So after all the hand-waving about why the letter-writer was wrong about probability calculations, they never did answer her direct question, which was, “What is the formula?”

If I'm remembering my statistics correctly, it's:

In other words, it's

1 - 365/365 * 364/365 * 363/365 * ... * (365-n)/365

I figured it would be something along those lines, which would have taken about half a minute for them to say on the show. Hey, if they’re going to read a letter that asks for a formula, they should state the formula. They didn’t even give a particularly good general description.

But it’s not often that in my opinion they fail so badly at a segment.

It would have taken considerably longer than half a minute to explain. It’s just the formula of my explanation, which would take a few minutes to explain.

I thought the later suggested method of calculation to be neater - with 23 people there’s 23 x 22/2 pairs of people ie 253 pairs.

Why 23 x 22/2? With the first person, there’s 22 pairs possible, with the 2nd person there’s 21 pairs, not already considered, possible, right down to the second last person for whom there’s only one possible pair not already mentioned, meaning there’s 22 + 21 + 20 + ... + 1 pairs, or rearranging the numbers (22 + 1) + (21 + 2) + (20 + 3) + ... + (12 + 11). In other ‘words’ 23 x 11 or 23 x 22/2.

There’s a 364/365 chance one pair won’t share a birthday, so the chance of none of the pairs sharing a birthday is (364/365)

^{253}, which equals 0.49952284596342 (roughly). So the chance of at least one of the pairs sharing a birthday (there could be more) is 0.50047715403658 (again roughly).

Neat isn’t it? I love numbers. It must appeal to my inner Asperger’s.