Banach's matchbox problem

Banach's match problem is a classic problem in probability attributed to Stefan Banach.

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 that the box picked is empty. If it is assumed that each of the matchboxes originally contained $N$ matches, what is the probability that there are exactly $k$ matches in the other box?

Solution

Let $E$ denote the event that the man discovers the matchbox in his right pocket is empty and there are $k$ matches in the matchbox in his left pocket. This event occurs only if the $(N+1)$st choice of the matchbox in his right pocket is made at the $(N+1+N-k)$th trial.

Hence $E$ is a random variable with the negative binomial distribution, with parameters

$p=1/2,\ r=N+1,\ n=2N-k+1,$ mmmmm

and so

$P(E)={\binom {2N-k}{N}}\left({\frac {1}{2}}\right)^{2N-k+1}.$

Since it is equally likely that the matchbox found to be empty is in the left pocket, the desired probability is

$2\cdot P(E)={\binom {2N-k}{N}}\left({\frac {1}{2}}\right)^{2N-k}.$

References

• Ross, Sheldon (2006). A First Course in Probability (Seventh edition ed.). pp. 176–177. ISBN 0131856626. {{cite book}}: |edition= has extra text (help)

External links

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