Usually I hate brain teasers. And this case is no exception. Except this one keeps preoccupying me, so I'm going to make you suffer through it too.
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13 Yakuza subordinates go to the Yakuza boss's house. Out of respect, they all take off their shoes on the way in. After all of them have gathered, the boss walks over to the collection of shoes, sniffs, makes an unpleasant face, and picks up one (matching) pair of shoes. "These shoes are dirty and disgusting! Who dishonors me with such shoes?!" No one raises their hand. Infuriated, the Yakuza boss immediately shoots one of the subordinates utterly at random, takes the body and the offending shoes, and throws them into the snow.
Since the meeting cannot continue after such a violent episode, the boss dismisses the remaining subordinates. Each leaves one by one. If the subordinate's shoes are in the pile, he takes them and leaves. If not, he takes a random (matching) pair of shoes and leaves.
The question: What is the probability that the last man to leave takes his own shoes?
Update: I'll give you a small hint. I think it's probably easier to intuit the answer and then prove it than try to derive the answer (as I tried to).
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