Question

Let A,B be events in a sample space. you are given the following probabilities:
P(A)=0.8, P(B|A)=0.3, P(B|A^c)=0.8
Find:
1) P(B)
2) P(A|B)
3) P(A^c|B^c)

Answer 1

In the following, A' will be used in place of A^c for ease of typography.

(1) since A and A' are partitions of A, we can apply the law of total probability
P(B)=P(B|A)*P(A)+P(B|A')*P(A')

(2) Use Bayes Theorem
P(A|B)
=P(A∩B)/P(B) ... def of cond. prob
=P(B∩A)/P(B) ... commutativity
=[P(B|A)*P(A)]/P(B) ... def of cond. prob

(3) We need to use

P(A'∩B')=1-P(A∪B) ... complement
and
P(A∪B)=P(A)+P(B)-P(A∩B) ... prob. of union of sets

P(A'|B')
=P(A'∩B')/P(B') .. def of cond.prob
=(1-P(A∪B))/(1-P(B)) ... compl
=(1-P(B∪A))/(1-P(B)) ... commutativity
=[1-(P(A)+P(B)-P(A∩B)]/(1-P(B)) ... union of sets
=[1-(P(A)+P(B)-P(B∩A)]/(1-P(B)) ... commutativity
=[1-(P(A)+P(B)-P(B|A)*P(A)]/(1-P(B)) ... def. of cond. prob.

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