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Q.

A ship is filled with three engines E₁, E₂ and E₃. The engines function independently of each other with respective probabilities $\frac{1}{2}, \frac{1}{4} and \frac{1}{4}$ For the ship to be operational at least two of its engines must function Let X denotes the event that the ship is operational and Let X₁, X₂ and X₃ denote respectively the events that the engine E₁, E₂ and E₃ are functioning, which of the following is (are) true?

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a

$P (\frac{X_{1}^{C}}{X}) = \frac{3}{16}$

b

P(exactly two engines of the ship functioning / $(X) = \frac{7}{8}$

c

$P (\frac{X}{X_{2}}) = \frac{5}{16}$

d

$P (\frac{X}{X_{1}}) = \frac{7}{16}$

answer is B, D.

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Detailed Solution

$\begin{array}{l} P (X_{1}) = \frac{1}{2}, P (X_{2}) = \frac{1}{4}, P (X_{3}) = \frac{1}{4} \\ P (X) = P (X_{1} \cap X_{2} \cap X_{3}) + P (X_{1} \cap X_{2}^{c} \cap X_{3}) + P (X_{1}^{c} \cap X_{2} \cap X_{3}) + P (X_{1} \cap X_{2} \cap X_{3}^{c}) = \frac{1}{4} \end{array}$

$a. P (X_{1}^{C} ∣ X) = \frac{P (X \cap X_{1}^{C})}{P (X)} = \frac{1 / 32}{1 / 4} = \frac{1}{8}$

$b) P [exactly two engines of the ship are functioning ∣ X] = \frac{7 / 32}{1 / 4} = \frac{7}{8}$

$c. P (\frac{X}{X_{2}}) = \frac{P (X \cap X_{2})}{P (X_{2})} = \frac{5 / 32}{1 / 4} = \frac{5}{8}$

$d. P (\frac{X}{X_{1}}) = \frac{P (X \cap X_{1})}{P (X_{1})} = \frac{7 / 32}{1 / 2} = \frac{7}{16}$

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