A town has two hospitals, one big and one small.

probability/hospital

Problem

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
as girls?

Show solution

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).