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Runs in $10$ flips

Expected valueEasy

A fair coin is flipped $10$ times. A run is a maximal block of equal consecutive results. What is the expected number of runs?

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Hint

A new run starts whenever a flip differs from the previous one.

Solution

The first flip always starts a run.
Each of the other $9$ flips starts a new run when it differs from its predecessor, which has probability $\frac{1}{2}$ .
The expectation is $1 + 9 \times \frac{1}{2} = \frac{11}{2}$ .

Answer: $\frac{11}{2}$