Let $A,B$ and $C$ be three points(destinations), with $r_1$ being a road between $A$ and $B$ and $r_2,r_3$ and $r_4$ being a road between $B$ and $C.$ Let $p$ be the probability that the road is blocked by snow. Find the probability of $r_1$ being blocked given that there is no route between $A$ and $C.$
My Attempt: Let $q_1,q_2,q_3$ and $q_4$ represent the probability that the roads are not blocked by snow. Thus $q_i=1-p$ for $1\leq i\leq 4.$ Then we have a conditional probability which I think should be equal to $$P(X|Y)=\frac{P(X\cap Y)}{P(Y)}=\frac{1-q_1}{1-q_1(q_2+q_3+q_4-q_2q_3-q_3q_4-q_4q_2+q_2q_3q_4)} $$ $$=\frac{p}{1-(1-p)^2(1+p+p^2)}.$$ Does this make sense?
The method and computation both look good to me.