A ship is fitted with three engines `E_1,E_2 and E_3`. The engines function independently of each other with respective probabilities `1/2, 1/4, and 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 function. Which of the following is/are true? (a) `P(X_1^c "|"X)=3/(16)` (b)`P`(exactly two engines of the ship are functioning `"|"X` ) `=7/8` (c) `P(X"|"X_2)=5/6` (d) `P(X"|"X_1)=7/(16)`

Options

A. `P[X_(1)^(@)|X] = 3//16`
B. P [exactly two engines of the ship are functioning [X] `= (7)/(8)`
C. `P[X|X_(2)] = (5)/(16)`
D. `P [X | X_(1)] = (7)/(16)`

Answer

D