Phillip Sheridan frequently flies between Washington, D.C. and Raleigh, North Carolina, and as the distance is short, the airline uses turboprop airplanes. He often has a choice between flying on a two-engine or a four-engine airplane. Phillip’s intuition is that the four-engine plane is safer as he knows that a plane needs at least one half of its engines working properly to land the plane safely. Thus, if one engine failed, both planes could land safely, but if two engines failed, then the four-engine plane would be okay and the two-engine plane would crash. Is Phillip’s intuition correct? Assume that each engine has the same probability p of not failing and that the engines are independent of one another.
a. Explain why PB (R ≥ 1 | n = 2; p) is the probability of the two-engine plane landing safely and PB (R ≥ 2 | n = 4; p) is the probability of the four-engine plane landing safely.
b. Compute the probability of a safe arrival for both plane types when the probability of an engine not failing is p = 0.95, p = 0.80, p = 0.67, and p = 0.10. Comment on which plane is safer as the probability of engine failure increases (that is, as p decreases).