 Then formula_3 is the number of successes before formula_4 failures in Bernoulli trials with formula_7, which has the negative binomial distribution and thus

Returning to the original problem, we see that the probability that the left pocket is found to be empty first is formula_9 which equals formula_10 because both are equally likely.We see that the number formula_11 of matches remaining in the other pocket is

The expectation of the distribution is approximately formula_13.