Independent Events

Two events are independent when the outcome of one does not change the probability of the other. Flip a coin, then flip it again: the first result tells you nothing about the second — the two flips are completely separate.

For independent events, the probability that both happen is found by multiplying their probabilities. This is the AND rule:

P(A \text{ and } B) = P(A) \times P(B)

Each coin lands heads with probability \tfrac{1}{2}, so two heads in a row has probability \tfrac{1}{2} \times \tfrac{1}{2} = \tfrac{1}{4}.

When two events do not affect each other:

they are independent, and the probability that both happen is the product P(A \text{ and } B) = P(A) \times P(B) — you multiply for AND;
contrast this with mutually exclusive events, where you add for OR: P(A \text{ or } B) = P(A) + P(B);
for several independent events you just keep multiplying — three independent heads is \tfrac{1}{2} \times \tfrac{1}{2} \times \tfrac{1}{2} = \tfrac{1}{8}.

Two coins, four outcomes

Flip two fair coins. Each can land heads or tails, so together they make four equally likely outcomes. Just one of them is two heads.

Counting gives P(\text{HH}) = \tfrac{1}{4} — exactly what the AND rule predicts, \tfrac{1}{2} \times \tfrac{1}{2}.