Here’s a problem that came up “naturally” on our family email list and that @bee and I have been having fun with:

How many people do you have to know before *probably* every day in April is someone-you-know’s birthday?

By a “person” of course we’ll mean a random number in {1, …, 365}. No leap babies or anything. Certainly not caring about whether April is a more common or less common birth month.

And by “probably” we mean >50%. So think of the question as finding the probability as a function of the number of people, n, that all 30 birthdays are hit. Then it’s easy to check what value of n takes us over the 50% threshold.

Warmup 1: How many distinct birthdays do you expect to hit if you know n people?

Warmup 2: In expectation, how many people do you have to ask the birthday of before you get at least one answer for every day in April?

Background:

It may help to first learn about the Coupon Collector’s Problem, which Matt Parker has a nice exposition of on YouTube.