Conditional probability on Venn diagrams
Restrict Venn diagrams to a known set or its complement, then count the favourable outcomes inside that group
- Restrict to the whole known set Given B includes B only and the overlap. Exclude everything outside B.
- Intersect with the target After restricting, retain the part that also satisfies the target
- Complements change the condition Given not B uses A only and neither as the denominator
- Use counts, not areas These Venn diagrams are schematic. Circle areas do not represent counts
- Check possibility The known group must contain at least one equally likely item. A zero favourable count can still give a valid answer
- Distinguish a union from a condition A or B keeps both sets; A given B restricts attention to B before checking A.
Worked example
A uniformly selected collection has A only 9, B only 8, both 6 and neither 7. Find the probabilities of A given B, not A given B, and A given not B.
- \(\displaystyle N(B)=8+6=14\)Use the whole known set
- \(\displaystyle \Pr(A\mid B)=\frac6{14}=\frac37\)The overlap is favourable inside B.
- \(\displaystyle \Pr(A^{\prime}\mid B)=\frac8{14}=\frac47\)B only is favourable for not A within B.
- \(\displaystyle N(B^{\prime})=9+7=16\)The complement condition includes neither
- \(\displaystyle \Pr(A\mid B^{\prime})=\frac9{16}\)A only is favourable within not B.
Common mistakes
Excluding the overlap from B
✕\(\displaystyle \Pr(A\mid B)=\frac68\)✓\(\displaystyle \Pr(A\mid B)=\frac6{14}\)
The condition includes every item in B.Keeping the old denominator after complementing B
✕\(\displaystyle \Pr(A\mid B^{\prime})=\frac9{14}\)✓\(\displaystyle \Pr(A\mid B^{\prime})=\frac9{16}\)
The known not-B group has a different sizeReversing the order
✕\(\displaystyle \Pr(A\mid B)=\frac6{15}\)✓\(\displaystyle \Pr(B\mid A)=\frac6{15}\)
The denominator fifteen belongs to A, not B.Difficulty levels
challengeIdentify the known set, count it and calculate either conditional direction
Questions about this topic
Is given B the same as B only?+
No. Given B includes its overlap with A.
Where does neither belong if not B is known?+
Neither is outside B, so it belongs in the reduced sample space
What supplies the numerator for A given B?+
The shared region A and B.
Can a nonempty condition give zero?+
Yes. If its overlap with the target is empty, none of its possible outcomes is favourable
Why do the circles not change size with the counts?+
They are schematic. The labels and counts, not area, carry the information
Does conditional mean independent?+
No. Conditional probability uses known information. Whether it changes the target probability is examined in the later independence section
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 10 Mathematics, chapter 4: Probability
In this chapter
Describe and order probability valuesSingle-step theoretical probability and complements4A Probability scale and experiments: Experimental probability from observed frequenciesFinite events, probability assignments and complements4A Probability scale and experiments: Design and interpret a probability simulation4B Venn diagrams and sets: Venn regions, set language and probabilityCompound events, the addition rule and mutual exclusivityComplete and use a two-way probability tableConstruct and complete two-way frequency tablesConvert between Venn diagrams and two-way tablesConditional probability from exact representations4D Conditional probability: Conditional probability on Venn diagramsTwo-step sample-space tables with and without replacementRepresent a multi-stage finite sample spaceTree-diagram probability for two-stage experimentsMulti-stage probability, multiplication and total probability4G Independence and simulations: Independence on Venn diagramsTest exact independence4G Independence and simulations: Design and interpret a probability simulation4G Independence and simulations: Experimental probability from observed frequencies