Today is my birthday, so a fitting topic is the Birthday Problem or Birthday Paradox. The basic formulation is: Given \(n\) random people, what is the probability there are at least 2 people who share the same birthday. Given there are 365 days in the typical year, intuitively it seems like you’d need maybe half that number (180 or so) to have a 50% chance of a pair of people having the same birthday. In reality, you need just 23. In fact, with 70 people the probability of at least 2 people having the same birthday is 99.9%.
Mathematically, this probability is determined by first calculating the probability of NOT seeing a birthday match given \(n\) people and subtracting that probability from 1 to get the opposite: the probability that there is a match. For 1 person, the probability of NO match is 100% (\(365/365\)) because all days of year are available since they’re the first person. For 2 people it’s \((365/365) \times (364/365)\) because all days except the first person’s birthday are possible without having a match. For 3, it’s \((365/365) \times (364/365) \times (363/365)\), and so on with each additional person having fewer and fewer days where their birthday could be without matching any previous person’s birthday.
Effectively, this is a hidden negative compounding problem. You start with a very high probability of no match (100%) and with each additional person it is whittled down by a small percentage amount. In fact, the effect here is even more extreme because that percentage decrease is actually increasing with each additional person (i.e. it’s harder and harder not to match with each additional person). When discussing Compound Interest, we saw that a daily 1% increase would lead to almost 40x growth over the course of a year. This is roughly the -1% growth version where instead of a value shooting off into space, it rapidly approaches 0 (0% probability of NO birthday match in this case).