A Princeton slot machine has $100$ pictures, each equally likely to occur. One is a picture of a tiger. Alice and Bob independently use the slot machine, and each repeatedly makes independent plays. Alice keeps playing until she sees a tiger, at which point she stops. Similarly, Bob keeps playing until he sees a tiger. Given that Bob plays twice as much as Alice, let the expected number of plays for Alice be $\tfrac{a}{b}$ with $a, b$ relatively prime positive integers. Find the remainder when $a + b$ is divided by $1000$.

Let's think step by step and output the final answer within \boxed{}.
user