Hard

Probability

You have 2 boxes and 100 distinct cards numbered $1-100$. At each turn, you deal a card from the top of the deck and place the card in a box uniformly at random. What is the expected value of the smallest numbered card in the box that has card $100$ in it? The answer is in the form $a(1-a^{-b})$ for integers $a$ and $b$. Find $ab$.

Notes

Hint