It's all the rage among the cocktail crowd.
A. "What are the odds that two people in this room share the same birthday?"
That's a somewhat hard thing to compute directly.
B. "What are the odds that there are no shared birthdays in this room"
is a simple thing to compute.
Often, you hear the solution to "A" represented as :
= 1 - "B"
Of course, that's an error. Statement "A" must be rewritten as:
A. "What are the odd that AT LEAST two people in this room share the same birthday?"
for the identity "A" + "B" = 1 to be true, that is, for the answer to our question to be 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
man (ball) is free. Put him in whatever urn (day) you desire.
The odds of the next ball missing his urn are 364/365. The next one's
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. (There's at least one hidden assumption in the MAN/BALL and BIRTHDAY/URN mapping described above. "FREE BEER" for the first person who spots a legit hidden assumption and explains it to your humble narrator. It's "FREE BEER" or gratitude.)
That's all for now. Generating Functions are "hard".
If you have comments or suggestions, email me at
INDEX to a few of Fred's pages