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A ship is fitted with three engines E(1)...

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?

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)`

Text Solution

Verified by Experts

The correct Answer is:
B, D
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)`
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