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), List the 23 other possible outcomes, {(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, l), (4, 1, 2, 3), (4, 1, 3, 2), (4, 2, 1, 3), (4, 2, 3, 1), (4, 3, 1, 2), (4, 3, 2, 1)} {(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)} {(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)} {(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)} {(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, l, 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)} which outcomes are contained in the event that exactly two of the books are returned to their correct owners? {(1, 2, 4, 3), (1, 4, 3, 2), (1, 3, 2, 4), (2, 1, 3, 4), (3, 2, 1, 4)} {(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)} {(1, 2, 4, 3), (1, 4, 3, 2), (2, 1, 3, 4), (3, 2, 1, 4), (4, 2, 3, 1)} {(1, 2, 4, 3), (1, 4, 3, 2), (1, 3, 2, 4), (2, 1, 3, 4), (3, 2, 1, 4), (4, 2, 3, 1)} {(l, 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? ____ What is the probability that exactly one of the four students receives his or her own book? (Give the answer to four decimals places, ) _____ What is the probability that exactly three receive their own books? (Give the answer to four decimals places.)

Explanation / Answer

Solution:-

a) 23 other possible outcomes are:-

(1,2,3,4), (1,2,4,3), (1,3,2,4), (1,3,4,2), (1,4,3,2), (1,4,2,3)
(2,1,3,4), (2,1,4,3), (2,3,1,4), (2,3,4,1), (2,4,3,1), (2,4,1,3)
(3,2,1,4), (3,2,4,1), (3,1,2,4), (3,1,4,2), (3,4,1,2), (3,4,2,1)
(4,2,3,1), (4,2,1,3), (4,3,2,1), (4,3,1,2), (4,1,3,2), (4,1,2,3)

b) Probability of this event = 0.25

The outcomes when two of the books are returned to the correct owners are:-
(1,2,4,3), (1,3,2,4), (1,4,3,2), (2,1,3,4), (3,2,1,4), (4,2,3,1)

Total number of such outcomes = 6

Probability of this event = 6/24 = 0.25

c) The probability that exactly one of the four student recieves his or her own book = 0.3333

Different outcomes = (1,3,4,2), (1,4,2,3), (2,3,1,4), (2,4,3,1), (3,2,4,1), (3,1,2,4), (4,2,1,3), (4,1,3,2)

Number of outcomes that exactly one of the four student recieves his or her own book = 8

The probability that exactly one of the four student recieves his or her own book = 8/24 = 0.3333

d) The probability that exactly three of the four student recieves his or her own book = 0

If three recieves their own book than 4th one would also recieve his or her book so there are no outcomes when exactly three recieves their own books.

Number of outcomes that exactly three of the four student recieves his or her own book = 0

The probability that exactly three of the four student recieves his or her own book =0/24 = 0

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