I wrote about this some time ago[0] and submitted it here on HN[1], where there was an interesting discussion. That article includes a complete derivation[2] that's simply quoted in this submission.

[0] http://www.solipsys.co.uk/new/TheBirthdayParadox.html?HN-201...

[1] https://news.ycombinator.com/item?id=4753014

[2] http://www.solipsys.co.uk/new/TheBirthdayParadox.html#toc_na...