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

Answer

B, D

Explanation

`P(X_(1))=1/2,P(X_(2))=1/4,P(X_(3))=1/4`
`P(X)=P(X_(1)nnX_(2)nnX_(3))+P(X_(1)nnX_(2)^(C)nnX_(3))`
`+P(X_(1)^(C)nnX_(2)nnX_(3))+P(X_(1)nnX_(2)nnX_(3)^(C))=1/4`
(1) `P(X_(1)^(C)//X)=(P(XnnX_(1)^(C)))/(P(X))=(1//32)/(1//4)=1/8`
(2) P [exactly two engines of the shp are functioning
`|X]=(7//32)/(1//4)=7/8`
(3) `P((X)/(X_(2)))=(P(XnnX_(2)))/(P(X_(2)))=(5//32)/(1//4)=5/8`
(4) `P((X)/(X_(1)))=(P(XnnX_(1)))/(P(X_(1)))=(7//32)/(1//2)=7/16`