Three letters are dictated to three persons and an envelope is addressed to each of them, the letters are inserted into the envelopes at random so that each envelope contains exactly one letter. Find the probability that at least one letter is in its proper envelope.

Let L₁, L₂, L₃ be three letters and E₁, E₂, E₃ be their corresponding envelopes respectively.

There are 6 ways of inserting 3 letters in envelopes. These are as follows:

L₁E₁, L₂E₃, L₃E₂
L₂E₂, L₁E₃, L₃E₁
L₃E₃, L₁E₂, L₂E₁
L₁E₁, L₂E₂, L₃E₃
L₁E₂, L₂E₃, L₃E₁
L₁E₃, L₂E₁, L₃E₂

There are 4 ways in which at least one letter is inserted in a proper envelope.

Thus, the required probability = ⁴/₆ = ²/₃.