At a craps table, I have a wager with the house that the shooter will make 4 'the hard way' i.e. {2,2}, at odds of 7:1 (with such bets, it doesn't matter what the shooter himself wants to score).
The shooter, an old man, tosses the dice. Unfortunately he is shortsighted, and can't see what he rolled, so he asks me, 'Can you see a 2 down there?'
To tell the truth, I am a bit nearsighted myself, and a touch hard of hearing, so I answer 'Could you please repeat that?'
The old man says, ''What do you see down there? ' and I duly reply, 'I can see a 2, but I'm not sure what the other is'
What is the probability that I won my wager?
What would have been the probability, had I answered the shooter's original question?
Consider every possible roll of two dice, and list only those where at least one 2 occurred:
2,1
2,2
2,3
2,4
2,5
2,6
1,2
3,2
4,2
5,2
6,2
All of these satisfy a "yes" answer to the first, unheard question (second question of the puzzle). The probability is 1/11 you won the wager.
But you didn't hear this. Based on the second question (subject of the first question of the puzzle), consider two repetitions of each of the possibilities listed above:
Reporting first die:
2,1 2
2,2 2
2,3 2
2,4 2
2,5 2
2,6 2
1,2 1
3,2 3
4,2 4
5,2 5
6,2 6
Same set, but this time happening to report the second die:
2,1 1
2,2 2
2,3 3
2,4 4
2,5 5
2,6 6
1,2 2
3,2 2
4,2 2
5,2 2
6,2 2
There are 12 cases where you report seeing a 2. In two of these cases you won your bet. The probability is 2/12 = 1/6 that you won.
Corrected as per tomarken's later comment.
Edited on April 20, 2016, 2:52 pm

Posted by Charlie
on 20160420 10:36:25 