Average number of runs in 100 tosses

Question

A fair coin is tossed $100$ times. A run is a maximal block of consecutive identical outcomes (for example, HHTHHH has three runs). What is the expected number of runs?

Short answer

$50.5$ . The first toss always opens a run, and each of the other $99$ tosses opens a new run exactly when it differs from the toss before it, which happens with probability $1/2$ . By linearity, $1+99/2=50.5$ .

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Lessons behind this question

Random Variables and Probability Mass Functions
Expectation, Mean, and Variance
Conditional Expectation and Variance Revisited

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