Conditional probability and the product rule

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.

The definition of conditional probability, P(A|B)=P(AB)/P(B), can be rewritten
as P(AB)=P(A|B)P(B). This is the product rule.

Example: If P(A)=.5 andP(B|A)=.4, P(BA)=.4 × .5 =.2. (of course AB=BA).

Two events A and B are called independent if P(A|B)=P(A), i.e., if
conditioning on one does not effect the probability of the other. Since
P(A|B)=P(AB)/P(B) by definition, P(A)=P(AB)/P(B) if A and B are independent,
hence P(A)P(B)=P(AB); this is sometimes given as the definition of
independence. Rearranging this last equation as P(AB)/P(A)=P(B), we see that
if P(A|B)=P(A), then also P(B|A)=P(B).

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.

• If A and B are independent, P(AB)=P(A)P(B) (because P(A|B)=P(A) for
  independent events). (Example: If A and B are independent and P(A)=.3 and
  P(B)=.6, then P(AB)=.3 × .6 = .18.)
• N.B.: If A and B are disjoint (which includes the case where A and B
  are complementary)
  • P(AB)=0
  • P(A|B)=0=P(B|A)

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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Questions?