2.4StatisticsStretch

Probability Formulae

Two formulae handle most probability problems: the addition formula for “or”, and the multiplication form of the conditional probability formula for “and”. Together they let you compute combined probabilities.

24 min Conditional Probability

Edexcel A level Maths: 2.4 Probability Formulae (Addition and Conditional)Curated by Kepler, not produced by us. Watch it before the notes below.

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What you'll be able to do

Use the addition formula
Use the multiplication formula
Rearrange the conditional formula
Apply to combined events

Addition formula

For any two events, $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ . Subtracting the overlap avoids double-counting.

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

Addition (“or”) formula.

Multiplication formula

Rearranging the conditional formula gives $P(A \cap B) = P(B)\,P(A \mid B)$ — useful for finding the probability of both events.

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

Multiplication (“and”) formula.

Example 1

P(A) = 0.5

P(B) = 0.4

P(A \cap B) = 0.2

. Find

P(A \cup B)

0.5 + 0.4 - 0.2

= 0.7

Answer

0.7

Tip — For mutually exclusive events P(A∩B) = 0, so P(A∪B) = P(A) + P(B).

Formula recap

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

Addition.

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

Multiplication.

Common mistakes to avoid

Forgetting to subtract the overlap in the addition formula.

Subtract P(A∩B) to avoid double-counting.

Using P(A)P(B) for P(A∩B) when events are not independent.

Use P(B)P(A | B) in general.

Key takeaways

P(A∪B) = P(A) + P(B) − P(A∩B).
P(A∩B) = P(B)P(A | B).
Mutually exclusive: P(A∩B) = 0.

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

The addition formula is $P(A \cup B) =$
- $P(A)+P(B)-P(A \cap B)$
- $P(A)+P(B)$
- $P(A)P(B)$
- $P(A)-P(B)$
Show answer and explanation
Answer: $P(A)+P(B)-P(A \cap B)$
Subtract the overlap.
With $P(A)=0.5$ , $P(B)=0.4$ , $P(A\cap B)=0.2$ , $P(A\cup B)=$
- $0.7$
- $0.9$
- $0.5$
- $1.1$
Show answer and explanation
Answer: $0.7$
0.5+0.4−0.2.
The multiplication formula is $P(A \cap B) =$
- $P(B)P(A \mid B)$
- $P(A)+P(B)$
- $P(A \cup B)$
- $P(A)-P(B)$
Show answer and explanation
Answer: $P(B)P(A \mid B)$
Rearranged conditional.
For mutually exclusive events, $P(A \cap B)$ is:
- $0$
- $1$
- $P(A)P(B)$
- $P(A)$
Show answer and explanation
Answer: $0$
They cannot both happen.
Why subtract P(A∩B) in the addition formula?
- to avoid double-counting
- to add the overlap twice
- for independence
- no reason
Show answer and explanation
Answer: to avoid double-counting
Overlap counted once.

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