Find the probability of at least one birthday match among a group of 48 people.

Answer 1

The probability is

1 - Probability of no matches

The probability of no matches is the ratio of the number of ways you can choose the birthdays such that each person has a different birthday (let's call this N) divided by the number of ways to distribute birthdays without restriction (let's call this M).

CLearly M = 365^48

and

N = 365!/(365 - 48)!

To see this, consider that the first person can have 365 birhtdays, the second can have 365-1 as his/her birthday bust be different from the first, The third can have 365 - 2 birthdays, etc. etc.

Then, to evaluate the numbers you have to be careful if your calculator does not display numbers larger than 10^100. You can then take logarithms and use Strirling's formula to evaluate the factorials, or simply sum the 48 terms in the expression for Log(N).

Yo should fiund that 1 - N/M = 0.9606