Many of you have seen The Birthday Problem: Given a group of n people, what is the probability that someone shares a birthday?
Here, we are only concerned with birth day and month (not year). The solution assumes that a person is equally born on any of the 365 days in the year, thus ignoring leap years.
Let P(n) = the probability that someone shares a birthday in a group of n people and let Q(n) = the probability that everyone has unique birthdays. There are 365^n ways for n people to be born on any of the 365 days.Then
P(n) = 1 – Q(n) = 1 – (365*364*…*(365-n+1))/365^n.
P(2) = 0.0028
P(5) = 0.0271
P(10) = 0.1169
P(20) = 0.4114
P(30) = 0.7063
P(40) = 0.8912
P(50) = 0.9704
P(60) = 0.9941 –> in a room with 60 people, you are almost certain to have at least two people that share a birthday!
The key assumption is that all birth dates are equally likely. This NPR article shows that humans have a “mating season” that makes July – September birthdays more likely. I posted the image below.
This will, of course, change our answer above. The probabilities depend on who is in the room. Have you simulated the Birthday Problem with an unequal birthday distribution? If so, please shed light on realistic numbers for P(n).
On a side note, the image below suggests that babies are induced on December 27-30 for a tax break. I’m not sure how I feel about that.