Let E and F be two independent events. The probability that exactly one of them occurs is `11//25` and the probability of none of them occurring is `2//25.` If P(T) denotes the probability of occurrence of the event T, then

Options

A. `P(E)=(4)/(5),P(F)=(3)/(5)`
B. `P(E)=(1)/(5),P(F)=(2)/(5)`
C. `P(E)=(2)/(5),P(F)=(1)/(5)`
D. `P(E)=(3)/(5),P(F)=(4)/(5)`

Answer

A, B

Explanation

`P(E uu F) - P( E nn F) = (11)/(25) " "…. (i) ["i.e. only E or only F"]`

Neither of them occurs ` = (2)/(25)`
`rArr P(bar(E) nn bar (F)) = (2)/(25) " "...(ii)`
From Eq. (i), `P(E) + P(F) - 2P (E nn F) = (11)/(25) " " .....(iii)`
From Eq. (ii), `(1-P(E))(1-P(F)) = (2)/(25)`
`rArr 1-P(E) - P(F) + P(E) * P(F) = (2)/(25) " "..(iv)`
From Eqs. (iii) and (iv),
`P(E) + P(F) = (7)/(5) "and " P(E) * P(F) = (12)/(25)`
`therefore P(E) * [(7)/(5) - P(E)] = (12)/(25)`
`rArr (P(E))^(2) - (7)/(5) P(E) + (12)/(25) = 0`
`rArr [P(E) - (3)/(5)] [P(E) - (4)/(5)] = 0`
`therefore P(E) = (3)/(5) "or" (4)/(5) rArr P(F) = (4)/(5) " or" (3)/(5)`