Venn and two-way table conversion
Translate the four disjoint groups between Venn diagrams and two-way tables without confusing regions with margins
- Interior cells match regions A-and-B is the overlap; A-and-not-B is A only; not-A-and-B is B only; not-A-and-not-B is neither
- Margins are whole sets The A margin combines A only with the overlap. A-only excludes the overlap
- Keep the universe Neither is still inside the universe and must be included in its total
- Convert counts consistently For uniform selection, divide every joint count by the same grand total. Multiply a joint probability by that total to recover a count
- Read labels after transposition A cell can move from row to column without changing the combination it represents
Worked example
A collection has A only 7, B only 4, both 5 and neither 8. Put A in rows and B in columns, then find the A margin and the probability of neither
- \(\displaystyle \begin{array}{c|cc}&B&\text{Not B}\\ A&5&7\\ \text{Not A}&4&8\end{array}\)Match each cell with both category conditions
- \(\displaystyle N(A)=5+7=12\)The margin includes the overlap
- \(\displaystyle N(S)=7+4+5+8=24\)Neither is part of the universe
- \(\displaystyle \Pr(\text{neither})=\frac8{24}=\frac13\)Use the whole collection as denominator
Common mistakes
Using the whole set as A only
✕\(\displaystyle N(A\text{ only})=12\)✓\(\displaystyle N(A\text{ only})=12-5=7\)
Remove the overlap from the A marginOmitting neither
✕\(\displaystyle N(S)=7+4+5=16\)✓\(\displaystyle N(S)=7+4+5+8=24\)
The universe includes items outside both circlesReversing an exclusive cell
✕\(\displaystyle N(A\text{ only})=4\)✓\(\displaystyle N(A\text{ only})=7\)
Read both table labels; B only and A only are different cellsDifficulty levels
introConvert Venn counts to four joint table cells
standardConvert Venn counts to joint probabilities and margins
challengeRecover a missing table cell and convert counts or probabilities to Venn counts
examRecover a missing Venn count and complete joint cells and margins
Questions about this topic
Where does the overlap go?+
In the cell whose row and column both name membership in the two events
Where does neither go?+
In the Not A and Not B cell. It remains inside the universe
Is an A margin the same as A only?+
No. The margin includes both A only and the overlap
Can rows and columns be swapped?+
Yes, if every label, cell and margin is transposed consistently
How do probabilities become counts?+
Multiply each joint probability by the collection size when the model is uniform selection from that collection
Does the area of a Venn circle represent its count?+
No. These diagrams are schematic; use the displayed data
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