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Runs of equal die values

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A fair $6$ -sided die is rolled $8$ times. A run is a maximal block of consecutive equal results. What is the expected number of runs?

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Hint

A new run starts when a roll differs from the previous roll.

Solution

The first roll starts a run.
Each of the other $7$ rolls differs from its predecessor with probability $\frac{5}{6}$ .
The expectation is $1 + 7 \times \frac{5}{6} = \frac{41}{6}$ .

Answer: $\frac{41}{6}$