Question

Three students accidentally leave copies of the textbook in their classroom after class. At the beginning of the next lecture, the professor distributes the three books in a completely random fashion to each of the three students (1, 2, and 3) who left their books. One possible outcome is that 1 receives 2’s book, 2 receives 1’s book, 3 receives his or her own book. This outcome can be abbreviated as (2, 1, 3).

a. List all other possible outcomes.

b. Let Y denote the number of students who receive their own book. Determine the PMF of Y.

Answer 1

a.

Possible outcomes :

1. (1,2,3)

2. (1,3,2)

3. (2,1,3)

4. (2,3,1)

5. (3,1,2)

6. (3,2,1)

b.

Y : Number of students who receive their own book .

For each of the above possible outcomes; we determine the value of Y:

1. (1,2,3) - Y = 3 ; Student 1 gets his own Book 1 ; Student 2 gets his own book 2 : Student 3 gets s his own the book 3 ; Hence Y=3

2. (1,3,2) - Y = 1 ; Only students 1 gets his own Book1

3. (2,1,3) - Y = 1 ; Only students 3 gets his own the Book3

4. (2,3,1) - Y = 0 ; Nobody gets their own book

5. (3,1,2) - Y = 0 ; Nobody gets their own book

6. (3,2,1) - Y = 1 ; Only student 2 gets s his own Book2

Probability of the each of the above outcomes = 1/6

Therefore,

P(Y=0) = P(2,3,1) + P(3,1,2) = 1/6 + 1/6 = 2/6 =1/3

P(Y=1) = P(1,3,2) + P(2,1,3) + P(3,2,1) = 1/6 + 1/6 + 1/6 = 3/6 =1/2

P(Y=3) = P(1,2,3) = 1/6

PMF of Y :

Y	P(Y)	P(Y)
0	1/3	0.333333333
1	1/2	0.5
3	1/6	0.166666667