Five married couples are seated around a table at random. Let $X$ be the number of wives who sit next to their husbands. What is the expectation of $X$ $(E[X])$?

Let $X_i$ be the event that wife $i$ seats next to her husband, $i=1,2, \dots, 5$. Indeed $X_i=1$ is she seats next to her husband and $0$ otherwise. 
\[X=X_1+ \dots +X_5.\]
Then 
\[E[X_i]=\frac{2}{9}\]
which is the probability that wife $i$ seats in the two sears next to her husband. Hence 
\[E[X]=E[X_1]+\dots +E[X_5]=5\times \frac{2}{9}=\times \frac{10}{9}.\]