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:

0.0177 days

Step-by-step 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