Two dice puzzle -- Part Trois

Quote: OnceDear

Proof that 1 in 6 is a perfectly valid answer in spite of Wizard insisting otherwise

My work is almost done

First, I restate the EXACT wording of the first post that raised this ugly mess

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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And now I stake my case completely for the potential validity of 1/6

Our calling rules are indeterminate!

I'm arguing that nowhere in the OP did it say that he was CONFIRMING twos. Wizard assumed that without good reason. He denies the possibility that he doesn't call a number at every roll, also without good reason.

Let's say he saw both dice but always called one of the dies values. Choosing randomly between the two dice seen.
That is not ruled out in the OP and is a perfectly reasonable assumption. He's not obligated to declare twos as that is not in the original premise. YES he sees both dice.

Here are all the 72 possibilities, equally weighted.

First 36 possibilities considering that he calls the first value to catch his eye, maybe being the one nearest to him

1,1 "At least one of the dice is a one!
1,2 "At least one of the dice is a one!
1,3 "At least one of the dice is a one!
1,4 "At least one of the dice is a one!
1,5 "At least one of the dice is a one!
1,6 "At least one of the dice is a one!
2,1 "At least one of the dice is a two!!
2,2 "At least one of the dice is a two!!
2,3 "At least one of the dice is a two!!
2,4 "At least one of the dice is a two!!
2,5 "At least one of the dice is a two!!
2,6 "At least one of the dice is a two!!
3,1 "At least one of the dice is a three!!
3,2 "At least one of the dice is a three!!
3,3 "At least one of the dice is a three!!
3,4 "At least one of the dice is a three!!
3,5 "At least one of the dice is a three!!
3,6 "At least one of the dice is a three!!
4,1 "At least one of the dice is a four!!
4,2 "At least one of the dice is a four!!
4,3 "At least one of the dice is a four!!
4,4 "At least one of the dice is a four!!
4,5 "At least one of the dice is a four!!
4,6 "At least one of the dice is a four!!
5,1 "At least one of the dice is a five!!
5,2 "At least one of the dice is a five!!
5,3 "At least one of the dice is a five!!
5,4 "At least one of the dice is a five!!
5,5 "At least one of the dice is a five!!
5,6 "At least one of the dice is a five!!
6,1 "At least one of the dice is a six!!
6,2 "At least one of the dice is a six!!
6,3 "At least one of the dice is a six!!
6,4 "At least one of the dice is a six!!
6,5 "At least one of the dice is a six!!
6,6 "At least one of the dice is a six!!

Now another 36 possibilities where he calls the second die to catch his eye, maybe the one furthest from him.
1,1 "At least one of the dice is a one!
1,2 "At least one of the dice is a two!
1,3 "At least one of the dice is a three!
1,4 "At least one of the dice is a four!
1,5 "At least one of the dice is a five!
1,6 "At least one of the dice is a six!
2,1 "At least one of the dice is a one!
2,2 "At least one of the dice is a two!
2,3 "At least one of the dice is a three!
2,4 "At least one of the dice is a four!
2,5 "At least one of the dice is a five!
2,6 "At least one of the dice is a six!
3,1 "At least one of the dice is a one!
3,2 "At least one of the dice is a two!
3,3 "At least one of the dice is a three!
3,4 "At least one of the dice is a four!
3,5 "At least one of the dice is a five!
3,6 "At least one of the dice is a six!
4,1 "At least one of the dice is a one!
4,2 "At least one of the dice is a two!
4,3 "At least one of the dice is a three!
4,4 "At least one of the dice is a four!
4,5 "At least one of the dice is a five!
4,6 "At least one of the dice is a six!
5,1 "At least one of the dice is a one!
5,2 "At least one of the dice is a two!
5,3 "At least one of the dice is a three!
5,4 "At least one of the dice is a four!
5,5 "At least one of the dice is a five!
5,6 "At least one of the dice is a six!
6,1 "At least one of the dice is a one!
6,2 "At least one of the dice is a two!
6,3 "At least one of the dice is a three!
6,4 "At least one of the dice is a four!
6,5 "At least one of the dice is a five!
6,6 "At least one of the dice is a six!
Of the 12 times that he calls "At least one of the dice is a two", they are both twos on two occasions.
Thus if he calls "At least one of the dice is a two" then the probability of both dice being a two is ONE in SIX
The answer 2/12 or 1/6
QED

That he peeks at both dice is irrelevant.
The calling rules MATTER and they are unstated.

bonus spoiler

Now..... If your friend had a declared hatred of using the word Two, after a nasty incident where 2 anvils fell on his 2 feet, but was honor bound to truthfully declare in the form "At least one of the dice is a X" and at one roll, he declared "At least one of the dice is a Two", what is the probability that the other dice is a two?

