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)/(10)`
C. `P(X//X_(2))=(5)/(16)`
D. `P(X//X_(1))=(7)/(16)`

Answer

D

Explanation

Given, `P(X_(1))=(1)/(2),P(X_(2))=(1)/(4)` and `P(X_(3))=(1)/(4)`
`therefore P(X)= {P(X_(1)nn X_(2)nn bar(X)_(3)+P(X_(1)nn bar(X)_(2)nn X_(3))+P(bar(X)_(1)nn X_(2)nn X_(3))+P(X_(1)nn X_(2)nn X_(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))=(1)/(4)`
(a) Now, `P(X_(1)^(c )//X)=P((X_(1)^(c )nn X)/(P(X)))`
`= (P(bar(X)_(1)nn X_(2)nn X_(3)))/(P(X))=((1)/(2).(1)/(4).(1)/(4))/((1)/(4))=(1)/(8)`
(b) P (exactly two engines of the ship are functioning)
`(P(X_(1)nn X_(2)nn bar(X)_(3))+P(X_(1)nn bar(X)_(2)nn X_(3))+P(bar(X)_(1)nn X_(2)nn X_(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(X nn X_(2)))/(P(X_(2)))`
`("P (ship is operating, when " E_(2) "functioning)")/(P(X_(2)))`
`(P(X_(1)nn X_(2)nn bar(X)_(3))+P (bar(X)_(1)nn X_(2) nn X_(3))+P(X_(1)nn X_(2)nn X_(3)))/(P(X_(2)))`
`=((1)/(2).(1)/(4).(3)/(4)+(1)/(2).(1)/(4).(1)/(4)+(1)/(2).(1)/(4).(1)/(4))/((1)/(4))=(5)/(8)`
(d) `P((X)/(X_(1)))=(P(X nn X_(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)`