1.

A and B are two events such that odds odds against A are 2 : 1 odds in favour of `A cup B` are 3 : 1. If `x le P(B) le y`, then the ordered pair (x, y) isA. `(5//12, 3//4)`B. `(2//3, 3//4)`C. `(1//3, 3//4)`D. none of these

Answer» Correct Answer - A
We have,
`P(A)=(1)/(3) " and" P(A cup B)=(3)/(4)`
`therefore P(A cup B)=P(A)+P(B)-P(A cap B)`
`implies (3)/(4)=(1)/(3)+P(B)-P(A cap B)`
`implies (5)/(12)=P(B)-P(A cap B)`
`implies P(B)=(5)/(12)+P(A cap B) implies P(B) ge 5//12`
Again,
`P(B)=(5)/(12)+P(A cap B)`
`implies P(B) le 5//12+P(A) " " [because P(A cap B) le P(A)]`
`implies P(B) le (5)/(2)+(1)/(3)=(3)/(4)`
From (i) and (ii), we obtain `5//12 le P(B) le 3//4`
Hence, `x=5//12 " and" y=3//4`


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