Expectation of seating arrangements Question: $5$ men and $5$ women are seated randomly in a single row of chairs. The expected number of women sitting next to at least $1$ man equals ? The answer is $35/9$. Could somebody please show the workings to this question because I'm getting stuck in the very first step of gauging the probability of the random variable $X$.

Pick a particular woman W. What's the probability she's sitting next to at least one man? Well this depends on whether she's sitting at the end of the row or not, so it's easier to split into two cases.

If W is sitting on the end (probability $1/5$), there is one person next to her, and $5$ men out of the $9$ other people, so the probability it's a man is $5/9$.

If W is not sitting on the end, (probability $4/5$), there are two people next to her. The number of possible pairs of people to sit next to her is $\binom 92=36$, and all of these are equally likely. The number of pairs of other women is $\binom 42=6$, so the probability she is sitting next to at least one man is $5/6$.

So the probability W is sitting next to at least one man is $\frac 15\times\frac59+\frac45\times\frac56=\frac79$. Now the expected number of women is just this probability times the total number of women.