Solution
First consider the case where the matchbox in his right pocket has an unlimited number of matches and let M be the number of matches removed from this one before the left one is found to be empty. When the left pocket is found to be empty, the man has chosen that pocket (N+1) times. Then M is the number of successes before (N+1) failures in Bernoulli trials with p=1/2, which has the negative binomial distribution and thus
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Returning to the original problem, we see that the probability that the left pocket is found to be empty first is which equals 1/2 because both are equally likely. We see that the number K of matches remaining in the other pocket is
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