The birthday paradox says, if you have 23 people in the same room, the odds are greater than half that two people will share a birthday. I tried to calculate it using
1 - (number of ways 23 people can have birthdays that are ALL DIFFERENT)/(number of ways 23 people can have birthdays).
Here's how 23 people can have birthdays that are all different: it is just
365 choose 23.
To calculate all the different ways 23 people can have birthdays is more complicated. In this case, order doesn't matter, but it is okay for multiple people to have the same birthdays. The formula for this is
(n + k - 1) choose k, so in this case:
(387 choose 23).
1 - (365 choose 23) / (387 choose 23) = 75%, not 50%. What am I doing wrong?