Problems

Oxtown and Camville are connected by three roads, which are at risk of being blocked by flooding. On two of the three roads there are two sections which may be blocked. On the third road there is only one section which may be blocked. The probability that each section is blocked is \(p\). Each section is blocked independently of the other four sections. Show that the probability that Oxtown is cut off from Camville is \(p^3 \l 2-p \r^2\). I want to travel from Oxtown to Camville. I choose one of the three roads at random and find that my road is not blocked. Find the probability that I would not have reached Camville if I had chosen either of the other two roads. You should factorise your answer as fully as possible. Comment briefly on the value of this probability in the limit \(p\to1\).

The four towns \(A,B,C\) and \(D\) are linked by roads \(AB,AC,CB,BD\) and \(CD.\) The probability that any one road will be blocked by snow on the 1st of January is \(p\), independent of what happens to any other \([0 < p < 1]\). Show that the probability that any open route from \(A\) to \(D\) is \(ABCD\) is \[ p^{2}(1-p)^{3}. \] In order to increase the probability that it is possible to get from \(A\) to \(D\) by a sequence of unblocked roads the government proposes either to snow-proof the road \(AB\) (so that it can never be blocked) or to snow-proof the road \(CB.\) Because of the high cost it cannot do both. Which road should it choose (or are both choices equally advantageous)? In fact, \(p=\frac{1}{10}\) and the government decides that it is only worth going ahead if the present probability of \(A\) being cut off from \(D\) is greater than \(\frac{1}{100}.\) Will it go ahead?