What is the probability that at least one letter is in the correct envelope? A secretary types three letters and the three corresponding envelopes. In a hurry, he places at random one letter in each envelope. What is the probability that at least one letter is in the correct envelope? My effort: There are three choices of an envelope for the first letter, then there are two choices of an envelope for the second letter, and finally there is one choice of an envelope for the third letter, thereby making a total of six possible choices. Now we should first try to compute the probability that none of the three letters is in the correct envelope. But how to compute this probability? Once we know this probability, then our required probability is one minus this probability.

For $3$ objects this is fairly trivial, as you can simply inspect each combination:

* $123:$ all letters are in the correct envelope
* $132:$ letter #$1$ is in the correct envelope
* $213:$ letter #$3$ is in the correct envelope
* $231:$ no letter is in the correct envelope
* $312:$ no letter is in the correct envelope
* $321:$ letter #$2$ is in the correct envelope



In $4$ out of $6$ combinations, there is at least one letter in the correct envelope.

Hence the probability of having at least one letter in the correct envelope is $\dfrac{4}{6}$.