Albert and Billy are playing a guessing game. They begin by filling a jar with $10$ white marbles and $10$ black marbles. Albert then draws out all $20$ marbles one by one, asking Billy each time to guess the color that will be drawn next.
Albert will give Billy a dollar for each correct guess. Billy, being a highly skilled marble guesser, plays optimally. Suppose he can expect to win $\$\frac{a}{b}$, where$a$and$b$are relatively prime positive integers. Compute the value of$a+b\$.