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A ship has three independent engines E1, E2, and E3 that operate with probabilities 1/2, 1/4, and 1/4 respectively. The ship is operational if at least two engines function. Let X be the event that the ship is operational. Which of the following are true?
- P(E1 does not function | ship is operational) = 3/16
- P(exactly two engines are functioning) = 7/8
- P(ship is operational | E2 is functioning) = 5/16
- P(ship is operational | E1 is functioning) = 7/16
Correct answer: P(ship is operational | E1 is functioning) = 7/16
Solution
P(X) = (1/2)(1/4)(1/4) + (1/2)(1/4)(3/4) + (1/2)(3/4)(1/4) + (1/2)(1/4)(1/4) = 1/32 + 3/32 + 3/32 + 1/32 = 8/32 = 1/4. Then evaluate each option using conditional probability.
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