Quote:gordonm888

1. what the known and unknown variables are in a problem statement

2. What parameter you are asked to calculate or define

3. Writing an equation that expresses the requested parameter in terms of the known and unknown variables

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Quote:DweenYou have two 6-sided dice in a cup. You shake the dice, and slam the cup down onto the table, hiding the result. Your partner peeks under the cup, and tells you, truthfully, "At least one of the dice is a 2."

What is the probability that both dice are showing a 2?

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Knowns:

- 2 dice, randomized, hidden (no funny business implied)

- nonzero quantity of 2's (1 or 2 inferred)

Unknowns:

- did the observer see one dice or two?

Given the people I know:

- some like the number 2, because even prime numbers are a peculiarity. They will report a 2, if possible.

- some dislike the number 2, because even prime numbers are a peculiarity. They will report anything else, if possible.

- some do not care about 2's.

OnceDear and others can continue to question "the partner's" gender identity, reasons for looking at the dice, whether 'the partner' has a health condition that affects his/her vision or cognitive ability, what the partner's motivation was for reporting on a 2 versus any other numeral, what prior agreements your partner had with you on signals and terminology and other issues. All of which I think is rubbish. My head isn't filled with hay and rags and I don't care to make a habit of conducting conversations that are uninteresting and of no importance to anyone anywhere ever.

I originally posted on this thread because I wanted to explain the mathematics. I am satisfied that I did that. I disagree with OnceDear on this matter. I don't think it's worth my time to further acknowledge his comments. I speculate he will respond with his usual hyperactivity by swamping us with another two dozen or so posts that rapturously crusade on this issue. But I won't read them. Because I have things to do and a life to live. And a partner who is asking me to come to bed.

Quote:gordonm888"Your partner peeks under the cup . . ."

OnceDear and others can continue to question "the partner's" gender identity, reasons for looking at the dice, whether 'the partner' has a health condition that affects his/her vision or cognitive ability, what the partner's motivation was for reporting on a 2 versus any other numeral, what prior agreements your partner had with you on signals and terminology and other issues. All of which I think is rubbish. My head isn't filled with hay and rags and I don't care to make a habit of conducting conversations that are uninteresting and of no importance to anyone anywhere ever.

I originally posted on this thread because I wanted to explain the mathematics. I am satisfied that I did that. I disagree with OnceDear on this matter. I don't think it's worth my time to further acknowledge his comments. I speculate he will respond with his usual hyperactivity by swamping us with another two dozen or so posts that rapturously crusade on this issue. But I won't read them. Because I have things to do and a life to live. And a partner who is asking me to come to bed.

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My challenge to you is open anytime. 20:1 is sweet odds since you are so sure 11:1 is the right answer under any circumstances.

The person who looks at the dice is forced every time to say “there is at least one [].” Where [] is any number 1-6 that the peeker sees on at least one dice. If the peeker sees two different numbers, he (or she) will randomly pick one of them.

Following that rule, the peeker calls out “I see at least one 2.”

What is the chance of two 2s?

Answer 1/6.

Do folk see how the rules the peeker follows can make the odds 1/6 or 1/11? Or frankly anything between 0% to 100%.

Be satisfied. Where ignorance is bliss, 'tis folly to be wise'Quote:gordonm888"Your partner peeks under the cup . . ."

I originally posted on this thread because I wanted to explain the mathematics. I am satisfied that I did that. I disagree with OnceDear on this matter.

Back at you.Quote:I don't think it's worth my time to further acknowledge his comments.

Wrong again.Quote:I speculate he will respond with his usual hyperactivity by swamping us with another two dozen or so posts that rapturously crusade on this issue.

Have a nice day.

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Quote:Dieter

(trimmed!)Quote:DweenYou have two 6-sided dice in a cup. You shake the dice, and slam the cup down onto the table, hiding the result. Your partner peeks under the cup, and tells you, truthfully, "At least one of the dice is a 2."

What is the probability that both dice are showing a 2?

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Knowns:

- 2 dice, randomized, hidden (no funny business implied)

- nonzero quantity of 2's (1 or 2 inferred)

Close Dieter, but a minor area where you and I disagree.

"- nonzero quantity of 2's (1 or 2 inferred) ON THIS ONE OBSERVATION

Quote:

Unknowns:

- did the observer see one dice or two?

No. We seem to graciously accept from the physics of a cup that he/she saw both dice.

