medium Probability
A release-checked medium problem for training Exchangeability, Counting With Indicators, Linearity of Expectation, Coins.
Fix a positive integer $n \geq 3$. $n$ people stand around in a circle, each having their own coin with probability $0 < p < 1$ of heads per flip. Each person flips their coin, and the outcomes are recorded. Find the expected number of people for which their flip differs from both people adjacent to them. Report the answer when $p = 2/3$ and $n = 18$.
This Probability problem is tagged Exchangeability, Counting With Indicators, Linearity of Expectation, Coins. State the random variables and conditioning information explicitly, then check the result against boundary cases before opening hints or a solution.
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