Thread: same birthday?

Hi everybody,
the other day I was watching a programme on the BBC, and someone said you only need to collect 23 people at random to have 50% chances that at least 2 of them have the same birthday.

So I set about calculating this probability for a generic N (number of people), and found this formula (leap year not considered):



It seems to work, and it predicts 99.9% chances of at least 2 matching birthdays with as little as 70 people, and certainty (1) for N>=366.

However, as the corresponding plot looks sigmoidal, my question is whether there is a nicer formula, where there is no index i.

Thanks
L

You can replace the product with a ratio of factorals, divided by a power of 365. However, the factorals are just disguised versions of products.

There was another thread on this just the other day (http://www.thescienceforum.com/Perce...ity-27015t.php) though the question brought up was different.

In that thread, we ended up working out a pretty good approximation to the final probability: . That always underestimates the actual probability by a little bit (past two people anyway).

Here's a video which makes it pretty simple to understand...

http://www.youtube.com/watch?v=98OTsYfTt-c


Note: It's in context of battling back religiot and spiritual claims, but the opening makes the concept being asked about in the OP quite clear.

Thank you for your answers. I like the approximated formula.
And I didn't know Wikipedia had an entry for it.

For medium size n (365 >> n >> 0), you could use Stirling's formula to approximate factorals.