In a country there are 4 cities, N is to the North, W is to the West, S is to th
ID: 3305428 • Letter: I
Question
In a country there are 4 cities, N is to the North, W is to the West, S is to the South and E is to the East. There are straight highways connecting the following pairs of cities: (N, W), (N, E), (S, W), (S, E) and (N, S). Because of a snow storm, each highway is blocked by snow with probability p, independently from one another. Find the probability that it is possible to get from E to W by car. (Extra question for no extra credit: can you find a simple explanation of your result in case p = 1 2 ?) Hint: use the total law of probability with the complete set of events {(N, S) is blocked }, {(N, S) is not blocked }
Explanation / Answer
There are four possible ways of getting from E to W:
(I) By using the path: E->S->W. Probability of going through this path is: (1-p)*(1-p)
(II) By using the path: E->N->W. Probability of going through this path is: (1-p)*(1-p)
(III) By using the path: E->S->N->W. Probability of going through this path is: (1-p)*(1-p)*(1-p)
(IV) By using the path: E->N->S->W. Probability of going through this path is: (1-p)*(1-p)*(1-p)
So,
Reqd probability = 2( (1-p)2 + (1-p)3 )
Hope this helps !