Conditional probability and independence

P(A | B) is the probability of A among the cases where B happened; A and B are independent when knowing B does not change the probability of A.

P(A \given B) = \frac{P(A \cap B)}{P(B)}, \quad P(B) > 0

Definition of conditional probability.

$P(A \cap B)$: Probability that both happen.
$P(B)$: Probability of the conditioning event.

A,\,B \text{ independent} \iff P(A \cap B) = P(A)\,P(B) \iff P(A \given B) = P(A)

Independence.

$P(A)$: Unconditional (marginal) probability of A.

When to use

Two-way tables: condition on a row or a column and read the proportion within it.
Screening, risk and conversion questions ("of customers who saw the offer, what fraction bought?").
Checking whether two categorical variables are related (formally: chi-square test of independence).

How to interpret

Conditioning shrinks the sample space to the cases where B occurred. Independence in a table means the conditional distributions are the same in every row.

Common mistakes

Confusing $P(A \given B)$ with $P(B \given A)$ — e.g. P(positive test | disease) vs P(disease | positive test). Use Bayes' theorem to turn one into the other.
Dividing by the wrong total in a two-way table (grand total instead of the row or column total).
Concluding causation from dependence.

Example

In a survey, 30% of customers are members ( $P(M) = 0.3$ ) and 12% are members who renewed early ( $P(R \cap M) = 0.12$ ). $P(R \given M) = 0.12/0.3 = 0.4$ : 40% of members renewed early. If 25% of all customers renewed early, $P(R \given M) = 0.4 \ne 0.25$ , so membership and early renewal are not independent.