Six guests at a party leave their hats in a pile…
Contributed by StemPy LabTry it on paper first. A hint, the key idea and the worked solution are here when you need them.
Six guests at a party leave their hats in a pile. When they leave, each guest takes one hat from the pile at random, and every way of giving out the six hats is equally likely. Find the number of ways in which nobody gets their own hat. Find the probability that exactly two guests get their own hats. Find the probability that exactly four guests get their own hats. Then find the probability that at least one guest gets their own hat. Give the probabilities to four places.
Need a reminder? The key ideaShow
All different: for objects in equally likely boxes. For matching, counts the arrangements with no match, and exactly matches have probability .
The worked solution
Have you tried it?
Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.
Step 1. Nobody. out of ways.
Step 2. Exactly two. Choose the two guests in ways. The other four must all be wrong: ways. So .
Step 3. Exactly four. ways, so .
Step 4. At least one. .
Answer: ways, , and .
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