Question

Highway engineers estimate that the main highway between two cities has a 1%

probability of being blocked in good weather, and a 5% probability of being blocked in

snowy weather. The detour route, on the other hand, is a smaller road which has a 2%

probability of being blocked in good weather and a 15% probability of being blocked when it

snows. On a certain day, forecasters predict a 60% chance of snow.

a.

Are the events that it snows and that the main highway is blocked

dependent or independent?

b.

Are the events that the highway is blocked and that the detour is blocked

dependent or independent?

c.

Find the overall probability that, on this day, the highway will be

blocked (whether or not the detour is blocked).

d.

Find the overall probability that, on this day, both the highway

and

the detour will be blocked.

e.

Find the probability that, on this day, either the highway, the detour, or both

will not be blocked.

Answer 1

Let S shows the event of snow and G shows the event of good weather. So

P(S) = 0.60, P(G) = 1 - P(S) = 0.40

The H shows the event that highway is blocked. So we have

P(H|S) = 0.05, P(H|G) = 0.01

The D shows the event that detour route is blocked. So we have

P(D|S) = 0.15, P(D|G) = 0.02

(a)

The probability that highway is blocked is

P(H) = P(H|S)P(S) + P(H|G)P(G) = 0.05 * 0.60 + 0.01 * 0.40 = 0.034

Since

So the events that it snows and that the main highway is blocked are dependent.

(b)

Yes because both routes are independent from each other.

(c)

The probability that highway is blocked is

P(H) = P(H|S)P(S) + P(H|G)P(G) = 0.05 * 0.60 + 0.01 * 0.40 = 0.034

(d)

The probability that the detour is blocked is

P(D) = P(D|S)P(S) + P(D|G)P(G) = 0.15 * 0.60 + 0.02 * 0.40 = 0.098

So the overall probability that, on this day, both the highway and the detour will be blocked is

P(H and D) = P(H)P(D) = 0.034 * 0.098 = 0.003332

(e)

The probability that, on this day, either the highway, the detour, or both will not be blocked is