# Probability question

1. Sep 17, 2007

### travis0868

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

1. The problem statement, all variables and given/known data
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.

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

3. 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.

2. Sep 18, 2007

### 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.

3. Sep 19, 2007

### travis0868

Thanks a lot, EnumaElish!

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