“Good, fast, cheap. Choose two.” In other words, quality is often expensive and takes time. And sometimes quick and cheap solutions of lower quality are sufficient. This choice is the point of this series on the birthday problem: What’s the probability that two students in a classroom share a common birthday? After failing in part 1 due to an unreasonably expensive method, we took a cheaper shortcut in part 2 and got approximate results as a trade-off. In this final part, we’ll spend some time understanding the problem to calculate precise results.

In this post, we will answer a question that we previously failed at: What is the probability that two students in a classroom share the same birthday? This is described as a paradox because for a classroom of realistic size, the probability is surprisingly higher than intuitively expected. Previously, we could only solve this for very small classrooms of 2 or 3 students. The reason? Too much data.

After the “toy problem” of the previous post, I decided to take a chance at a more tangible, real-world question. Consider this one: What is the probability that two students in a classroom share the same birthday? Intuitively (and correctly), you might think it depends on the total number of students. But for a realistic classroom of around 20 students, the probability should be quite low - whatever “low” means. After all, how many times have you witnessed that?