2.3StatisticsStretch

Conditional Probabilities in Venn Diagrams

A Venn diagram makes conditional probability concrete. To find P(A | B) you restrict to the part of the diagram inside B, then look at how much of it is also in A.

24 min Conditional Probability

Edexcel A level Maths: 2.3 Conditional Probabilities In Venn DiagramsCurated by Kepler, not produced by us. Watch it before the notes below.

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What you'll be able to do

Read probabilities from a Venn diagram
Compute conditional probabilities from regions
Fill in missing regions
Combine with the conditional formula

Reading conditional probabilities

To find $P(A \mid B)$ , take the probability in $A \cap B$ and divide by the total probability inside $B$ . The diagram supplies both numbers directly.

P(A \mid B) = \frac{P(A \cap B)}{P(B)} = \frac{\text{overlap}}{\text{all of } B}

Restrict to B, then look at the overlap.

Example 1In a Venn diagram,

A \cap B = 0.15

and the whole of

B = 0.4

. Find

P(A \mid B)

\dfrac{0.15}{0.4}

= 0.375

Answer

0.375

Tip — The denominator is the TOTAL probability in B, including the overlap.

Completing the diagram

All regions (including outside both sets) must sum to 1. Use this to find a missing region before computing probabilities.

Formula recap

P(A \mid B) = \frac{P(A \cap B)}{P(B)}

From the diagram.

\text{all regions sum to } 1

Completing the diagram.

Common mistakes to avoid

Using only the “A only” region as the numerator.

The numerator is the overlap A∩B.

Dividing by the overlap instead of all of B.

Denominator is the total inside B.

Key takeaways

P(A | B) = (A∩B) / (all of B).
Numerator is the overlap; denominator is everything in B.
All regions sum to 1 — use this to find a missing one.

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.

For P(A | B) from a Venn diagram, the denominator is:
- all of B
- the overlap
- all of A
- the whole diagram
Show answer and explanation
Answer: all of B
Restrict to B.
The numerator for P(A | B) is:
- $A \cap B$
- A only
- B only
- $A \cup B$
Show answer and explanation
Answer: $A \cap B$
The overlap.
If $A \cap B = 0.15$ and all of B = 0.4, P(A | B) =
- $0.375$
- $0.15$
- $0.4$
- $0.6$
Show answer and explanation
Answer: $0.375$
0.15/0.4.
All regions of a Venn diagram sum to:
- $1$
- $0$
- $P(A)$
- $2$
Show answer and explanation
Answer: $1$
Total probability.
The region outside both circles represents:
- neither event
- both events
- A only
- B only
Show answer and explanation
Answer: neither event
(A∪B)′.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

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Put the methods into practice: see how A-Level maths duels work. An account is required to play; the overview and sample are public.