Probability-33-Baye's theorem

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Event A occurs with a probability of 0.4. The conditional probability that A occurs given that B occurs is 0.5, while the conditional probability that A occurs given that b does not occur is 0.2. What is the conditional probability that b occurs given that A occurs?

Solution: P(A)=0.4 , P(A|B) = 0.5 and P(A|Bc) = 0.2 given. we need to find P(B|A) = ?.

By Baye's theorem, P(B|A) = (P(B)P(A/B))/P(A) ----------(1) .

so, we need to find P(B) in order to get the value of P(B|A).

P(A|B) = 0.5

or,p(AᴖB)/p(B)=0.5

so p(AᴖB) = 0.5 p(B).

so, p(A ᴖBc)=p(A)-p(AᴖB)=0.4-0.5 p(B)

now, P(A|Bc) = 0.2 given. so, P(AᴖBc)/P(Bc) = 0.2 or, [0.4-0.5 P(B)]/P(Bc) = 0.2

or, [0.4-0.5 P(B)]/[1-P(B)] = 0.2

or, 0.4-0.5 P(B) = 0.2-0.2P(B)

or, -0.5P(B) + 0.2P(B) = 0.2-0.4

or, 0.3P(B) = 0.2 or, P(B) = 0.2/0.3 = 0.7

Now putting P(B)=0.7 in (1) we get, P(B|A) =(0.7×0.5)/0.4 = 0.8

So, the probability that b occurs given that A occurs is 0.8 .