Two special relationships between events make the probability formulas much simpler. Mutually exclusive events can’t both happen — so their intersection is zero and the union is just an addition. Independent events don’t affect each other — so their intersection is just a multiplication. The trap to avoid: these are different ideas, not the same one. Knowing which formula belongs to which is most of the battle.
📘 What you need to know
Mutually exclusive: the events cannot both occur. So P(A ∩ B) = 0.
For mutually exclusive events: P(A ∪ B) = P(A) + P(B) (no overlap to subtract).
Independent: one occurring does not affect the probability of the other. So P(A ∩ B) = P(A)P(B).
For independent events: P(A|B) = P(A) and P(B|A) = P(B).
Test for independence: check whether P(A ∩ B) = P(A)P(B).
Both formulas are in the formula booklet.
For any two events: P(A) = P(A ∩ B) + P(A ∩ B′).
Don’t confuse them: mutually exclusive ≠ independent. They’re separate properties.
Mutually exclusive vs independent
These two words get mixed up constantly. One is about overlap, the other about influence.
Mutually exclusive
can’t both
No overlap — if one happens the other can’t. P(A ∩ B) = 0. E.g. rolling a 6 vs rolling a prime.
Independent
don’t affect
One has no effect on the other. P(A ∩ B) = P(A)P(B). E.g. tails on flip 1 vs tails on flip 2.
mutually exclusive: no overlap
Mutually exclusive circles never touch — there’s no region where both happen.
🧠 Memory aid — “exclusive = add, independent = multiply”
Mutually exclusive → ADD the probabilities (P(A) + P(B), because the overlap is 0). Independent → MULTIPLY them (P(A)P(B), for the intersection). If a question says “or” with exclusive events, add; if it says “and” with independent events, multiply.
The two formulas
Mutually exclusive — unionP(A ∪ B) = P(A) + P(B)
because P(A ∩ B) = 0 — in the formula booklet ✓
Independent — intersectionP(A ∩ B) = P(A)P(B)
in the formula booklet ✓
🤔 Can two events be both mutually exclusive AND independent?
Almost never (only in trivial cases where one event has probability 0). If they’re mutually exclusive, then A happening means Bcan’t happen — so knowing A occurred drops B‘s chance to 0. That’s a big effect, which is the opposite of independent. So treat them as opposite situations: exclusive events strongly influence each other, independent events don’t influence each other at all.
Testing for independence
To check if two events are independent, see whether the multiplication rule actually holds.
🧭 Recipe — is it independent?
Work out P(A)P(B) from the given probabilities.
Compare with the actual P(A ∩ B).
Equal → independent; not equal → not independent.
Useful identity: for any two events, P(A) = P(A ∩ B) + P(A ∩ B′) — the part of A inside B plus the part outside B.
Worked examples
WE 1
Find a probability using independence
Two events Q and R have P(Q) = 0.8 and P(Q ∩ R) = 0.1. Given Q and R are independent, find P(R).
independent → P(Q ∩ R) = P(Q)P(R)0.1 = 0.8 × P(R)P(R) = 0.1 / 0.8P(R) = 0.125 (or 1/8)use the multiply rule and solve for the unknown.
WE 2
Mutually exclusive with a relationship
Two events S and T have P(S) = 2P(T). They are mutually exclusive and P(S ∪ T) = 0.6. Find P(S) and P(T).
mutually exclusive → P(S ∪ T) = P(S) + P(T)0.6 = P(S) + P(T)substitute P(S) = 2P(T)0.6 = 2P(T) + P(T) = 3P(T)P(T) = 0.2P(T) = 0.2, P(S) = 0.4add (exclusive), then use the relationship to solve.
WE 3
Test whether events are independent
P(A) = 0.5, P(B) = 0.4, P(A ∩ B) = 0.2. Are A and B independent?
compute P(A)P(B)0.5 × 0.4 = 0.2compare with P(A ∩ B)P(A ∩ B) = 0.2 = P(A)P(B) ✓yes — independentthey match, so the multiply rule holds.
WE 4
Not independent
P(A) = 0.6, P(B) = 0.5, P(A ∩ B) = 0.2. Are A and B independent?
compute P(A)P(B)0.6 × 0.5 = 0.3compare with P(A ∩ B) = 0.20.3 ≠ 0.2 ✗no — not independentthe product doesn’t match the actual intersection.
WE 5
Mutually exclusive “or”
When rolling a fair dice, A = “roll a 6” and B = “roll a prime (2, 3, 5)”. Find P(A ∪ B).
can both happen? no → mutually exclusiveP(A) = 1/6, P(B) = 3/6add (exclusive)P(A ∪ B) = 1/6 + 3/6 = 4/6P(A ∪ B) = 2/36 isn’t prime, so no overlap — just add.
Mutually exclusive ⇒ P(A ∩ B) = 0; independent ⇒ P(A ∩ B) = P(A)P(B).
Test independence by checking P(A ∩ B) = P(A)P(B).
Read the wording: “cannot both” = exclusive; “does not affect” = independent.
They’re not the same property — don’t assume one implies the other.
Set up an equation when a relationship like P(S) = 2P(T) is given, then solve.
⚠ Common mistakes
Confusing mutually exclusive with independent. They’re different — and usually opposites.
Using P(A) + P(B) when events overlap. That only works if they’re mutually exclusive.
Multiplying when events aren’t independent.P(A)P(B) needs independence.
Assuming “independent” means P(A ∩ B) = 0. That’s mutually exclusive, not independent.
Not checking the test before declaring independence — verify with the formula.
Forgetting to use a given relationship (e.g. P(S) = 2P(T)) to form an equation.
Next up — Conditional Probability. So far the events have been independent or exclusive; next you’ll handle events that do affect each other. P(A|B) — “A given B” — uses the formula P(A|B) = P(A ∩ B)P(B), and is the key to “without replacement” problems.
Need help with Probability?
Get 1-on-1 help from an IB examiner who knows exactly what Paper 1 & 2 are looking for.
