In my experience, the birthday paradox — or “birthday problem” — is a great introduction to probabilities. It generally is an effective way to convey the idea that our intuition can be misguided in its evaluation of the odds of something happening. The theory of probability will then appear as a reasonable crutch for our failing intuition… Provided we can come up with a reasonable model, of course.
Anyway, to get people hooked on the problem, I like to carry out the experiment with the class themselves. Unfortunately, the number of students I have is generally less than the coveted 23. But that doesn’t mean that it’s not worth a try… I’m generally quite risk-averse, but that kind of risk I can handle.
First, even if there are only 20 students, the chances of success (identical birthdays) remain relatively high 41%. That’s a lot better than most gambling games. (And think how valuable the reward is in terms of interactions in the class room!)
Second, there’s a good chance that two students will have an identical birthday or at least adjacent birthdays (like 3 May and 4 May). This is generally seen as remarkable enough and will work.
But how about 15 students? or 12 students?
This leads to a “weak birthday problem”, which consists in looking what is the maximum distance between the closest birthdays in a group of (n) people.
Interestingly, you have more than a 50% chance of finding two people with birthdays one day apart in a room of twelve people (8 Dec and 10 Dec), a lot less than the expected one-month difference.