Independent Events

Flip a fair coin and it comes up heads. Flip it again — is the second flip any more likely to be tails now, to "even things up"? No. The coin has no memory. The first result tells you nothing about the second. Two events like this, where one happening does not change the other's probability, are called independent.

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}. Multiplying by a number below 1 always makes things smaller — which fits, since asking for two things at once is harder than asking for one.

When two events do not affect each other:

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}. Counting outcomes and multiplying probabilities agree, which is a good sign the rule is trustworthy.

Three worked examples

1. A six and a head. Roll a die and flip a coin. The die doesn't care what the coin does, so they're independent:

P(\text{six and heads}) = \tfrac{1}{6} \times \tfrac{1}{2} = \tfrac{1}{12}.

2. Drawing with replacement. A bag holds 3 red and 2 blue counters. You draw one, put it back, shake, and draw again. Because you replaced the first counter, the second draw faces the exact same bag — the draws are independent, each with P(\text{red}) = \tfrac{3}{5}:

P(\text{two reds}) = \tfrac{3}{5} \times \tfrac{3}{5} = \tfrac{9}{25} = 0.36.

3. Several events in a row. Keep multiplying for as many independent events as you like. The chance of throwing four heads in a row is

\tfrac{1}{2} \times \tfrac{1}{2} \times \tfrac{1}{2} \times \tfrac{1}{2} = \tfrac{1}{16}.

Where this shows up

Multiplying independent chances powers a huge amount of real life. A password of several independently chosen characters is hard to guess because each position multiplies the possibilities. Backup systems on a plane or in a data centre are built so that two independent parts failing at once — say 0.01 \times 0.01 = 0.0001 — is vanishingly rare. Even genetics leans on it: the chance of inheriting two particular independent traits is the product of their separate chances.

The habit worth building: whenever a question asks for several things happening together and one doesn't sway the others, multiply — and expect the answer to shrink fast as you pile on more events.

Two traps catch almost everyone here.

Trap one: the multiply rule needs independence. Suppose you draw two cards but do not put the first back. Now the second draw is different: after taking one ace, only 3 aces remain among 51 cards, so the odds have changed. You cannot just do \tfrac{4}{52} \times \tfrac{4}{52}. Events where one outcome shifts the other's probability need conditional probability. The tell-tale phrase is "without replacement".

Trap two: don't confuse the two rules. Independent (one doesn't affect the other) goes with multiply, for AND. Mutually exclusive (can't both happen) goes with add, for OR. They are completely different ideas — in fact two events that can't both happen are about as non-independent as it gets, because if one happens the other's chance drops to zero.

At a roulette wheel, red comes up five times running. The table buzzes — "black is due!" — and people pile their chips on black. This is the famous gambler's fallacy, and the multiplication rule shows exactly why it's wrong.

Each spin is independent: the wheel has no memory of the last five reds. Black's chance on the next spin is still just under one half, precisely as it was on spin one. Five reds in a row is unlikely up front — (\tfrac{18}{37})^5 \approx 0.027 — but that's the chance before you start. Once five reds have already landed, the next spin doesn't know or care.

The same multiply-the-tiny-numbers rule explains the flip side: winning a big lottery twice, or a string of rare independent things all going wrong at once (as in some disasters), is astronomically unlikely — because you multiply already-tiny probabilities together and the result becomes almost unimaginably small.

See it explained