Three highways connect city a with city b. two highways connect city b with city c.
during a rush hour, each highway is blocked by a traffic accident with probability 0.2,
independently of other highways.
(a) compute the probability that there is at least one open route from a to c.
(b) how will a new highway, also blocked with probability 0.2 independently of other
highways, change the probability in (a) if it is built
(α) between a and b?
(β) between b and c?
(γ) between a and c?

(a) The probability that there is no open route from A to B is (0.2)^3 = 0.008.
Therefore the probability that at least one route is open from A to B is given by: 1 - 0.008 = 0.992.
The probability that there is no open route from B to C is (0.2)^2 = 0.04.
Therefore the probability that at least one route is open from B to C is given by:
1 - 0.04 = 0.96.
The probability that at least one route is open from A to C is:


(b)
α The probability that at least one route is open from A to B would become 0.9984. The probability in (a) will become:

β The probability that at least one route is open from B to C would become 0.992. The probability in (a) will become:


Gamma: The probability that a highway between A and C will not be blocked in rush hour is 0.8. We need to find the probability that there is at least one route open from A to C using either a route A to B to C, or the route A to C direct. This is found by using the formula:


Therefore building a highway direct from A to C gives the highest probability that there is at least one route open from A to C.