Question

Suppose you are shopping for a private jet, you have to decide whether to get a jet with one engine or two. Let p denote the probability that an engine fails during a flight. A “catastrophic engine failure” is an engine failure that makes the plane unable to fly.

(a) What is the probability of a catastrophic engine failure during a flight of the plane with only one engine?

(b) You are also considering a jet with two engines, however the jet is unable to fly with only functioning engine. Assuming engine failures are independent events, what is the probability of a catastrophic engine failure with this jet?

(c) Which of the two jets is safer?

(d) Suppose instead that the jet with two engines can fly with only one functioning engine. What is its probability of a catastrophic engine failure? Which jet is safer?

Answer 1

a) With only 1 engine, probability of catastrophic engine failure is if that one engine fails. Hence,

Probability of a catastrophic engine failure during a flight of the plane = p

b) Since the jet is unable to fly with only functioning engine, failure instances would be if

• 1st engine fails and 2nd engine is working fine
• 1st engine is working fine and 2nd engine fails
• Both 1st and 2nd engine fail

This can also be represented as Total scenarios- Both engines work fine

Hence,

Probability of a catastrophic engine failure = 1 - (1-p)*(1-p)

= 1 - [1 - 2p + p²] = 2p - p²

c) Safety of the two jets will depends on value of p. For any value of p between 0 and 1, Probability of a catastrophic engine failure for second jet is higher, and hence 1st jet is safer.

d) Now, if the jet can fly with one engine, failure will occur only when both engines fail.

Hence, probability of a catastrophic engine failure = p²

For any value of p between 0 and 1, probability of a catastrophic engine failure for 1st jet is always higher, and hence, jet 2 is safer.