Posted by Preet on Monday, July 23, 2012 at 1:11am.
In the following, A' will be used in place of Ac 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.
Please check and understand every line. Post if you have questions.
Related Questions
Probability - Let A and B be events in a sample space S such that P(A) = 0.6, P...
Math - Can someone please explain this problem? Consider the experiments ...
probability - Let A and B two mutually exclusive events of a sample space S. If ...
Math - ) In an experiment, a pair of dice is rolled and the total number of ...
math - Assume that your sample space is S = {O1, O2, O3, O4, O5, O6 } with ...
stats - you are given the probabilities listed below P(A)= 0.25, P(B) = 0.30, P(...
statistics - Assume that your sample space is S = {O1, O2, O3, O4, O5, O6 } with...
Math - In an experiment, a pair of dice is rolled and the total number of points...
m - If events A and B are from the same sample space, and if and , can events A ...
math - Below is a sample space for a family with 3 children. BGG stands for the...
For Further Reading
