Percentage of 2 different Sets of Birthday Out of 14 People

Question

I am trying to calculate the odds of in a group of 14 people, the percentage of there being 2 different birthday matches (2 people having one Date of Birth, then another 2 people having a different Date of Birth).

I know the odds are about 20% of 2 people out of 14 having the same birthday.

So what would the odds be of there being 2 different birthday matches out of a group of 14?

Answer 1

We make the usual assumption that birthdays are uniformly distributed in a 365-day year.

The number of ways to assign birthdays to people such that there are exactly two pairs is the product of

• the number of ways to choose the two duplicated birthdays: $\binom{365}2$
• the number of ways to assign the earlier birthday to two people, and the later birthday to two other people: $\binom{14}2\binom{12}2$
• the number of ways to assign 10 out of the remaining 363 days to the rest of the people: $\frac{363!}{353!}$

Dividing this product by the number of ways to distribute birthdays without restrictions $365^{14}$ yields the final probability of around $0.018776$, or 1.9%.