“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.