h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

I think the "break even" point is 20, not 23.

If you think about it his way...
In order for the likelihood that a birthay is shared to be greater than the likelihood that it is not, you need to be able to make more than 365/2 = 182.5 "pairs" from the group.

In a group of 19, you have (19*18)/2 = 171 pairs (47%)
In a group of 20, you have (20*19)/2 = 190 pairs (52%)

In a group of 23, you have (23*22)/2 = 253 pairs (69%)

While I don't know that your maths is wrong, I do know (now) that mine was !

Although I had the idea that 190 pairs (i.e. 20 people) was a better than even money bet for years, I could not remember if I was shown a proof, took it on faith, or worked it out for myself.

Also while I haven't checked your calculations, your presentation of the odds seemed right, so rather than try and disprove your "proof", I tried to disprove mine.

I could't go through every possible combination of "pairs" for 365 days of the year, (well not without far too much effort) so tested it out for days of the week.

If my theory was correct, then since three people give you three "pairs", the odds of three people sharing at least one "birthday of the week" would be 3/7 (43%), but it's not, it's 133/343 (39%).

cheers