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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).
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Solution
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`
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