Probability Question: Parzen's Modern Probability Theory, Ch.2 Ex.6.1

[travis0868]

This question is from Parzen (Modern Probability Theory), chapter 2, exercise 6.1

Homework Statement

Suppose that we have M urns, numbered 1 to M and M balls, numbered 1 to M. Let the balls be inserted randomly in the urns, with one ball in each urn. If a ball is put into the urn bearing the same number as the ball, a match is said to have occurred. Show that for j = 1,...,M the conditional probability of a match in the jth urn, given that there are m matches is m/M.

Homework Equations

Probability that there are exactly m matches in M urns is:
(1/m!)* \Sigma^{M-m}_{k=0}(-1)^k * (1/k!)

The Attempt at a Solution

The answer doesn't make sense. Suppose that j = M. The probability that there will be a match in the Mth urn given that there are M-1 matches is 1. Not (M-1)/M.

 

[EnumaElish]

I think m is the total number of matches. You may think of it as a referee looking into each urn, then reporting m matches, without disclosing which urns had matches.

 

[travis0868]

Thanks a lot, EnumaElish!