# Average Number of Matches

## Problem

The following are two versions of the matching problem:

(a) From a shuffled deck, cards are laid out on a table one at a time, face up from left to right, and then another deck is laid out so that each of its cards is beneath a card of the first deck. What is the average number of matches on the card above and the card below in repetitions of this experiment?

(b) A typist types letters and envelopes to n different persons. The letters are randomly put into the envelopes. On the average, how may letters are put into their own envelopes?

## Part (a) Solution

We will apply the property of Linearity of Expectation. Let $$X = X_1 + X_2 + \ldots + X_{52}$$ be the total number of card matches, where each $$X_i$$ represents the number of matches (either $$0$$ or $$1$$) for the $$i$$th card. We know $$E[X_i] = \frac{1}{52}$$. So, we can easily calculate the solution as follows:

$\begin{split} E[X] &= E[X_1 + X_2 + \ldots + X_{52}] \\ &= E[X_1] + E[X_2] + \ldots + E[X_{52}] \\ &= 52\left(\frac{1}{52}\right) \\ &= 1 \end{split}$

In conclusion, we can expect $$1$$ card to match on average.

## Part (b) Solution

The same logic from Part (a) applies here, giving a solution of $$n\left(\frac{1}{n}\right) = 1$$ matching letters on average.