Independence

P(A ∩ B) = P(A) P(B) — the multiplication rule.

The idea

Two events are independent when learning that one occurred tells you nothing about the probability of the other. In terms of conditional probability, $\mathbb{P}(A \mid B) = \mathbb{P}(A)$: conditioning on $B$ leaves the probability of $A$ unchanged.

Multiplying both sides by $\mathbb{P}(B)$ clears the denominator and gives the form usually taken as the definition: $\mathbb{P}(A \cap B) = \mathbb{P}(A)\,\mathbb{P}(B).$ Two events are independent if and only if their probabilities multiply. This form treats $A$ and $B$ symmetrically, so neither is singled out as the one conditioned on, and it still makes sense when $\mathbb{P}(B) = 0$, where the conditional form is undefined.

Independence is an arithmetic test, not a judgement about the experiment. Events produced by separate physical mechanisms — two coins, two dice — do multiply, which is what makes the rule useful in practice. But two events in a single experiment can multiply by numerical accident, and two events that feel unrelated can fail to. To decide, compute $\mathbb{P}(A \cap B)$ and compare it with $\mathbb{P}(A)\,\mathbb{P}(B)$.

Ways to work on it

• Walkthrough. Coin flips — the canonical independent events.
• Practice. One die, two events — independent or not?
• Hardest. Use independence to find both P(A ∩ B) and P(A ∪ B).

Not sure where to start? Take the ten-question placement test.