## Some Birthday Problems

One of the most famous counter-intuitive problems in probability is, “How many people must there be in a room before the probability that at least two of them have the same birthday exceeds one half?”. Most likely, you’ve heard this problem and its solution (which is 23) before. I seem to remember doing it in the fourth grade. But test your intuition against the following:

1. How many people must there be in a room before the probability of at least one of them having your birthday exceeds one half?
2. How does the birthday problem change if we’re not looking for two people with the same birthday, but two people with adjacent birthdays?
3. How does the birthday problem change if we account for the distribution of birthdays not being flat?
4. How many people must there be in a room before the probability that every day of the year has at least one person with that birthday exceeds one half? (Ignore Feb. 29)
5. People walk into a room one by one, and continue doing so until there’s a pair of people in the room with the same birthday. What is the expectation value of the number of people who walk into the room? Is it more, less, or the same as the 23 that solve the birthday problem?
6. How do the answers to these questions (including the original birthday problem) scale with the length of the year? For example, if the year were twice as long (and days were the same length), would the answer to the birthday problem be 46?

There are other obvious questions, such as how many people to have a 1/2 chance of three people with the same birthday, or how many to have two pairs of repeat birthdays, but these seem more tedious than interesting to me. Here is one more, though, which I got from a problem book:

Labor laws in Erewhon require factory owners to give every worker a holiday whenever one of them has a birthday and to hire without discrimination on grounds of birthdays. Except for these holidays they work a 365-day year. The owners want to maximize the expected total number of man-days worked per year in a factory. How many workers do factories have in Erewhon?