Events A1 A2 and A3 form a partition of the sample space wit

Events A1, A2 and A3 form a partition of the sample space with probabilities P(A1)=0.3, P(A2)=0.1, P(A3)=0.6. If E is an event in with P(E|A1)=0.3, P(E|A2)=0.5, P(E|A3)=0.6, compute

P(E) =

P(A1/E)

P(A2/E)

P(A3/E)

Solution

As

P(E) = P(A1)P(E|A1) + P(A2)P(E|A2) + P(A3)P(E|A3)

= 0.3*0.3 + 0.1*0.5 + 0.6*0.6

P(E) = 0.5 [ANSWER]

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Now,

P(A1|E) = P(A1)P(E|A1) / P(E)

= 0.3*0.3/0.5 = 0.18 [answer]

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Now,

P(A2|E) = P(A2)P(E|A2) / P(E)

= 0.1*0.5/0.5 = 0.1 [answer]

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Now,

P(A3|E) = P(A3)P(E|A3) / P(E)

= 0.6*0.6/0.5 = 0.72 [answer]