Ken's Puzzle of the Week
More Balls and Urns

There are two empty urns in a room. You have 50 white balls
and 50 black balls. After you place the balls in the urns,
two random balls will be picked, each from a random urn.
The first ball will not be returned to the urns after it is removed. Distribute the balls (all of them) into the urns to maximize
the chance of picking:
 two white balls.
 one white and one black ball (in any order.)

Repeat problem 1 if the balls will both be taken from the same urn.

What is the smallest number of balls (and their distribution) to make the chance
of pulling out two white balls, without replacement, each from a random urn,
exactly 1/2? Exactly 1/3?
Source: Original. Similar to puzzle on February 17, 1998.
Solution
Mail to Ken