You say it's your birthday?

It's all the rage among the cocktail crowd.

Q1.  "What are the odds that  two or more people in this room share the same birthday?"

That's a somewhat hard thing to compute directly.

However,

Q2.  "What are the odds that there are no shared birthdays in this room"

is a simple thing to compute.

Since Q1+Q2 = 1  , Q1=1-Q2 , that is, the answer to our question is the simple negation of a question that's easy to answer. For the record, you compute the chances that there are no shared birthdays among your "N" participants (assume a 365 day year) as being equivalent to the problem of distributing "N" balls among "365" urns with no shared urns.

If you really need to be walked through it, here's the deal: Your first birthday (ball) is free.  Put the first ball in whatever urn (day) you desire.  The odds of the next ball missing that urn are 364/365.  The next one's odds are
363/365.  After that, it's (365-(N-1))/365 as the odds that any given birthday won't land on one of our "N" days. The simple product rule tells you how to multiply out the odds of these events .

That's all for now.  Generating Functions are "hard".

Fred