Independent and Mutually Exclusive Events

Two ideas that are easy to mix up. Mutually exclusive events cannot happen together, so their probabilities add. Independent events do not affect each other, so their probabilities multiply.

Cambridge 9709 Paper 5Edexcel YMA01 WST01Edexcel 9MA0 Paper 3AQA 7357 Paper 3≈ 30 minRevision 5 · updated 2026-10-27

Section 1 of 14

Learning objectives

Know that mutually exclusive events have $P(A\cap B)=0$ and $P(A\cup B)=P(A)+P(B)$ .
Know that independent events have $P(A\cap B)=P(A)\,P(B)$ , equivalently $P(A\mid B)=P(A)$ .
Test two events for independence with numbers.
Explain why events with non-zero probabilities cannot be both exclusive and independent.

Pause and think

Roll one die. $A$ : "the score is even". $B$ : "the score is $5$ ". They can't both happen. Does knowing $B$ happened tell you anything about $A$ ?

Section 2 of 14

Core idea

Start hereDoes one event tell you anything about the other?

A coin landing heads tells you nothing about tomorrow's weather: independent. Rolling a 6 and rolling a 1 on the same throw can't both happen: mutually exclusive. People confuse the two, but they're almost opposites.

The big idea

Independent: $P(A\cap B)=P(A)P(B)$ , equivalently $P(B\mid A)=P(B)$ . Mutually exclusive: $P(A\cap B)=0$ , so $P(A\cup B)=P(A)+P(B)$ .

1. Definitions

Mutually exclusive (left) versus overlapping (right)

Mutually exclusive events never overlap. Independent events (with non-zero probabilities) always overlap, by exactly

P(A)P(B)

Why the product test

$\text{independent}\iff P(B\mid A)=P(B)$
Knowing $A$ doesn't change $B$ 's chance.
$\frac{P(A\cap B)}{P(A)}=P(B)$
Conditional formula.
$P(A\cap B)=P(A)P(B)$
Multiply by $P(A)$ .

Worked example $P(A)=0.4$ , $P(B)=0.5$ , $P(A\cap B)=0.2$

Independent? Exclusive? $P(A\cup B)$ ?

Goal: Classify.
Strategy: Test numerically.

$0.4\times0.5=0.2=P(A\cap B)$
Independent ✓.
$P(A\cap B)\neq0$
Not exclusive.
$0.4+0.5-0.2=0.7$
Addition rule.

AnswerIndependent, not exclusive;

0.7

✓ Check

P(B\mid A)=\frac{0.2}{0.4}=0.5=P(B)

✓.

Worked exampleDie: A = even, B = greater than 4

Are A and B independent?

Goal: Yes/no.
Strategy: List outcomes.

$P(A)=\tfrac12,\ P(B)=\tfrac13,\ A\cap B=\{6\}\Rightarrow\tfrac16$
Count.
$\tfrac12\times\tfrac13=\tfrac16$
Product matches.

AnswerIndependent.

✓ CheckAmong

\{5,6\}

, half are even, same as overall ✓.

Edge caseBoth? Two events with non-zero probabilities can't be both independent and exclusive: exclusive gives

P(A\cap B)=0

but independence needs

P(A)P(B)>0

Pause and think

$P(A)=0.3$ , $P(B)=0.5$ , independent. $P(A\cup B)$ ?

Summary

Independent: product rule.
Exclusive: no overlap.

Independent: One does not affect the other.
Mutually exclusive: Cannot both occur.

HistoryDe Moivre defines independence. The French mathematician Abraham de Moivre, who fled to London as a Protestant refugee, gave one of the first clear definitions of independent events in The Doctrine of Chances (1718), a book written partly for gamblers. He earned a living in coffee houses solving probability problems for a fee.

Fun factThe roulette wheel has no memory. On 18 August 1913 at the Monte Carlo casino, the ball landed on black

26

times in a row. Gamblers lost fortunes betting on red, sure it was "due". But spins are independent: after

25

blacks, red is no more likely than before. Believing otherwise is called the gambler's fallacy, or the Monte Carlo fallacy.

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