Independence

Mathematical definition of the conditional probability of A given that B is true:

$$P(A | B) = \frac{P(A \text{ and } B)}{P(B)} .$$

Definition. Two variables are independent if knowing one has no effect on your knowledge of the other, that is:

$$\text{A, B independent} \Rightarrow P(A | B) = P(A) ,$$

and similarly for B.

Math lovers prefer the to take the definition of independence to be

$$P(A \text { and } B) = P(A)\cdot P(B),$$

because it is symmetrical in A and B. See if you can see why the two definitions above are the same.

Think. What if $P(A \text{ and } B) \approx P(A)\cdot P(B)$ but they are not exactly equal? Later on we will learn tools for deciding the probability of your observations happening when A and B are actually independent.

Thinking Question. Are “race” and “wealth” independent variables? What evidence would you consider to decide?

Question. Why is independence important?