After all students have left the classroom, a statistics professor notices that

Question

After all students have left the classroom, a statistics professor notices that four copies of the text were left under desks. At the beginning of the next lecture, the professor distributes the four books at random to the four students (1, 2, 3, and 4) who claim to have left books. One possible outcome is that 1 receives 2's book, 2 receives 4's book, 3 receives his or her own book, and 4 receives 1's book. This outcome can be abbreviated (2, 4, 3, 1).

(a) List the 23 other possible outcomes.

a) { (1, 2, 3, 4), (1, 2, 4, 3), (1, 3, 4, 2), (1, 4, 3, 2), (1, 4, 2, 3), (2, 1, 3, 4), (2, 3, 4, 1), (2, 3, 1, 4), (2, 1, 4, 3), (2, 4, 1, 3), (3, 1, 2, 4), (3, 1, 4, 2), (3, 4, 1, 2), (3, 4, 2, 1), (3, 2, 1, 4), (3, 2, 4, 1), (4, 1, 2, 3), (4, 1, 3, 2), (4, 2, 1, 3), (4, 2, 3, 1), (4, 3, 1, 2), (4, 3, 2, 1) }
b) { (1, 2, 3, 4), (1, 2, 4, 3), (1, 3, 4, 2), (1, 4, 3, 2), (1, 4, 2, 3), (1, 3, 2, 4), (2, 1, 3, 4), (2, 4, 3, 1), (2, 3, 4, 1), (2, 3, 1, 4), (2, 1, 4, 3), (2, 4, 1, 3), (3, 1, 4, 2), (3, 4, 1, 2), (3, 4, 2, 1), (3, 2, 1, 4), (3, 2, 4, 1), (4, 1, 2, 3), (4, 1, 3, 2), (4, 2, 1, 3), (4, 2, 3, 1), (4, 3, 1, 2), (4, 3, 2, 1) }
c) { (1, 2, 3, 4), (1, 2, 4, 3), (1, 3, 4, 2), (1, 4, 3, 2), (1, 4, 2, 3), (1, 3, 2, 4), (2, 1, 3, 4), (2, 4, 3, 1), (2, 3, 4, 1), (2, 3, 1, 4), (2, 1, 4, 3), (2, 4, 1, 3), (3, 1, 2, 4), (3, 1, 4, 2), (3, 4, 1, 2), (3, 4, 2, 1), (3, 2, 1, 4), (3, 2, 4, 1), (4, 1, 2, 3), (4, 1, 3, 2), (4, 2, 1, 3), (4, 2, 3, 1), (4, 3, 1, 2), (4, 3, 2, 1) }
d) { (1, 2, 3, 4), (1, 2, 4, 3), (1, 3, 4, 2), (1, 4, 3, 2), (1, 4, 2, 3), (1, 3, 2, 4), (2, 1, 3, 4), (2, 3, 4, 1), (2, 3, 1, 4), (2, 1, 4, 3), (2, 4, 1, 3), (3, 1, 2, 4), (3, 1, 4, 2), (3, 4, 1, 2), (3, 4, 2, 1), (3, 2, 1, 4), (3, 2, 4, 1), (4, 1, 2, 3), (4, 1, 3, 2), (4, 2, 1, 3), (4, 2, 3, 1), (4, 3, 1, 2), (4, 3, 2, 1) }
e) { (1, 2, 3, 4), (1, 2, 4, 3), (1, 3, 4, 2), (1, 4, 3, 2), (1, 4, 2, 3), (1, 3, 2, 4), (2, 1, 3, 4), (2, 3, 4, 1), (2, 3, 1, 4), (2, 1, 4, 3), (2, 4, 1, 3), (3, 1, 2, 4), (3, 1, 4, 2), (3, 4, 1, 2), (3, 4, 2, 1), (3, 2, 1, 4), (3, 2, 4, 1), (4, 1, 2, 3), (4, 2, 1, 3), (4, 2, 3, 1), (4, 3, 1, 2), (4, 3, 2, 1) }

(b) Which outcomes are contained in the event that exactly two of the books are returned to their correct owners?
a){ (1, 2, 4 ,3), (1, 4, 3, 2), (1, 3, 2, 4), (2, 1 ,3, 4), (3, 2, 1, 4) }

b) { (1, 2, 4 ,3), (1, 2, 3, 4), (1, 4, 3, 2), (1, 3, 2, 4), (2, 1 ,3, 4), (3, 2, 1, 4), (4, 2, 3, 1) }    

c) { (1, 2, 4 ,3), (1, 4, 3, 2), (2, 1 ,3, 4), (3, 2, 1, 4), (4, 2, 3, 1) }

d) { (1, 2, 4 ,3), (1, 4, 3, 2), (1, 3, 2, 4), (2, 1 ,3, 4), (3, 2, 1, 4), (4, 2, 3, 1) }

e) { (1, 2, 4 ,3), (1, 2, 3, 4), (1, 3, 2, 4), (2, 1 ,3, 4), (3, 2, 1, 4), (4, 2, 3, 1) }

Assuming equally likely outcomes, what is the probability of this event?


(c) What is the probability that at least one of the four students receives his or her own book? (Give the answer to four decimals places.)


(d) What is the probability that exactly three receive their own books? (Give the answer to four decimals places.)


(e) What is the probability that at least two of the four students receive their own books? (Give the answer to four decimals places.)

Explanation / Answer

Solutions :

(a) The 23 other possible outcomes are :

1,2,3,4
1,2,4,3
1,3,2,4
1,3,4,2
1,4,2,3
1,4,3,2
2,1,3,4
2,1,4,3
2,3,1,4
2,3,4,1
2,4,1,3
3,1,2,4
3,1,4,2
3,2,1,4
3,2,4,1
3,4,1,2
3,4,2,1
4,1,2,3
4,1,3,2
4,2,1,3
4,2,3,1
4,3,1,2
4,3,2,1

(b) Answer :   d) { (1, 2, 4 ,3), (1, 4, 3, 2), (1, 3, 2, 4), (2, 1 ,3, 4), (3, 2, 1, 4), (4, 2, 3, 1) }

The probability of this event = 6/24 = 1/4

(c) The probability that at least one of the four students receive their own books is 15/24 .

(d) The probability that exactly three receive their own books is 0 .

(e) The probability that at least two of the four students receive their own books is 7/24 .

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