Question

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

Answer 1

Let the envelope be denoted by A, B, C

and the corresponding letter is a, b, c.

The letters are inserted into the envelopes at random, and each envelope contains exactly one letter.

Possible combinations can be

Aa, Bb, Cc, Aa, Bc, Cb, Ab, Ba, Cc, Ab, Bc, Ca, Ac, Bb, Ca, Ac, Ba, Cb

So total number of possible cases =$n\left(S\right)=6$

Let A be event that at least one letter is in its proper envelope,

A={(Aa, Bb, Cc), (Aa, Bc, Cb), (Ac, Bb, Ca), (Ab, Ba, Cc)}

$n\left(A\right)=4$

So,

The probability that at least one letter is in its proper envelope $P\left(A\right)=\frac{n\left(A\right)}{n\left(S\right)}=\frac{4}{6}=\frac{2}{3}$