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).

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

Explanation / Answer

Consider the outcome ( 2,4,3,1)

Which gives 1 receives 2's book , 2 receives 4's book , 3 receives their own book and 4 receives 1's book.

Total number of possible outcomes

n = 4! = 24

( Consider 4 boxes

The first box can be filled in 4 ways i.e .the first student get any of the 4 books.

since repetition is not allowed ( Every student get a single book)

The second box can be filled in 3 ways i.e the second tudent get any of the 3 books.

The third box can be filled in 2 way . i.e the third student get any of the 2 books.

The fourth box can be filled in 1 way. the last student get the remaining book.

Hence by multiplication principle

Total number of ways to distribute 4 books to 4 student is

= 4*3*2*1 = 4! = 24 )

The outcome ( 1,2,3,4) gives exactly four receive their own book.

i.e only one possible outcome gives exactly four receives their own book out of 24 outcomes.

Hence

P( Exactly four receives their own books ) = 1/24 = 0.0416

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