Runs of red and blue marbles

HardCombinatorics Expected Value~8m

You draw all $100$ marbles ( $50$ red, $50$ blue) from a bag one at a time, recording the color sequence. A "run" is a maximal block of same-colored marbles. What is the expected number of runs?

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Runs $= 1 + (\#\text{ color changes})$ . Use linearity over the $99$ adjacent pairs.
A pair differs in color with probability $2\cdot\tfrac{50}{100}\cdot\tfrac{50}{99}=\tfrac{50}{99}$ .

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Asked at: Jane Street, Two Sigma

Runs of red and blue marbles

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