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

Solution:

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 envelops as follows:

L₁ E₁, L₂ E₃, L₃ E₂

L₁ E₁, L₂ E₂, L₃ E₃

L₁ E₂, L₂ E₁, L₃ E₃

L₁ E₂, L₂ E₃, L₃ E₁

L₁ E₃, L₂ E₂, L₃ E₁

L₁ E₃, L₂ E₁, L₃ E₂

We can see that there are 4 ways (the ways with L₁ E₁ (or) L₂ E₂ (or) L₃ E₃)  in which at least one letter is inserted in a proper envelope.

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

NCERT Solutions Class 11 Maths Chapter 16 Exercise ME Question 6

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.

Summary:

The probability that at least one letter is in its proper envelope is 2/3