Question

Suppose there are 15 different color crayons in a box. Each time one obtains a crayon, it is equally likely to be any of the 15 types. Compute the expected # of different colors that are obtained in a set of 5 crayons. (Hint: use indicator variables and linearity of expectation)

Answer 1

4.38

Explanation:

There 15 different color crayon in a box.

We will enumerate the crayon from 1 to 15

Let Xi be the indicator random variable for the event that the ith crayon is among the set of 5 crayon.

E[Xi] = Pr(at least one type i crayon is in the set of 5)

E[Xi] = 1 - Pr(no type i crayons are in the set of 5)

= 1 - (14/15)^5

The expected number of crayon is

E(summation of Xi for the 15 crayons where i = 1) = 25(1- (14/15)^5)

= 4.38