Answer to bonus spoiler

100% certain. Unless both dice were two, he would have called the other number: He only calls a two when he absolutely can't avoid it

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Quote: unJon

Quote: OnceDear

Proof that 1 in 6 is a perfectly valid answer in spite of Wizard insisting otherwise

My work is almost done

First, I restate the EXACT wording of the first post that raised this ugly mess

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?
link to original post

And now I stake my case completely for the potential validity of 1/6

Our calling rules are indeterminate!

I'm arguing that nowhere in the OP did it say that he was CONFIRMING twos. Wizard assumed that without good reason. He denies the possibility that he doesn't call a number at every roll, also without good reason.

Let's say he saw both dice but always called one of the dies values. Choosing randomly between the two dice seen.
That is not ruled out in the OP and is a perfectly reasonable assumption. He's not obligated to declare twos as that is not in the original premise. YES he sees both dice.

Here are all the 72 possibilities, equally weighted.

First 36 possibilities considering that he calls the first value to catch his eye, maybe being the one nearest to him

1,1 "At least one of the dice is a one!
1,2 "At least one of the dice is a one!
1,3 "At least one of the dice is a one!
1,4 "At least one of the dice is a one!
1,5 "At least one of the dice is a one!
1,6 "At least one of the dice is a one!
2,1 "At least one of the dice is a two!!
2,2 "At least one of the dice is a two!!
2,3 "At least one of the dice is a two!!
2,4 "At least one of the dice is a two!!
2,5 "At least one of the dice is a two!!
2,6 "At least one of the dice is a two!!
3,1 "At least one of the dice is a three!!
3,2 "At least one of the dice is a three!!
3,3 "At least one of the dice is a three!!
3,4 "At least one of the dice is a three!!
3,5 "At least one of the dice is a three!!
3,6 "At least one of the dice is a three!!
4,1 "At least one of the dice is a four!!
4,2 "At least one of the dice is a four!!
4,3 "At least one of the dice is a four!!
4,4 "At least one of the dice is a four!!
4,5 "At least one of the dice is a four!!
4,6 "At least one of the dice is a four!!
5,1 "At least one of the dice is a five!!
5,2 "At least one of the dice is a five!!
5,3 "At least one of the dice is a five!!
5,4 "At least one of the dice is a five!!
5,5 "At least one of the dice is a five!!
5,6 "At least one of the dice is a five!!
6,1 "At least one of the dice is a six!!
6,2 "At least one of the dice is a six!!
6,3 "At least one of the dice is a six!!
6,4 "At least one of the dice is a six!!
6,5 "At least one of the dice is a six!!
6,6 "At least one of the dice is a six!!

Now another 36 possibilities where he calls the second die to catch his eye, maybe the one furthest from him.
1,1 "At least one of the dice is a one!
1,2 "At least one of the dice is a two!
1,3 "At least one of the dice is a three!
1,4 "At least one of the dice is a four!
1,5 "At least one of the dice is a five!
1,6 "At least one of the dice is a six!
2,1 "At least one of the dice is a one!
2,2 "At least one of the dice is a two!
2,3 "At least one of the dice is a three!
2,4 "At least one of the dice is a four!
2,5 "At least one of the dice is a five!
2,6 "At least one of the dice is a six!
3,1 "At least one of the dice is a one!
3,2 "At least one of the dice is a two!
3,3 "At least one of the dice is a three!
3,4 "At least one of the dice is a four!
3,5 "At least one of the dice is a five!
3,6 "At least one of the dice is a six!
4,1 "At least one of the dice is a one!
4,2 "At least one of the dice is a two!
4,3 "At least one of the dice is a three!
4,4 "At least one of the dice is a four!
4,5 "At least one of the dice is a five!
4,6 "At least one of the dice is a six!
5,1 "At least one of the dice is a one!
5,2 "At least one of the dice is a two!
5,3 "At least one of the dice is a three!
5,4 "At least one of the dice is a four!
5,5 "At least one of the dice is a five!
5,6 "At least one of the dice is a six!
6,1 "At least one of the dice is a one!
6,2 "At least one of the dice is a two!
6,3 "At least one of the dice is a three!
6,4 "At least one of the dice is a four!
6,5 "At least one of the dice is a five!
6,6 "At least one of the dice is a six!
Of the 12 times that he calls "At least one of the dice is a two", they are both twos on two occasions.
Thus if he calls "At least one of the dice is a two" then the probability of both dice being a two is ONE in SIX
The answer 2/12 or 1/6
QED

