# Probability question

• Sep 10th 2013, 11:16 AM
feferon11
Probability question
Hello, could you help me

Probability of failure of each airplane engine is equal to $p$.
Airplane engines break down independently of each other.
Airplane can fly, if work at least half of the existing engines.
For which values ​​of $p$ is a twin-engined plane safer than four-engined plane?

How to get the inequality?

Thank you
• Sep 10th 2013, 12:04 PM
HallsofIvy
Re: Probability question
If an airplane has two engines then "at least half of the engines working" is 1 or 2 engines working. The probability of exactly one engine breaking down is 2p(1- p) and the probability of neither breaking down is (1- p)^2 so the probability or "at least half of the engines working" is 2p(1- p)+ (1- p)^2= 2p- 2p^2+ 1- 2p+ p^2= 1- 2p- p^2.

If an airplane has four engines, then "at least half of the engines working" is 2 or three or four engines working. The probability of exactly two engines breaking down is (4!/(2!)(2!)p^2(1- p)^2= 12p^2(1- p)^2. The probability of exactly one engine breaking down (so three working) is 3p(1- p)^3. The probability of no engine breaking down (so all four working) is (1- p)^4. Add those to find the probability of "at least half of the engines working".