In California, it "never" rains during the summer (one summer when I was there it rained one day every month, and not very hard). If I am planning a picnic, I do not care that it rains one eighth of the days in California; rather that it rains one quarter of the days in September, or one thirtieth of the days in June, depending on when I want my picnic. This is the essence of conditional probability.
The probability of A conditioned on B, denoted P(A|B), is equal to P(AB)/P(B). The division provides that the probabilities of all outcomes within B will sum to 1. Conditioning restricts the sample space to those outcomes which are in the set being conditioned on (in this case B). Note that P(A|B) is not equal to P(B|A); the set after the vertical bar is the set one is conditioning on.
Example: If P(A)=.5, P(B)=.4, and P(AB)=.2 (hence P(AUB)=.7 and P(A'B')=.3), P(A|B)=.2/.4=.5 and P(B|A)=.2/.5=.4.
Example: If P(A)=.5 andP(B|A)=.4, P(BA)=.4 × .5 =.2. (of course AB=BA).
Examples: If P(A)=.5, P(B)=.4, and P(AB)=.2, then P(A|B)=.2/.4=.5 = P(A) and A and B are independent. If P(A)=.6, P(B)=.4, and P(AB)=.2, then P(A|B)=.2/.4=.5 which is not equal to .6=P(A), and A and B are not independent.
Competencies: If P(A)=.5, P(B)=.4, and P(AB)=.3, what is P(A|B)? Are A and B independent?
If P(A)=.6, P(B)=.4, and P(A|B)=.5, what is P(AB)?
If A and B are independent and P(A)=.3, P(B)=.6; P(AB)=?
Reflection: What are the relationships among independence, complementary, and mutually exclusive (disjoint)?
Challenge: If P(A)=.4, P(B)=.7, and P(AUB)=.9; what is P(A|B)?
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