Banach’s matchbox problem was posed by mathematician Hugo Steinhaus as an affectionate honor to fellow mathematician Stefan Banach, who was a heavy pipe smoker. A smoker has two matchboxes, one in each pocket. Each box has n matches in it. Whenever the smoker needs a match he reaches into a pocket at random and takes a match from the box. Suppose he reaches into a pocket and finds that the matchbox is empty. Find the probability that the box in the other pocket has exactly k matches.
(a) Let X be the number of matches in the right box when the left box is found empty. Show that X has a negative binomial distribution with parameters n + 1 and 1/2.
(b) Show that the desired probability is 2 × P(X = 2n + 1 − k).
(c) Work out the problem in detail for the case n = 1.