Birthdays are in the news this month as the last of the Baby Boomers hit age 50—most notably Michelle Obama, but also my youngest sibling—brother Paul. A little game I’ve played with my larger statistics classes is to poll them for their birthday—month and day (one mustn’t dare to ask for the year). It turns out that with 23 people coming together at random the odds tilt in favor of at least two sharing this special date. Somehow that just does not seem likely but all one needs to do for working this out is calculate the probability of all having different birthdays, and then subtract the answer from 1.
By the way, it takes 88 people to achieve a good chance of 3 sharing a birthday.
This last statistic (88 for 3) comes from statistician Mario Cortina Borja in an article he wrote for the latest issue of Significance detailing “The strong birthday problem,” that is, not just one person but everybody in a group sharing a birthday with at least one other. By assuming that the birthdays follow a uniform distribution,* Borja worked out this complex problem. His results are somewhat counter-intuitive in the way probabilities decrease from 2 to 365 and rise thereafter—quickly gaining at 2000 and beyond. (Of course if only “me, myself and I” are gathered, that is, one person, the probability is technically 100 percent of a birthday match.) The answer to this strong birthday problem is 3064. At 4800 people there’s a 99% chance that everyone will share a birthday with another.
Borja suggests that it might be fun for a large celebration to award a prize to anyone with a lone birthday. If one won such a contest, it would really be a lonely experience.
*P.S. Borja provides the math for birthdays being distributed non-uniformly, but leaves it at that because the computational cost of solving it is “fiendish.” That’s OK because other statisticians who studied this problem found that the results change very little with deviations from the uniform distribution.