Five bald men wearing hats go to a party, and when they arrived they put their hats in a dark closet. During the party, someone yells, “Fire!” and the men rush to the closet, grab a hat at random, and leave. What is the probability that
(a) exactly four get the right hats?
(b) all five get the wrong hat?
(c) more than one man gets the right hat?
Best Answer
Hints:
Exactly four right is obviously impossible. If four get the right hat, the fifth must also. The probability is $0$.
All wrong is a famous problem, the problem of counting Derangements.
The Wikipedia article is quite helpful. Note that there are $5!$ equally likely ways that the hats can be distributed among the $5$ people. From the formula in the article, you will find there are $44$ derangements of $5$ objects, so the required probability is $\dfrac{44}{5!}$.
For the probability that more than one gets the right hat, it is easier to find the probability that one or fewer gets the right hat. The probability of none is the answer to (b). For the probability of one, the lucky person can be chosen in $5$ ways. Multiply this by the probability none of the remaining $4$ get the right hat. This again is a derangements problem. For $4$, you can easily list and count the derangements. There are $9$. So the probability exactly one person gets the right hat is $\dfrac{45}{5!}$.
Thus the probability of (c) is $1-\dfrac{44}{5!}-\dfrac{45}{5!}$.