Suppose that five guests check their hats when they arrive at the Cigar Parlor and that these hats are returned randomly when they leave. Determine the probability that no guest will receive the proper hat

The probability that no guest gets their own hat back = (44/120) = 0.3667

Step-by-step explanation:

The hats can be returned to the guests in

5! different ways. That is, there are 120 difderent permutations of people getting their hats back.

And the number of combinations in which no guest gets their own hat back is 44. (Using the derangement theory)

A derangement is an arrangement of n objects in which no object retains its original position.

It is represented by !n and it is evaluated as the closest whole number to (n!/e)

That is,

!n = (n!/e) (note that e is the rulers number we're used to in mathematics)

e = 2.7183

So, the derangement of 5 guests would be

!5 = (5!/e) = (120/2.7183) = 44.1 = 44.

So, the probability that no guest gets their own hat back = (44/120) = 0.3667

Hope this Helps!!!