Let $X_i$ be the indicator random variable such that:

• $1$ if the $i$th letter ends up in the $i$th envelope.
• $0$ otherwise

$E[X_i] = P(X_i = 1) = 1/n$ for any $i$

let $X$ be the number of letters that ended up in their respective envelopes.

Now, $X$= $X_1 + X_2+ \ldots +X_n$

$E[X] = E[\sum_{i=1}^{n} X_i]$

$= \sum_{i=1}^{n} E[X_i]$ (Using Linearity of Expectations)

$=\sum_{i=1}^{n} (1/n) = 1$

Therefore, we expect on average one letter to be in the correct envelope.