Independent events

Independence is about influence, not about whether events can happen together. Events A and B are independent when P(A and B) = P(A) × P(B), which is equivalent to saying P(A given B) = P(A). Flipping a coin and rolling a die are independent: the coin result tells you nothing about the die. Drawing two cards from a deck without replacement is not independent, because removing the first card changes what is left for the second.

The multiplication rule makes calculations straightforward. The probability of flipping heads twice is (1/2)(1/2) = 1/4. The probability of rolling a 6 and then drawing a red card is (1/6)(1/2) = 1/12. Extending to several events, the probability of getting at least one success in n independent trials is often easiest to find as 1 minus the probability of no successes — for example, the chance of at least one head in four flips is 1 − (1/2)⁴ = 15/16.

Independent is not the same as mutually exclusive. Mutually exclusive events cannot happen at the same time, so P(A and B) = 0; knowing that A happened tells you B definitely did not, which means mutually exclusive events with nonzero probabilities are always dependent. Sampling with replacement produces independent draws; sampling without replacement produces dependent ones, though the dependence becomes negligible when the population is very large relative to the sample.

Independence appears throughout competition and certification math. The AMC 8 and AMC 12 test it in combinatorics and probability problems, often requiring you to decide whether trials are independent before multiplying, and the AMC 12 pushes into conditional probability and advanced independence. The Praxis Core Math exam tests the same multiplication rule in its data analysis and probability section.