A ship is fitted with three engines \(E_{1}, E_{2}\) and \(E_{3}\). 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 \(X\) denote the event that the ship is operational and let \(X_{1}, X_{2}\) and \(X_{3}\) denote respectively the events that the engines \(E_{1}, E_{2}\) and \(E_{3}\) are functioning. Which of the following is(are) true?

Given that \(P\left(X_{1}\right)=\frac{1}{2}, \mathrm{P}\left(X_{2}\right)=\frac{1}{4}, \mathrm{P}\left(X_{3}\right)=\frac{1}{4}\) \(P(X)=P\) (at least 2 engines are functioning)

\(\begin{aligned}=P\left(X_{1} \cap X_{2}\right.&\left.\cap X_{3}^{C}\right)+P\left(X_{1} \cap X_{2}^{C} \cap X_{3}\right) \\ &+P\left(X_{1}^{C} \cap X_{2} \cap X_{3}\right)+P\left(X_{1} \cap X_{2} \cap X_{3}\right) \end{aligned}\)

\(=\frac{1}{2} \times \frac{1}{4} \times \frac{3}{4}+\frac{1}{2} \times \frac{3}{4} \times \frac{1}{4}+\frac{1}{2} \times \frac{1}{4} \times \frac{1}{4}+\frac{1}{2} \times \frac{1}{4} \times \frac{1}{4}=\frac{1}{4}\)

(a) \(P\left(X_{1}^{C} / X\right)=\frac{P\left(X_{1}^{C} \cap X\right)}{P(X)}=\frac{P\left(X_{1}^{C} \cap X_{2} \cap X_{3}\right)}{P(X)}\)

\(=\frac{\frac{1}{2} \times \frac{1}{4} \times \frac{1}{4}}{\frac{1}{4}}=\frac{1}{8}\)

\(\therefore\) (a) is not true.

(b) \(P\) [Exactly two engines are functioning \(/ X]\)

\(=\frac{P[(\text { Exactly two engines are functioning }) \cap X]}{P(X)}\)

\(=\frac{P\left(X_{1}^{C} \cap X_{2} \cap X_{3}\right)+P\left(X_{1} \cap X_{2}^{C} \cap X_{3}\right)+P\left(X_{1} \cap X_{2} \cap X_{3}^{C}\right)}{P(X)}\)

\(=\frac{\frac{1}{2} \times \frac{1}{4} \times \frac{1}{4}+\frac{1}{2} \times \frac{3}{4} \times \frac{1}{4}+\frac{1}{2} \times \frac{1}{4} \times \frac{3}{4}}{\frac{1}{4}}=\frac{7}{8}\)

\(\therefore\) (b) is true.

(c) \(P\left(X / X_{2}\right)=\frac{P\left(X \cap X_{2}\right)}{P\left(X_{2}\right)}\)

\(=\frac{P\left(X_{1} \cap X_{2} \cap X_{3}\right)+P\left(X_{1}^{C} \cap X_{2} \cap X_{3}\right)+P\left(X_{1} \cap X_{2} \cap X_{3}^{C}\right)}{P\left(X_{2}\right)}\)

\(=\frac{\frac{1}{2} \times \frac{1}{4} \times \frac{1}{4}+\frac{1}{2} \times \frac{1}{4} \times \frac{1}{4}+\frac{1}{2} \times \frac{1}{4} \times \frac{3}{4}}{\frac{1}{4}}=\frac{5}{8}\)

\(\therefore\) (c) is not true.

(d) \(P\left(X / X_{1}\right)=\frac{P\left(X \cap X_{1}\right)}{P\left(X_{1}\right)}\)

\(=\frac{P\left(X_{1} \cap X_{2} \cap X_{3}\right)+P\left(X_{1} \cap X_{2}^{C} \cap X_{3}\right)+P\left(X_{1} \cap X_{2} \cap X_{3}^{C}\right)}{P\left(X_{1}\right)}\)

\(=\frac{\frac{1}{2} \times \frac{1}{4} \times \frac{1}{4}+\frac{1}{2} \times \frac{3}{4} \times \frac{1}{4}+\frac{1}{2} \times \frac{1}{4} \times \frac{3}{4}}{\frac{1}{2}}=\frac{7}{16}\)

\(\therefore\) (d) is true.