Two Matchboxes

Question

A smoker carries a matchbox in each pocket, each starting with 3 matches. Every time he needs a match he picks a pocket uniformly at random. At some point he reaches into a pocket and finds the box empty. What is the probability that at that moment the other box is also empty?

Accepted Answer

Track reaches into pockets as fair coin flips. The other box is empty at discovery iff the first 6 reaches split exactly 3–3 between the pockets: that uses up both boxes, and then the 7th reach, whichever pocket he picks, discovers an empty box. The 7th reach needs no probability at all; discovery is certain once both are empty. P = P(textfirst 6 reaches are 3–3) = fracbinom632^6 = frac2064 = frac516 = 0.3125. The trap is mistaking "uses the last match" for "discovers the box empty." Discovery is a later reach into the dead pocket; conflating the two events produces wrong answers for the general case below. General distribution (Banach's matchbox problem) With N matches per box, the other box holds exactly k matches at discovery with probability P(K = k) = binom2N - kN left(frac12right)^2N - k, derived by the same picture: the first 2N - k reaches contain N into the doomed pocket and N - k into the other, the next reach hits the doomed pocket again (factor frac12), and a factor 2 chooses which pocket is doomed. At k = 0 the two correction factors cancel, recovering the central binomial binom2NN2^-2N. For N = 50 that is approx frac1sqrt50pi approx 0.08, the same central-binomial asy

Two Matchboxes

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