MCQ
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 $X _3$ denotes respectively 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 $\mid X ]=\frac{7}{8}$
$(C)$ $P\left[X \mid X_2\right]=\frac{5}{16}$
$(D)$ $P\left[X \mid X_1\right]=\frac{7}{16}$
- A$(B,D)$
- B$(B,C)$
- C$(A,D)$
- D$(C,D)$