Question

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 was put in which envelope. What is the probability that exactly one of the students received his or her own card?

Answer 1

Hint : To solve this question, we will start with finding the favourable outcomes of exactly one of the students receiving his or her own card, then will find the total outcomes of arranging three envelopes and three report cards. Then after applying the probability formula, we will get our required answer.

Complete step-by-step answer :
We have been given that a clerk was asked to mail three report cards to three students. It is given that he addresses three envelopes but unfortunately paid no attention to which report card be put in which envelope. We need to find the probability that exactly one of the students received his or her own card.
Let there be three envelopes A, B and C, and three report cards a, b and c.
Now let us see the cases where exactly one of the students received his or her own card.
Case $1.$ Where A got a, B might get c, and C might get b.
Case $2.$ Where B got b, A might get c, and C might get a.
Case $3.$ Where C got c, A might get b, and B might get a.
So, three are three possible cases where exactly one of the students received his or her own card.
i.e., Number of favourable outcomes where exactly one of the students received his or her own card $= {\text{ }}3$
Since, there are three envelopes and three report cards, therefore, number of total ways of arranging three report cards into three envelopes is $3!.$
So, the number of total outcomes of arranging three envelopes and three report cards $= {\text{ }}3!$
We know that, Probability $= \dfrac{{{\text{favourable outcomes}}}}{{{\text{total outcomes}}}}$
On putting the value in the above formula, we get
Probability of exactly one of the students received his or her own card $= \dfrac{3}{{3!}}$
$\begin{gathered} = \dfrac{3}{{3 \times 2}} \\\ = \dfrac{1}{2} \\\ \end{gathered}$

Thus, the probability of getting exactly one of the students received his or her own card is $\dfrac{1}{2}.$

Note : Students should carefully consider the cases here. We have taken three possible cases, in all the cases, only one student got his or her own report card, so that is the favourable outcomes here. And total outcomes will be the number of total ways of arranging three report cards into three envelopes.