Hard

Probability

Two players, say $A$ and $B$, play the following game: Both players have $100$ marbles and may put anywhere between $1$ and $100$ marbles in the box each. This decision is not revealed to the other player. Then, they draw $2$ marbles with replacement between trials. If the marble belongs to $A$, then assuming that $A$ put $a$ marbles in the box, $A$ is paid $100-a$ monetary units from a third party. Similarly if the marble belongs to $B$, then assuming $B$ but $b$ marbles in the box, $B$ is paid $100-b$ monetary units from a third party. Assume both players play optimally. Find the expected total payout of player $A$.

Notes

Hint