Birthday probability It is well known that in a room of 20 p

Birthday probability: It is well known that in a room of 20 people, the probability of 2 people with the same birth date (may be different year but the same month and the day in the month) is not low.
(a) Get the formula (using combinations) for this probability (which is about 0.4)
(b) compute this probability to 3 digits after the decimal point. Generalize this to n people.

Solution

No of people = 20

Each person having birthday on a particular day is independent of the other and there are only two trials either having birthday on that day or not.

Hence X no of persons having the same day of a month as birthday is binomial with p = 1/365 and n = 20

a) P (x=2) = 20C2 (1/365)²(364/365)¹⁸

=0.001357

b) For three persons x =3

Hence Prob = P(X=3) = 20C3 (1/365)³(364/365)²⁰

= 2.238x10^-5

= 0.000 (upto 3 decimals rounded off)