Smallest card sharing a box

HardProbability Expected Value~8m

You deal $100$ distinct cards numbered $1$ through $100$ one at a time, each placed into one of two boxes independently and uniformly at random. Consider the box that ends up containing card $100$ , and let $S$ be the smallest card number in that box. The expected value $\mathbb{E}[S]$ can be written as $a\left(1 - a^{-b}\right)$ for integers $a$ and $b$ . What is $a\cdot b$ ?

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Every card $1..99$ is in card $100$ 's box independently with probability $1/2$ . Use $\mathbb{E}[S]=\sum_{k\ge1}P(S\ge k)$ .
$S \ge k$ requires cards $1,\dots,k-1$ to all miss the box, probability $(1/2)^{k-1}$ . Sum the geometric series.

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Smallest card sharing a box

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