A ship is fitted with three engines `E_(1),E_(2),and E_(3)` the engines function independently of each othe with respectively probability `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 functioning. Which of the following is (are) true?

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`