Problem source

Bar magnets are placed randomly end-to-end in a straight line. If adjacent magnets have ends of opposite polarities facing each other, they join together to form a single unit. If they have ends of the same polarity facing each other, they stand apart. Find the expectation and variance of the number of separate units in terms of the total number $N$ of magnets.

Solution source

There are $N-1$ gaps between the magnets which are independently gaps or not gaps.

Therefore the total number of gaps is $X \sim Binomial(N-1, \frac12)$ and

\begin{align*}
\mathbb{E}(X) &= \frac{N-1}{2} \\
\textrm{Var}(X) &= \frac{N-1}{4}
\end{align*}