucky?
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Re: A roll of the dice
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.
 SweetPea
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Re: A roll of the dice
Donnageddon wrote:I am still waiting for MM to eat crow.
To say that, means you've actually understood nada.
. .
How do the Deniers get so lucky?
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