Last spring while working on data about BC graduates I happened to notice that there was at least one graduate for every possible birthday except for one. This made me think of the birthday problem which asks: given a certain number of people in a group, what is the probability that two of them have the same birthday. Probability theory shows that if there are 23 people in a group there is a greater than 50% chance that two of them have the same birthday and if there are 70 people in the room the chance that two of them have the same birthday is greater than 99.9%. These numbers are much smaller than most people expect [1].
This made me wonder how many people one would need to have in group for there to be a 50% or 90% chance that at least one person would have every possible birthday (except Feb 29th). I spent some time thinking about it but didn’t make much headway.
A few days later I saw that the Royal Statistical Society (RSS) in the UK has an ask a statistician blog. So I sent in my question and, to my surprise, the RSS agreed to answer it. You can find the question and an explanation here. It turns out that one needs 2,287 group members to have a 50% chance of covering all birthdays and 2,972 group members to have a 90% chance of doing so. Last spring we had 2,254 graduates [2] so having one birthday with no graduate is not too surprising.
[1] These results hold under certain simplifying assumptions such as ignoring leap years, twins and the fact that not all birthdays are equally likely.
[2] This is the number who had filed for graduation at the time I did the analysis. The actual number of graduates was different.
Last Updated November 30, 2016