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), (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) }

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, 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) }

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, 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) }

d) { (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) }

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, 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) }

b) 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 at least two of the four students receive their own books? (Give the answer to four decimals places.)

Answer 1

(a)

{(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)} is the correct answer

(b) The probability that at least one of the four students receives his or her own book = 15/24 = 0.6250

(c) The probability that at least two of the four students receive their own books = 7/24 = 0.2917