You have 10 envelopes labeled 1 through 10 and 20 letters la

You have 10 envelopes, labeled 1 through 10, and 20 letters, labeled 1 through 10 (with exactly 2 letters having the same label). You randomly put two letters in each envelope. What is the expected number of envelopes that contain at least one of the two letters with the same number as the envelope?

Solution

let us first find out E[X] = Number of envelopes NOT having any matching letter.

Let us define an indicator variable Xk = 1 if the kth envelope doesn\'t have a matching letter
and 0 otherwise

Therefore, we have,
E( number of envelopes with no matching letter) = E?(Xk) = ?E(Xk)

The expectation of an indicator random variable is just the probability of the event it indicates,
so E(Xk) = Probability that the kth envelope doesn\'t have a matching letter = (9/10)