That he peeks at both dice is irrelevant.
The calling rules MATTER and they are unstated.

bonus spoiler

Now..... If your friend had a declared hatred of using the word Two, after a nasty incident where 2 anvils fell on his 2 feet, but was honor bound to truthfully declare in the form "At least one of the dice is a X" and at one roll, he declared "At least one of the dice is a Two", what is the probability that the other dice is a two?

Answer to bonus spoiler

100% certain. Unless both dice were two, he would have called the other number: He only calls a two when he absolutely can't avoid it

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You’re wasting time. 1 in anything is a valid answer to the original wording.
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Quote: Dween

I'm interested to see how people solve this one.

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Quote: OnceDear

BUT You assume that the peeker is confirming twos. Exactly the mistake Wizard makes.
The peeker is calling the value of one of the dice. We don't know which one. We don't know how he decides which one. We don''t know what he declares if no two is rolled. See my reply to Gordon.
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Quote: Dieter

Quote: OnceDear

BUT You assume that the peeker is confirming twos. Exactly the mistake Wizard makes.
The peeker is calling the value of one of the dice. We don't know which one. We don't know how he decides which one. We don''t know what he declares if no two is rolled. See my reply to Gordon.
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This is exactly what I came to as I mulled it over for the last 3 or 4 hours.

Is there something special about twos in this context?

Until I know that, I am making the assumption that a dice value is randomly revealed.

This is fun!

I am greatly enjoying the opportunity to analyze my assumptions.

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Quote: OnceDear

Quote: unJon

Quote: billryan

I appreciate the explanation and accept it, but don't fully grasp it.
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Potentially an easier way to think about it:...

So fair odds to the two dice problem are 1 / (1+10) = 1/11.
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BUT You assume that the peeker is confirming twos. Exactly the mistake Wizard makes.
The peeker is calling the value of one of the dice. We don't know which one. We don't know how he decides which one. We don''t know what he declares if no two is rolled. See my reply to Gordon.
link to original post

Quote: gordonm888

Quote: OnceDear

Quote: unJon

Quote: billryan

I appreciate the explanation and accept it, but don't fully grasp it.
link to original post

Potentially an easier way to think about it:...

So fair odds to the two dice problem are 1 / (1+10) = 1/11.
link to original post

BUT You assume that the peeker is confirming twos. Exactly the mistake Wizard makes.
The peeker is calling the value of one of the dice. We don't know which one. We don't know how he decides which one. We don''t know what he declares if no two is rolled. See my reply to Gordon.
link to original post

The peeker might have said "one of the dice is a 5." There is nothing special about a two. Then the odds would have been 1/11 that the other die is a 5.

This 2-dice puzzle challenges you to understand the additional information that has been conveyed to you and its effect on your ability to assign probabilities to the unknown outcomes. I am teaching senior-level high-school math at home and the textbooks constantly reinforce the need to understand
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

Approximately half the people posting on this thread seem to have never been taught the above approach in their education and don't seem to understand how to interpret a word problem. The problem statement isn't War and Peace, you don't need to analyze motivations of the "friend," you are simply asked to straightforwardly interpret new information from your friend (at face value.)

For some reason, some otherwise fairly intelligent people seem determined on this thread to provoke others by pretending to be obtuse. I understand that BillRyan has strengths other than math -that's okay, I'm cool with that. But for OnceDear to constantly denounce people on other threads who don't understand "Expected Value" and then to play at being obtuse about dice probabilities on this thread is very discouraging. No wonder Wizard lost his cool with Alan and the other flat-earthers.
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Quote: gordonm888

The peeker might have said "one of the dice is a 5." There is nothing special about a two. Then the odds would have been 1/11 that the other die is a 5.

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Quote: oncedear to wizard a few minutes ago

Stop speed reading. ! Stop seeing what you want to see. You're doing an [redacted]

If you only read one sentence here, read the red one.

If the dice had been a two and a four, or a four and a two, what would the peeker have said? And why?