A clerk was asked to mail three report cards to three students. He addresses three envelopes but unfortunately paid no attention to which report card be put in which envelope. What is the probability that exactly one of the students received his or her own card?

Explanation

Let A be the first report card in correct envelope.

Let B be the second report card in correct envelope.

Let {C} be the third report card in correct envelope.

So,` P(A)=P(B)=P(C)=frac{1}{3}`

Required probability `=P[(A cap B^{prime} cap C^{prime}) cup(A^{prime} cap B cap C^{prime}) cup(A^{prime} cap B^{prime} cap C)]`
`
=P[(A cap B^{prime} cap C^{prime})+(A^{prime} cap B cap C^{prime})+(A^{prime} cap B^{prime} cap C^{prime})]`
`
=P(A) P(frac{B^{prime}}{A}) P(frac{C^{prime}}{A cap B^{prime}})+P(B) P(frac{A^{prime}}{B}) `
`P(frac{C^{prime}}{B cap A^{prime}})+
P(C) P(frac{A^{prime}}{C}) P(frac{B^{prime}}{A cap A^{prime}})`
`
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