Question

For a group of 50 people, assuming that each person is equally likely to have a birthday on each of 365 days in the year, compute

(a) The expected number of days of the year that are birthdays of exactly 3 people:

Answer 1

0.0177 days

Explanation:

Solution:-

- The group has n = 50 people

- Each person is equally likely to have a birthday on each of 365 days in the year.

- The probability of success p = 1 / 365

- The probability of failure q = (1 - 1/365)

- We will denote random variable X as a given day has k number of birthdays.

- X follows binomial distribution:

X ~ Bi ( 50 , 1/365)

- The probability that there would be 3 birthdays on a day would be:

P ( X = 3 ) = 50C3 ( 1 / 365)^3 * ( 1 - 1/365)^47

P ( X = 3 ) = 0.00035

- The expected number of days for exactly three people to have same birth date:

E( X = 3 ) = P (X=3)*n = 0.00035*50 = 0.0177 days