Probability question: names in hats

[robvalue]

I had forgotten about it for a long time, I just remembered it recently. It was about.... 10 years ago I heard it!

OK, I've drawn a series of 5 diagrams. The first diagram starts off with the full probability tree, if players could just pick any number and stick with it. It gives the expected result, that each of the possible results has a probability of 1/6. (1/3 * 1/2 * 1).

Then I've gone through and removed, bit by bit, the possibilities that the rules forbid. I've removed the whole branch that starts with player 1 picking 1, because he cannot. Then I've removed the branch for player 2 picking 2, because again he cannot.

Because several options are being removed, this causes the probability of the remaining options to increase. They must now total to 1, from each branch. So 3 options of 1/3 each drop down to 2 with 1/2; and 2 options of 1/2 each will drop down to 1 option with probability 1 (certainty). And it doesn't alter the probabilities of getting to those points in the first place.

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