A ship is fitted with three engines \( E_{1}, E_{2} \) and \( E_{3}...

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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
\( \mathrm{P} \) function. Let \( X \) denote the event that the ship is operational and let \( X_{1}, X_{2} \) and \( X_{3} \) denotes respectively

W the events that the engines \( E_{1} E_{2} \) and \( E_{3} \) are functioning. Which of the following is (are) true?
(A) \( P\left[X_{1}^{c} \mid X\right]=\frac{3}{16} \)
(B) \( P[ \) Exactly two engines of the ship are functioning \( \mid X]=\frac{7}{8} \)
(C) \( P\left[X \mid X_{2}\right]=\frac{5}{16} \)
(D) \( P\left[X \mid X_{1}\right]=\frac{7}{16} \)
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