Question

For flights from a particular airport in January, there is a 30 percent chance of a flight being delayed because of icy weather. If a flight is delayed because of icy weather, there is a 10 percent chance the flight will also be delayed because of a mechanical problem. If a flight is not delayed because of icy weather, there is a 5 percent chance that it will be delayed because of a mechanical problem. If one flight is selected at random from the airport in January, what is the probability that the flight selected will have at least one of the two types of delays?

(A) 0.065 (B) 0.335 (C) 0.350 (D) 0.450 (E) 0.665

Answer is B but why

Answer 1

Let C shows the event that  flight being delayed because of icy weather and M shows the event that flight is delayed due to mechanical problem. So

P(C) = 0.30

P(M |C) = 0.10, P(M|C') = 0.05

By the complement rule,

P(C') = 1 - P(C) = 0.70

Now,

P(M and C) =P(M|C)P(C) = 0.10 * 0.30 = 0.03

And by the law of total probability

P(M) = P(M|C)P(C) + P(M|C')P(C') = 0.10 * 0.30 + 0.05 * 0.70 = 0.03 + 0.035 = 0.065

By the addition rule of probability,  the probability that the flight selected will have at least one of the two types of delays is

P(M or C) = P(M) + P(C) - P(M and C) = 0.30 + 0.065 - 0.03 = 0.335

Hence, correct option is B.