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Question

# If A and B are two events such that 2P(A)=3P(B), where 0<P(A)<P(B)<1, then which one of the following is correct?

A
P(A|B)<P(B|A)<P(AB)
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B
P(AB)<P(B|A)<P(A|B)
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C
P(B|A)<P(A|B)<P(AB)
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D
P(AB)<P(A|B)<P(B|A)
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Solution

## The correct option is A P(A∩B)<P(B|A)<P(A|B)Given:−2P(A)=3P(B)⇒ P(A)=3P(B)2 →⟨1⟩⇒∴P(A)>P(B)Now we know:−P(A|B)=P(A∩B)P(B) Similarly,P(B|A)=P(A∩B)P(A)⇒P(B|A)=2P(A∩B)3P(B) [Using⟨1⟩]∴ From above we conclude P(A|B)>P(B|A)Now also we know, 0<P(A)<1 and 0<P(B)<1∴P(B|A)=P(A∩B)P(A)>P(A∩B)Similarly, P(A|B)=P(A∩B)P(B)>P(A∩B)∴P(A|B)>P(B|A)>P(A∩B)

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