Two roads join Ayton to Beaton, and two further roads join Beaton to the City. Ayton is directly connected to the City by a railway. All four roads and the railway are each independently blocked by snow with probability $p$. I am at Ayton.
(a) Find the probability that I can drive to the City.
(b) Find the probability that I can travel to the City.
(c) Given that I can travel to the City, what is the probability that the railway is blocked?
MY ATTEMPT
(a) There are two possibilities to drive from Ayton to Beaton. The probability that both roads are blocked by snow is given by $p^{2}$. Therefore the probability to reach Beaton from Ayton by road is given by $(1-p^{2})$. The same applies to the travel from Beaton to City. Since both events are independent, the sought probability is given by $(1-p^{2})^{2}$.
However I am unable to proceed from here. Can somebody help me to solve it? Thanks!
Following on from what you have done, the probability you cannot drive to the city is $1-(1-p^2)^2$ So the probability you cannot travel to the city by any means is $$p(1-(1-p^2)^2)$$ Then the probability you can travel to the city by some means or other is 1 - this.
Then use conditional probability formula for the last part...