Problem Solving

BTGmoderatorDC wrote: ↑

Tue Jan 17, 2023 11:08 pm

A small, experimental plane has three engines, one of which is redundant. That is, as long as two of the engines are working, the plane will stay in the air. Over the course of a typical flight, there is a 1/3 chance that engine one will fail. There is a 75% probability that engine two will work. The third engine works only half the time. What is the probability that the plane will crash in any given flight?

(A) 7/12
(B) 1/4
(C) 1/2
(D) 7/24
(E) 17/24


OA D

Source: Manhattan Prep

Solution:

We are given that a plane will stay in the air as long as at least two of the three engines are working. So, to determine the probability that the plane will crash, we need to determine the probability that at least two of the three engines are not working.

The probability of engine one not working is 1/3, the probability of engine two is not working is 1/4, and the probability of engine 3 not working is 1/2.

Scenario 1:

1 does not work, 2 does not work, 3 works

1/3 x 1/4 x 1/2 = 1/24

Scenario 2:

1 does not work, 2 works, 3 does not work

1/3 x 3/4 x 1/2 = 3/24

Scenario 3:

1 works, 2 does not work, 3 does not work

2/3 x 1/4 x 1/2 = 2/24

Scenario 4:

1 does not work, 2 does not work, 3 does not work

1/3 x 1/4 x 1/2 = 1/24

Thus, the probability that the plane will crash is 1/24 + 3/24 + 2/24 + 1/24 = 7/24.

Answer: D