Expected colour runs drawing 50 red and 50 blue

Question

A bag holds $50$ red and $50$ blue marbles. You draw them all one by one in random order and record the colour sequence. A run is a maximal block of consecutive same-coloured marbles; for example the sequence RBBRRRBRR is made up of $5$ runs. Find the expected run count.

Short answer

$51$ . Runs $=1+$ number of colour changes between neighbours. Each of the $99$ adjacent pairs differs in colour with probability $2\cdot\tfrac{50}{100}\cdot\tfrac{50}{99}=\tfrac{50}{99}$ , so the expectation is $1+50=51$ .

Adapted from QuantQuestions.io; question and answer rewritten by SelfMadeQuant.com.

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Lessons behind this question

Expectation, Mean, and Variance
Counting and Combinatorics
Conditional Expectation and Variance Revisited

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