TITLE: Interesting probability question about islands and bridges
QUESTION [6 upvotes]: The inhabitants of the beautiful and ancient canal city of Pentapolis live on 5 islands separated from each other by water. Bridges cross from one island to another as shown.

On any day, a bridge can be closed, with probability $p$, for restoration work. Assuming that the 8 bridges are closed independently, find the mean and variance of the number of islands which are completely cut off because of restoration work.

REPLY [2 votes]: By linearity of expectation, it's straightforward to see that the mean is $4p^3+p^4$.
Since the cutoff probabilities are correlated, the variance of their sum can be computed as the sum of all possible covariances. By repeatedly using the formula $\operatorname{Cov}(X,Y)=E[XY]-E[X]E[Y]$, notice that:

The variance of the first four islands' cutoffs is $p^3-p^6$.
The variance of the middle island's cutoff is $p^4-p^8$.
The covariance of the middle island and any outer island is $p^6-p^7$.
The covariance of any two adjacent outer islands is $p^5-p^6$.
The covariance of any two opposite outer islands is $0$.

If we add up the $25$ possible covariances, we'll get: $4$ of type 1., $1$ of type 2., $8$ of type 3., $8$ of type 4., and $4$ of type $5$. So the total variance is
$$
4(p^3-p^6)+p^4-p^8+8(p^6-p^7+p^5-p^6)=4p^3+p^4+8p^5-4p^6-8p^7-p^8 \, .
$$