Splitting black and white balls between two sacks

You have $50$ white and $50$ black balls and two sacks, and must place every ball in one of the sacks. Afterwards a sack is chosen with probability $\tfrac12$ each and a ball is drawn at random from it. You win if the ball is white. How should you distribute the balls to maximise your winning chance, and what is that chance?

Put one white ball alone in one sack and all $99$ others in the second: winning chance $\tfrac{74}{99}\approx0.747$. The first sack wins for sure and the second gives $\tfrac{49}{99}$, so the total is $\tfrac12\left(1+\tfrac{49}{99}\right)$. Adding more balls to the first sack only dilutes its certainty for a negligible gain in the second.