More unknowns

Whether the observer is ONLY reporting for 2's and says ?what? if 2 is not rolled? Would he/she be silent or report something? That affects the denominator in our eventual formula.

Whether the observer is ALWAYS reporting 2's or might he/she report a different number if possible.

Those two unknowns, on their own, give us many alternative answers.

Lot's of trivial stuff we don't know, such as preferences as to number called. I'm happy to ignore such trivia as out of scope.

I think that it is generally accepted that the observer will make a report of some sort every time a two is rolled.

What we cannot agree on are one simple but critical unknown.... WHAT, if anything does he/she report if no two is rolled. Wizard seems to guess that he she will report something like "Neither of the dice are twos". Others guess that he/she would report SOMETHING truthful.

There is a precedent, I believe in craps, Monopoly, snakes and ladders, ludo, etc, All rolls of the dice, once observed are declared and acted upon.

Quote:

Given the people I know:

- some like the number 2, because even prime numbers are a peculiarity. They will report a 2, if possible.

- some dislike the number 2, because even prime numbers are a peculiarity. They will report anything else, if possible.

- some do not care about 2's.The second dice has a 1/6 chance of being a 2.The second dice has a 1/11 chance of being a 2.Since she had no choice but to report a 2, both dice are 2's. 1.If they are not special, then by implication he/she is not only reporting 2's1/11.There is a 6/36 (1/6) chance of rolling doubles; 1/36 that they are both 2's. This case does not fall within the constraints of the problem.

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I think that "peeks under the cup" does not necessarily imply "observes both dice" or "randomly selects an observed value to report".

There is a difference between "The house is green" and "The house appears green on this side."

It really is a simple question which asks for a simple answer. The simple answer is best.

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Quote: Dween

You have two 6-sided dice in a cup. You shake the dice, and slam the cup down onto the table, hiding the result. Your partner peeks under the cup, and tells you, truthfully, "At least one of the dice is a 2."

What is the probability that both dice are showing a 2?

_______________

After hearing the question, don't you put yourself into the position of the partner who sees under the cup?

The question is begging you to take the partner's point of view.

And the partner truthfully sees at least one of the dice is 2.

Everyone agrees that for the partner the answer is 1/6.

The only question is are you taking on the role of the partner?

Yes, you are.

If you weren't asked to take on the role of the partner the question would have been phrased differently. For example, it could have been phrased:

Two dice are shaken. How many combinations of the two dice contain at least one 2? How many combinations contain 2-2?

In this wording you are not led to take on the role of the partner who actually looks and sees the result of at least one die.

Quote:AlanMendelsonGuys thanks for the interesting discussion but I think you're all going way too far in picking apart the question.

It really is a simple question which asks for a simple answer. The simple answer is best.

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Quote: Dween

You have two 6-sided dice in a cup. You shake the dice, and slam the cup down onto the table, hiding the result. Your partner peeks under the cup, and tells you, truthfully, "At least one of the dice is a 2."

What is the probability that both dice are showing a 2?

_______________

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Quote trimmed.

The only answer to the question as posed is: do not have enough information to determine the chance of 2 twos.

Quote:DieterI am seeing a range of answers that might not be wrong.

I think that "peeks under the cup" does not necessarily imply "observes both dice" or "randomly selects an observed value to report".

There is a difference between "The house is green" and "The house appears green on this side."

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The problem statement says that your partner is reporting truthfully, although it does not state that the partner is reporting accurately. But this line of inquiry (the unreliable reporter) does not allow one to conjure up an answer to the problem without making assumptions that are wholly unsupported by the statement of the problem.

All of this discussion is about whether the so-called truthful statement "at least one of the two dice is a 2" can be interpreted to mean something else because of the manner in which the information is obtained. You all are asking "If a partner does has seen both dice, why doesn't the partner report on both dice?" "Why is the information limited in such a bizarre way?" "Why would my partner only report on 2s and not report on the outcome of both dice? Why did he select 2's?"

My response is that this was presented as a logic problem, challenging you to understand the implication of incomplete information about the outcome of rolling two dice. If you don't buy the premise of the problem statement then you are free to ignore the problem. But all you are saying is that "I believe my partner wouldn't respond that way after looking at the dice" and complaining that Wizard didn't include statements about the peeker being a perfect logician or a Monty Hall-style game host. Meh. Get over it. We all have burned too many electrons on this.