What is important to know about Combined events?

AND rule (independent events): P(A and B) = P(A) × P(B).

What is important to know about Combined events?

OR rule (mutually exclusive events): P(A or B) = P(A) + P(B).

What is important to know about Combined events?

General OR rule: P(A or B) = P(A) + P(B) − P(A and B).

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Statistics & Probability · Higher

Combined events

Combined events involve finding the probability of two or more events occurring. At Higher tier you use the AND rule (multiply) and OR rule (add) and must be able to work with independent and mutually exclusive events.

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Key facts to remember

1AND rule (independent events): P(A and B) = P(A) × P(B).
2OR rule (mutually exclusive events): P(A or B) = P(A) + P(B).
3General OR rule: P(A or B) = P(A) + P(B) − P(A and B).
4Two events are independent if one does not affect the other.
5Two events are mutually exclusive if they cannot both happen at the same time.
6All probabilities in a sample space must sum to 1.

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Formulas

AND rule (independent)

P(A ∩ B) = P(A) × P(B)

OR rule (mutually exclusive)

P(A ∪ B) = P(A) + P(B)

General OR rule

P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

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Worked examples

Example 1

A bag contains 3 red and 7 blue balls. A ball is chosen, replaced, then another is chosen. Find P(both red).

Working

P(red) = 3/10
Since replaced, the events are independent.
P(both red) = P(red) × P(red) = 3/10 × 3/10
= 9/100

Answer9/100 or 0.09

Example 2

P(A) = 0.4, P(B) = 0.3. A and B are mutually exclusive. Find P(A or B).

Working

Mutually exclusive means P(A and B) = 0.
P(A or B) = P(A) + P(B) = 0.4 + 0.3 = 0.7

Answer0.7

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Common mistakes

✗Adding probabilities when you should multiply (AND means multiply).

✗Using the simple OR rule when events are NOT mutually exclusive.

✗Not replacing when the question says "without replacement" — events become dependent.

✗Leaving answers greater than 1 — always check your final probability is between 0 and 1.

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Exam tips

✓"And" → multiply. "Or" (mutually exclusive) → add.

✓Always check if events are independent (with replacement) or dependent (without replacement).

✓Draw a sample space diagram or tree diagram to organise your working.

✓Check probabilities sum to 1 as a sense-check.

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