All - that was not a math puzzle segment. It was a name that logical fallacy. The whole point was to discuss the cognitive error she was making, not review the math. Most people don't care about the actual formula or wouldn't remember it - but they would get the logical point. Anyone interested in the actual formula could easily look it up, as we said.

I can see your point, and it’s difficult to explain the equation, but the logic is reasonably simple. The chance of person 1 being born is 365/365. The chance of person 2 having a birthday on any date other than person 1’s is 364/365, the chance of person 3 having a birthday on any date other than person 1’s and 2’s is 363/365, and so on. And the chance of person 23 having a birthday on any date other than persons 1 to 22 is 342/365.

So the chance of 23 people not sharing a birthday is the product of all those 23 numbers all less than 1 and becoming progressively smaller. And when you multiply a large number of numbers less than one, the result progressively becomes much smaller. To find the exact result, you need the equation. And a very good calculator (my scientific calculator isn’t good enough). To do it, you need something like WolframAlpha, which is easy to use. And very neat.

And the chance of having at least one common birthday is 1 minus the chance of having no common birthdays. Without having and using the equation it’s a laborious business calculating the chance of having no common birthdays (365/365 times 364/365 times 363/365 times 362/365 times...times 342/365), but it can be done.