Shared Birthdays Among Any Group Of People (Probability; Odds)

[oxo66]

Oooh ooh! This thread reminds me of a nicely odd occurrence I had when travelling on a local bus about 10-12 years ago.

I was travelling with my mum on the bus back from the local large town to our respective homes along the route and we were talking about birthdays. The bus was packed out, and so maybe we were chatting a little more loudly than usual due to the noise level from all the passengers and the rattly rural bus (we get all the urban cast-offs down here in the deep south west!).

I remarked that in my 40-or-so-odd years I'd never met anyone with the same birthday as me and mentioned the date. The lady behind us (I recognised her slightly, she worked in a local library I used occasionally) tapped me on the shoulder and said, "that's my birthday" and I was stupidly delighted to finally meet someone else with the same birthday. Then the little girl sitting in front of us with her Gran turned around to tell me it was her birthday too! I had my passport with me for some official reason and she was as happy as me to meet two other unrelated people in the same place at the same time with that birthday.

Maybe someone can work out the odds of 3 people on a bus holding about 45 people sharing the same birthday?

When things like this happen it makes me irrationally happy

Oh yes I thought, that someone could be me...

Turns out calculating the odds for 3 people is hard. A general exact formula for any given number of people has been worked out, but it's horrible to use. While I was looking for a less horrible approximation, I came across some code to run monte carlo simulations of birthdays. Plugging in your bus capacity of 45 and running it for 2 million buses, I got

No shared birthday(s) on bus of 45 people 5.9% of the time,
Birthday(s) shared by at most two people 84.7%
Birthday(s) shared by at most three people 9.1%
Birthday(s) shared by at most four people 0.3%

so the chances of three people sharing a birthday on a bus of 45 people are about one in eleven. It's more likely that three share a birthday than that none do.

oxo