Lecture 2 - Ch 4 - Independency

A and B are independent events if and only if P(A|B) = P(A).

This is basically saying that A and B are independent only if the probability of A occurring given that B has occurred is the same probability as A happening in any case (even if B had not occurred). In other words, the probability of A occurring is not influenced by whether B happens or not. I.e. A is independent of B.

We also learned that if A is independent of B, then B is independent of A. Meaning,
P(A|B) = P(A) implies P(B|A) = P(B).

The proof the statement above was given as a pop quiz. See next post for the solution. (I know... you can't wait!)

The Multiplication Rule
Another check for independency can be derived as follows:
Since we know from the definition of conditional probability that

By multiplying both sides by P(B), we get: P(A|B)*P(B) = P(A/\B)
And we know that if the events are independent that P(A|B) = P(A), so substituting this on the left side of the equation, we get

The multiplication rule is more commonly used as a test of independence than the first definition, but we need to know and understand both definitions.

We went over an example of possible discrimination in promotions within the police department. That example is a good sample midterm question, so review it!