Suppose there are 6 people in a room Calculate the probabili

Suppose there are 6 people in a room. Calculate the probability that at least two people share the same birthday?

Solution

First, find the probability that all of them have different birthdays.
Suppose there are three people. The probability that the second person has a different
birthday from the first person is 364/365, since there’s only one day that’s forbidden.
The probability that the third person has a different birthday from the first two, assuming
the first two had different birthdays, is , since there are two days that are forbidden.
Now by the laws of conditional probability, the probability that all three birthdays are
different
= Prob(first two people have different birthdays AND third person has a different birthday
from the first two)
= Prob(first two have different birthdays) * Prob(third birthday is different GIVEN the
first two are different)
=364/365 * 363/365
So if there are six people, the probability of all six birthdays being different is

= 364/365 * 363/365 * 362/365 * 361/365 * 360/365 = 0.9595
So the probability of at least two people sharing the same birthday is 1 — 0.9595 = 0.0405