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Matching neighbours in coin row

Expected valueEasy

A fair coin is flipped $20$ times independently, giving a row of outcomes. Count the adjacent pairs of flips that show the same side. What is the expected value of this count? Give a fraction a/b in lowest terms.

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Hint

Count the adjacent pairs, then find the chance that one pair matches.

Solution

A row of $20$ flips has $19$ adjacent pairs.
For one pair, the second flip equals the first with probability $\frac{1}{2}$ .
Each pair has an indicator with expectation $\frac{1}{2}$ , and linearity applies even though neighbouring pairs overlap.
The total is $19 \times \frac{1}{2} = \frac{19}{2}$ .

Answer: $\frac{19}{2}$

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