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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 Probability
Edexcel Statistics Y1 — Probability playlist (Zeeshan Zamurred)Curated by Kepler, not produced by us. Watch it 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
1

Mutually exclusive events

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

P(A∪B)=P(A)+P(B)(mutually exclusive)P(A \cup B) = P(A) + P(B) \quad (\text{mutually exclusive})
No overlap to subtract.
Example 1On one die roll, are “roll a 2” and “roll a 5” mutually exclusive?
1They cannot both occur on one roll.
Answeryes
2

Independent events

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

P(A∩B)=P(A)×P(B)(independent)P(A \cap B) = P(A) \times P(B) \quad (\text{independent})
Multiply for “and” when independent.
Example 1P(A) = 0.5, P(B) = 0.2, independent. Find P(A and B).
10.5×0.20.5 \times 0.2.
Answer0.10.1

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

3

Telling them apart

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

Formula recap

P(A∩B)=0P(A \cap B) = 0
Mutually exclusive.
P(A∪B)=P(A)+P(B)P(A \cup B) = P(A) + P(B)
Add (mutually exclusive).
P(A∩B)=P(A)P(B)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”.

Test yourself

Answer a few questions, then review anything you missed.

Next step

Check your understanding

Work through questions on Mutually Exclusive and Independent Events. Use the feedback to decide what needs another look.

3 past-paper questions · 4 marks

Practise this topic

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