This article is from the Puzzles FAQ, by Chris Cole firstname.lastname@example.org and Matthew Daly email@example.com with numerous contributions by others.
A town has two hospitals, one big and one small. Every day the big
hospital delivers 1000 babies and the small hospital delivers 100
babies. There's a 50/50 chance of male or female on each birth.
Which hospital has a better chance of having the same number of boys
The small one. If there are 2N babies born, then the probability of an
even split is
(2N choose N) / (2 ** 2N) ,
where (2N choose N) = (2N)! / (N! * N!) .
This is a DECREASING function.
If there are two babies born, then the probability of a split is 1/2
(just have the second baby be different from the first). With 2N
babies, If there is a N,N-1 split in the first 2N-1, then there is a
1/2 chance of the last baby making it an even split. Otherwise there
can be no even split. Therefore the probability is less than 1/2
overall for an even split.
As N goes to infinity the probability of an even split approaches zero
(although it is still the most likely event).