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Question:

A ship is fitted with three engines E1, E2, and E3. The engines function independently of each other with respective probabilities 1/2, 1/4, and 1/4. For the ship to be operational, at least two of its engines must function. Let X denote the event that the ship is operational and let X1, X2, and X3 denote respectively the events that the engines E1, E2, and E3 are functioning. Which of the following is (are) true?

P[X|X1]=7/16

P[Xc1|X]=3/16

P[X|X2]=5/16

P[Exactly two engines of the ship are functioning|X]=7/8

Solution:

P(X1)=1/2, P(X2)=1/4, P(X3)=1/4
P(X) = P(X1∩X2∩Xc3) + P(X1∩Xc2∩X3) + P(Xc1∩X2∩X3) + P(X1∩X2∩X3) = 1/4 (A)
P(Xc1/X) = P(X∩Xc1)/P(X) = (1/2×1/4×1/4)/(1/4) = 1/8
P(exactly 2 engines|X) = [(1/2×1/4×3/4) + (1/2×3/4×1/4) + (1/2×1/4×1/4)] / (1/4) = 7/8
P(X|X2) = P(X∩X2)/P(X2) = [(1/2×1/4×1/4) + (1/2×1/4×3/4) + (1/2×1/2×1/4)]/(1/4) = 5/32/(1/4) = 5/8
P(X|X1) = P(X∩X1)/P(X1) = [(1/2×1/4×1/4) + (1/2×1/4×3/4) + (1/2×1/2×1/4)]/(1/2) = 7/32/(1/2) = 7/16