
Probability question
Hello, could you help me
Probability of failure of each airplane engine is equal to $\displaystyle p$.
Airplane engines break down independently of each other.
Airplane can fly, if work at least half of the existing engines.
For which values of $\displaystyle p$ is a twinengined plane safer than fourengined plane?
How to get the inequality?
Thank you

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".