# 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.