If A and B are two events such that P(A) = 3/4 and P(B) = 5/8, then (correct options may be more than one)

Options

A. `P(A uu B) ge 3//4`
B. `P(A' nn B) le 1//4`
C. `1//8 le P(A nn B') le 3//8`
D. `3//8le P(A nn B) le 5//8`

Answer

A, B, C, D

Explanation

`A sube A uu B`
implies `P(A) le P(A uu B)`
`implies P(A uu B) ge (3)/(4)`
Also `P(A nn B) = P(A) + P(B) - P(A uu B) ge P(A) + P(B) - 1`
= `(3)/(4) + (5)/(8) - 1 = (3)/(8)`
Now, `A nn B sube B`
`implies P(A nn B) le P(B) = (5)/(8)`
`therefore (3)/(8) le P(A nn B) le (5)/(8)" (1)"`
and `P(A nn B') = P(A) - P(A nn B)`
or `(3)/(4) - (5)/(8) leP(A nn B') le (3)/(4) - (3)/(8)`
or `(1)/(8) le P(A nn B') le(3)/(8)`
`therefore P(A nn B)= P(B) - P(A' nn B) " "["Using Eq. (1)"]`
or `(3)/(8) le P(B) - P(A' nn B) le (5)/(8)`
`or 0 le P(A' nn B) le (1)/(4)`