S5.3StatisticsCore

Mutually Exclusive and Independent Events

Two special relationships between events simplify probability calculations. Mutually exclusive events cannot both happen; independent events do not affect each other. Each has its own “add” or “multiply” rule.

25 min Video by Zeeshan Zamurred Probability

Edexcel Statistics Y1 — Probability playlist (Zeeshan Zamurred)Watch the full walkthrough before the notes below.

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What you'll be able to do

Define mutually exclusive events
Define independent events
Use the addition rule for mutually exclusive events
Use the multiplication rule for independent events

Mutually exclusive events

Two events are $mutually exclusive$ if they cannot happen at the same time — their intersection is empty, so $P (A \cap B) = 0$ . For these, the addition formula simplifies.

P (A \cup B) = P (A) + P (B) (mutually exclusive)

No overlap to subtract.

1They cannot both occur on one roll.

Answeryes

Independent events

Two events are $independent$ if one happening does not change the probability of the other. For these, the probability of both is the product.

P (A \cap B) = P (A) \times P (B) (independent)

Multiply for “and” when independent.

0.5 \times 0.2

Answer

0.1

Tip — Mutually exclusive → ADD (for “or”). Independent → MULTIPLY (for “and”).

Telling them apart

Do not confuse the two: mutually exclusive events have $P (A \cap B) = 0$ (they never coincide), whereas independent events generally $do$ overlap, with $P (A \cap B) = P (A) P (B)$ .

Formula recap

P (A \cap B) = 0

Mutually exclusive.

P (A \cup B) = P (A) + P (B)

Add (mutually exclusive).

P (A \cap B) = P (A) P (B)

Multiply (independent).

Common mistakes to avoid

Treating mutually exclusive and independent as the same thing.

Mutually exclusive: P(A∩B)=0. Independent: P(A∩B)=P(A)P(B). Usually not both.

Multiplying probabilities for events that are not independent.

P(A∩B)=P(A)P(B) only holds when the events are independent.

Key takeaways

Mutually exclusive: cannot co-occur, P(A∩B)=0, so P(A∪B)=P(A)+P(B).
Independent: no influence, P(A∩B)=P(A)×P(B).
Add for mutually exclusive “or”; multiply for independent “and”.

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