Expectation

Question

Suppose 100 people all toss a hat into a box and then proceed to randomly pick out a hat. What is the expected number of people to get their own hat back.

Hint: express the number of people who get their own hat as a sum of random variables whose expected value is easy to compute.

Comments

I have only one doubt here:

One way :

We know probability for every person getting right hat is 1/100 hence

Expectation = 1*(1/100) + 1*(1/100) + 1*(1/100) + ... 100 times = 1

Its the answer.

Another way :

If there are 100 person from 1,2...100

then it should be  : 1*(1/100) + 2*(1/100) + 3*(1/100) + ........ = 50.5

 

How to know which one to use

Answer is 1 though

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@Na462 Only 1 way they get back their own hat. right?

1,2, 3........ is that all a value for a person?

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@Na462 I think your second equation is wrong.

It should be

1*(1/100) + 2*(2/100) + 3*(3/100)+.....+ 100*(100/100)