An expectation question

Question

there are four couples that are sat around a round table. What are the expected number of couples that sit together. I did this via linearity of expectation, denoting the probability that a fixed seat has his/her partner sitting next to him/her. Since there are two adjacent seats and seven total seats this would be $\frac{2}{7}$. So then the expected number of couples is $8*\frac{2}{7}$. Is this correct? It seems very high.

Answer 1

You want to multiply it by $4$, not $8$.

Seat the first girl. Then her boyfriend has a probability of $\frac{2}{7}$ that he is sitting next to her, as you correctly stated. Do this for each of the four couples to get an expected value of

$$4\cdot\frac{2}{7}=\frac{8}{7}$$