An airplane has four engines. On a flight from NY to Paris, each engine has a 0.001 chance of failing. The plane will crash if at any time two or fewer engines are working properly. Assume failures of different engines are independent.

a) what is the probability that the plane will crash?
b) given that the engine 1 will not fail during flight, what is the probability that the engine will crash?
c) given that engine 1 will fail during the flight, what is the probability that the plane will not crash?

Explanation:

airplane has four engines

n=4

p = 0.001

This problem is subject to binomial distribution

The plane will crash if 2 or fewer engines are working properly

X - number of engines failed

a) P(crash) = p(x=2 engines fail) + p(x=3 engines fail) + p(x=4 engines fail)

= 4C2 (0.001)^2(0.999)^2 +  4C3 (0.001)^3(0.999)^1 +  4C4 (0.001)^4(0.999)^0

= 0.000005991

[4C2 = 4!/2!(4-2)!]

b)  engine 1 will not fail, 3 engines can fail

P(crash) = p(2 engines fail) + p(3 engines fail)

= 3C2 (0.001)^2(0.999)^1 +  3C3 (0.001)^3(0.999)^0

= 0.000002998

c) engine 1 will fail, 3 engines work properly

If even 1 engines fails, plane will crash

P(will not crash) = 3C0 (0.001)^0(0.999)^3 +  3C1 (0.001)^1(0.999)^2

= 0.9999