## Conditional Probability, Bayes’ Rule, and Independence

1.10.1. Conditional Probability

Consider a statistical experiment with probability space { ^, &, P}, and suppose it is known that the outcome of this experiment is contained in a set B with P (B) > 0. What is the probability of an event A given that the outcome of the experiment is contained in B? For example, roll a dice. Then ^ = {1, 2, 3, 4, 5, 6}, & is the a-algebra of all subsets of ^, and P({«}) = 1/6 for rn = 1,2, 3, 4, 5, 6. Let B be the event The outcome is even (B = {2, 4, 6}), and let A = {1, 2, 3 }. If we know that the outcome is even, then we know that the outcomes {1, 3} in A will not occur; if the outcome is contained in A, it is contained in A П B = {2}...

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