Banach's matchbox problem

Banach's match problem is a classic problem in probability attributed to Stefan Banach. Feller [1] says that the problem was inspired by a humorous reference to Banach's smoking habit in a speech honouring him by Hugo Steinhaus, but that it was not Banach who set the problem or provided an answer.

Suppose a mathematician carries two matchboxes at all times: one in his left pocket and one in his right. Each time he needs a match, he is equally likely to take it from either pocket. Suppose he reaches into his pocket and discovers for the first time that the box picked is empty. If it is assumed that each of the matchboxes originally contained matches, what is the probability that there are exactly matches in the other box?

Solution

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Without loss of generality consider the case where the matchbox in his right pocket has an unlimited number of matches and let   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   times. Then   is the number of successes before   failures in Bernoulli trials with  , which has the negative binomial distribution and thus

 
Distribution of probability of having k matches remaining in the other pocket.
 .

Returning to the original problem, we see that the probability that the left pocket is found to be empty first is   which equals   because both are equally likely. We see that the number   of matches remaining in the other pocket is

 .

The expectation of the distribution is approximately  . (This is shown using Stirling's approximation.[2]) So starting with boxes with   matches, the expected number of matches in the second box is  .

See also

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References

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  1. ^ Feller, William, An Introduction to Probability Theory And Its Applications, Third Edition, Wiley, 1968, Chapter VI, section 8
  2. ^ Feller, page 238.
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