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Suppose you have two fair dice. You throw the dice in a box. You ask a friend to look in the box. Your friend then tells you that one of the dice is a 4. Given this information, what is the probability that the other dice is a 6?

I found this problem on youtube and it baffles me that the answer given is not $1/6$. I would love to see how you think about this problem and what your conclusion is.

JustANoob
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    To emphasize what is said in the linked post... it matters very much how it was decided to tell us that there was a four. A less ambiguous phrasing of the problem would have been "You have two fair dice, you throw them in a box and ask your friend to look inside. You then ask your friend 'is there at least one 4 showing?' to which your friend replies yes. Given this information, what is the probability that one of the dice shows a 6?" – JMoravitz Feb 09 '22 at 17:17
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    The punchline is that had we imagined the scenario that the dice were colored, one as red and one as blue, the probabilities don't change but we can more accurately keep track of the outcomes. Listing the red first and blue second, among the possible outcomes we might have been in given that there was a 4 would be $(1,4),(2,4),(3,4),(4,4),(5,4),(6,4),(4,1),(4,2),(4,3),(4,5),(4,6)$ for a total of eleven equally likely outcomes, two of which have a six showing, for a probability of $\frac{2}{11}$ – JMoravitz Feb 09 '22 at 17:19
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    A completely different problem would be where your friend was not prompted by a specific question from you and merely looked in the box and saw some numbers and chose one of the dice at random and said the number he saw. That is not the same problem and so has a different answer. A different problem still is where your friend prefers to always state the highest number he saw, or if he always prefers to say the lowest number he saw, or if he follows some other strategy... all of which giving potentially different answers to what the probability is. – JMoravitz Feb 09 '22 at 17:22
  • Thanks. I'm convinced now that 2/11 is the right answer. I just find it extremely non intuitive. – JustANoob Feb 09 '22 at 17:36

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