# Suspects and cops

16 cops and 8 suspects are sitting in a row, waiting for a train.

It would be desirable that each suspect is flanked on both sides by cops.

If they sit randomly, what is the expected value of the # of suspects so flanked ?

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What about extremities? If $C$ is a cop and $S$ is a suspect, do you consider $$SCC \dots CS$$ the suspects at both ends flanked or not? – Patrick Da Silva Feb 27 '12 at 1:57
@Patrick: No, they have to be flanked on both sides. – true blue anil Feb 27 '12 at 2:25

Expectations add, so you can compute an individual suspect's odds of being flanked and then multiply by 8. Each suspect has a $\frac{22}{24}$ chance of sitting somewhere other than the ends (where said suspect cannot be flanked), and then there is a $\frac{16}{23}$ chance the person to his left will be a cop, and then a $\frac{15}{22}$ chance the person to his right will be a cop (given that the person to his left is a cop). $$\frac{22}{24}\cdot\frac{16}{23}\cdot\frac{15}{22}=\frac{10}{23}$$ The expected number of suspects flanked by cops is $\frac{80}{23}$