Question

# If X and Y are two events such that P(X|Y)=12,P(Y|X)=13 and P(X∩Y)=16. Then, which of the following is/are correct ?

A
P(XY)=23
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B
X and Y are independent
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C
X and Y are not independent
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D
P(XCY)=13
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Solution

## The correct options are A P(X∪Y)=23 B X and Y are independent(i) Conditional probability. i.e. P(A|B)=P(A∩B)P(B) (ii) P(A∪B)=P(A)+P(B)−P(A∩B) (iii) Independent event, then P(A∩B)=P(A).P(B) Here, P(X|Y)=12,P(Y|X)=13 and P(X∩Y)=16 ∴P(X|Y)=P(X∩Y)P(Y)⇒12=16P(Y)⇒P(Y)=13 ...(i) P(Y|X)=13⇒P(X∩Y)P(X)=13 ⇒16=13P(X) ∴P(X)=12 ...(ii) P(X∪Y)=P(X)+P(Y)−P(X∩Y) =12+13−16=23 ...(iii) P(X∩Y)=16 and P(X).P(Y)=12.13=16⇒P(X∩Y)=P(X).P(Y) i.e. independent events ∴P(XC∩Y)=P(Y)−P(X∩Y)=13−16=16

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