Question

For reasons not entirely clear, Doomsday Airlines books a daily shuttle service from Altoona to Hoboken. They offer two round-trip flights, one on a two-engine prop plane, the other on a four-engine prop plane. Suppose that each engine on each plane will fail independently with the same probability p and that each plane will arrive safely at its destination only if at least half its engines remain in working order. Assuming that you want to continue living, for what values of p would you prefer to fly in the two-engine plane?

Answer 1

One should choose to fly in the 2-engine plane if the value of p is such that the probability of a maximum of half of the engines failing, i.e. probability of 0 or 1 engines failing is greater than that of more than half (that is, 2) engines failing.

P(no engine fails) = (1-p)^2

P(Any one engine fails ) = p*(1-p) + (1-p)*p = 2*p*(1-p)

And, P(both engines fail) = p^2

So, we need p to be such that,

P(no engine fails) + P(Any one engine fails ) > P(both engines fail)

=> (1-p)^2 + 2*p*(1-p) > p^2

=> 1 + p^2 - 2p + 2p - 2p^2 > p^2

=> 1 - p^2 > p^2

=> p^2 < 1/2

=> -0.707 < p < 0.707

But, p should be positive,

Hence, we get 0 < p < 0.707 for us to prefer flying in the two-engine plane.