Independence means that knowing one event or variable gives no information about another. For independent events, their joint probability equals the product of their individual probabilities: p(X, Y) = p(X)p(Y). section 5.3.3 section 5.4
For random variables, test every valid combination: independence holds exactly when p(X = x, Y = y) = p(X = x)p(Y = y). Equivalently, conditioning on one variable does not change the probability of the other. section 6.1.1 section 6.1.3
Use these distinctions when reading probability tables or equations:
- Joint probability describes a particular combination occurring together; marginal probability sums over the other variable. section 6.1.1
- Dependent variables violate the product condition because knowing one helps predict the other. section 5.4.2
- Independence of events differs from linear independence: independent equations or rows concern matrix rank and degeneracy, not probability. section 4.5.5
Weight and home distance
Suppose weight categories have probabilities 0.25, 0.5, and 0.25, while home-distance categories have probabilities 0.2, 0.6, and 0.2. Under independence, the joint probability of weight below 60 kg and a home less than 5 km away is 0.25 × 0.2 = 0.05. The same multiplication produces every cell in the joint table. section 5.4
Weight and height
Weight and height are dependent when a joint entry differs from the product of its marginals. For example, p(E1, F1) = 0.2, whereas p(E1)p(F1) = 0.26 × 0.26, so the variables are not independent. section 6.1.1
