A ship is fited with three engines \( \mathrm{E}_{1}, \mathrm{E}_{2} \) and \( \mathrm{E}_{3} \). 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
\( P \)
W its engines must function. Let \( \mathrm{X} \) denote the event that the ship is operational and \( \mathrm{X}_{1}, \mathrm{X}_{2}, \mathrm{X}_{3} \) denotes respectively the events that the engines \( \mathrm{E}_{1}, \mathrm{E}_{2} \) and \( \mathrm{E}_{3} \) are functioning. Which of the following is (are) true?
(A) \( \mathrm{P}\left[\mathrm{X}_{1}^{\mathrm{c}} \mid \mathrm{X}\right]=\frac{3}{16} \)
(B) \( \mathrm{P}[ \) Exactly two engines of ship are functioning \( \mid \mathrm{X}]=\frac{7}{8} \)
(C) \( \mathrm{P}\left[\mathrm{X} \mid \mathrm{X}_{2}\right]=\frac{5}{16} \)
(D) \( \mathrm{P}\left[\mathrm{X} \mid \mathrm{X}_{1}\right]=\frac{7}{16} \)
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