(Banach's Matchbox) A man carries in each of his two front pockets a box of matches originally containing [math]N[/math] matches. Whenever he needs a match, he chooses a pocket at random and removes one from that box. One day he reaches into a pocket and finds the box empty.

• Let [math]p_r[/math] denote the probability that the other pocket contains [math]r[/math] matches. Define a sequence of counter random variables as follows: Let [math]X_i = 1[/math] if the [math]i[/math]th draw is from the left pocket, and 0 if it is from the right pocket. Interpret [math]p_r[/math] in terms of [math]S_n = X_1 + X_2 +\cdots+ X_n[/math]. Find a binomial expression for [math]p_r[/math].
• Write a computer program to compute the [math]p_r[/math], as well as the probability that the other pocket contains at least [math]r[/math] matches, for [math]N = 100[/math] and [math]r[/math] from 0 to 50.
• Show that [math](N - r)p_r = (1/2)(2N + 1)p_{r + 1} - (1/2)(r + 1)p_{r + 1}[/math]\ .
• Evaluate [math]\sum_r p_r[/math].
• Use (c) and (d) to determine the expectation [math]E[/math] of the distribution [math]\{p_r\}[/math].
• Use Stirling's formula to obtain an approximation for [math]E[/math]. How many matches must each box contain to ensure a value of about 13 for the expectation [math]E[/math]? (Take [math]\pi = 22/7[/math].)

Notes