Question

If A and B are two independent events such that `P(barA∩ B) =2/15 and P(A ∩ barB) = 1/6`, then find P(A) and P(B).

Answer 1

It is given that A and B are independent events.

`P(barA∩B)=2/15`

`∴P(barA) P(B)=2/15         .....(1)`

Also, `(P∩barB)=1/6`

`∴P(A) P(barB)=1/6`

`⇒P(A)=1/(6[1−P(B)])        .....(2)`

From (1), we have

`[1−P(A)]P(B)=2/15`

`[1−1/(6[1−P(B)])]P(B)=2/15`

`{(6−6P(B)−1)/(6[1−P(B)])}P(B)=2/15`

`5 P(B)−6[P(B)]^2=(12[1−P(B)])/15`

`25P(B)−30[P(B)]^2=4−4P(B)`

`30[P(B)]^2−29P(B)+4=0`

`30[P(B)]^2−24P(B)−5P(B)+4=0`

`6P(B)[5P(B)−4]−1[5P(B)−4]=0`

`[5P(B)−4] [6P(B)−1]=0`

`P(B)=4/5, 1/6`

For P(B) = 4/5, using (2), we have

`P(A)=1/(6[1−P(B)] )          `

`\=1/(6[1−4/5])       `

`=5/6`

For P(B) = 1/6, using (2), we have

`P(A)=1/(6[1−16] )     `

`=1/5`

`∴ P(A)=5/6, P(B)=4/5 or P(A)=1/5, P(B)=1/6`