A company classifies its customers into k classes according to the frequency by which they place their orders. The probability that a customer of class j makes no orders in a period of one month is (n − j)∕n, for j = 1, 2,…, k, where 1 ≤ k < n. The proportion of customers who are classified into class j is pj, so that p1 + p2 + · · · + pk = 1.

If a certain customer of the company has placed at least one order during the last month, find the probability that he belongs to class j, for j = 1, 2,…, k.

Application: Consider a motor insurance company which classifies car drivers into k = 5 classes. Suppose the proportion of insured car drivers that fall into class j is pj = j∕15, for j = 1, 2,…, 5, and assume further that a driver that belongs to class j has a probability 1 − j∕100 not to make a claim in a given time period. Find the probability that a driver who makes at least one claim during that period belongs to class j, for j = 1, 2,…, 5.