A class of 10 students hang up their coats when they arrive at school. Just before recess, the teacher hands one coat selected at random to each child. What is the expected number of children who get his or her own coat?

Question

Mathematics

Bryan Hicks

4 June, 16:42

A class of 10 students hang up their coats when they arrive at school. Just before recess, the teacher hands one coat selected at random to each child. What is the expected number of children who get his or her own coat?

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Answers (1)

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Alyssa · Answer 1

0.1274

Step-by-step explanation:

Let X be the random variable that measures the number of children who get their own coat.

Then, the expected value of X is

E[X] = 1P (X=1) + 2P (X=2) + 3P (X=3) + ... + 10P (X=10)

The probability that a child gets her or his coat is

P (X=1) = 1/10

To compute the probability that 2 children get their own coat, we notice that there are 10! possible permutations of coats. The two children can get their coat in only one way, the other 8 coats can be arranged in 8! different positions, so the probability that 2 children get their own coat is

P (X=2) = 8!/10! = 1 / (10*9) and

2P (X=2) = 2 / (10*9)

Similarly, we can see that the probability that 3 children get their own coat is

P (X=3) = 7!/10! = 1 / (10*9*8) and

3P (X=3) = 3 / (10*9*8*7)

and the expected value of X would be

E[X] = 1/10 + 2 / (10*9) + 3 / (10*9*8) + ... + 10/10! = 0.1274