I have the following question:
Four men went to a party and hung their coats in a closet. When they left, each of them randomly and uniformly picked a coat. What is the probability that no one got the coat they came with to the party?
I tried to solve it with inclusion/exclusion:
A_i
- Man #i
took not his coat.
I know the probabilty space is 4!
.
Hence I tried to calculate: $$P=\frac{|A_1\cup A_2\cup A_3\cup A_4|}{4!}$$
The problem is in the inclusion/exclusion formula.
I don't know how to calculate the cardinality of the intersections. i.e.:$$|A_2\cap A_3|$$
Because if man #2
took a coat that doesn't belong to man #3
then the cardinality is 3*2 (the man #2
can take 3 coats and man #2
can take 2).
But, there is also an option man #2
took a coat that belongs to man #3
so I dont know how really calculate the cardinalities. It gets more complicated when there are more than 2 sets in the intersection.
Maybe inclusion/exclusion isn't the right tool here?