Question

For a group of 100 people, compute(a) the expected number of days of the year that are birthdays of exactly 3 people.(b) the expected number of distinct birthdays.

Answer 1

Explanation:

Leaving leap years, a year contains 365 days.

For a group of 100 people, each person is independent of the other and probability of any day being his birthday has a chance of

`(1)/(365)`

a) Probability that exactly 3 people have same birthday =
`(1)/(365^3)`

Each day is independent of the other

And hence no of days having exactly 3 persons birthday out of 100 persons is binomial with n = 365 and p =
`(1)/(365^3)`

Expected value of days = np =
`(1)/(365^2)`

b) Distinct birthdays is binomail with p =1/365 and n = 365

Hence

expected value = np =1