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` ,and 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" ]=(7)/(8)`
C. `P[X|X_(2)]=(5)/(16)`
D. `P[X|X_(1)]=(7)/(16)`

Answer

B, D

Explanation

PLAN It is based on law of total probability and Bay's Law.
Description of Situation It is given that ship would work of atleast two of engines must work. If X be event that the ship works. Then, `X rArr` either any two of `E_(1),E_(2),E_(3)` works or all three engines `E_(1),E_(2),E_(3)` works.
Given, `P(E_(1))=(1)/(2),P(E_(2))=(1)/(4),P(E_(3))=(1)/(4)`
`P(X) = {{:(P(E_(1) nn E_(2)nnbarE_(3))+P(E_(1)nn barE_(2) nnE_(3))),(+P(barE_(1)nnE_(2)nnE_(3))+P(E_(1)nn E_(2)nnE_(3))):}}`
`=((1)/(2)*(1)/(4)*(3)/(4)+(1)/(2)*(3)/(4)*(1)/(4)+(1)/(2)*(1)/(4)*(1)/(4))+((1)/(2)*(1)/(4)*(1)/(4))`
Now, `(a)P(X_(1)^(e)//X)`
`=P((X_(1)^(e)nnX)/(P(X)))=(P(barE_(1)nnE_(2)nnE_(3)))/(P(X))=((1)/(2)*(1)/(4)*(1)/(4))/((1)/(4))=(1)/(8)`
(b) (exactly two engines of the ship are functioning)
`=(P(E_(1)nnE_(2)nnbarE_(3))+P(E_(1)nnbarE_(2)nnE_(3))+P(barE_(1)nnE_(2)nnE_(3)))/(P(X))`
`=((1)/(2)*(1)/(4)*(3)/(4)+(1)/(2)*(3)/(4)*(1)/(4)+(1)/(2)*(1)/(4)*(1)/(4))/((1)/(4))=(7)/(8)`
(c) `P((X)/(X_(2)))=(P(XnnX_(2)))/(P(X_(2)))`
`=(P("Ship is operating with" E_(2) " Function"))/(P(X_(2)))`
`=(P(E_(1)nnE_(2)nnbarE_(3))+P(barE_(1)nnE_(2)nnE_(3))+P(E_(1)nnE_(2)nnE_(3)))/(P(E_(2)))`
`=((1)/(2)*(1)/(4)*(3)/(4)+(1)/(2)*(1)/(4)*(1)/(4)+(1)/(2)*(1)/(4)*(1)/(4))/((1)/(4))=(5)/(8)`
` P(X//X_(1))=(P(XnnX_(1)))/(P(X_(1)))=((1)/(2)*(1)/(4)*(1)/(4)+(1)/(2)*(3)/(4)*(1)/(4)+(1)/(2)*(1)/(4)*(3)/(4))/(1//2)`
`(7)/(16)`