Two-way count and probability tables
Build and complete count and joint-probability tables, then select the cells for an event
- Read both labels A cell belongs to the intersection of its row and column categories
- Margins are sums A row or column total includes every interior cell along that row or column
- Use a complete denominator For uniform selection from the collection, divide a favourable count by the grand total \(\displaystyle \Pr(E)=\frac{N(E)}{N(S)}\)
- Probability tables total one The four joint probabilities sum to one. Each margin is a probability for a whole event
- Recover a blank Use an available margin minus its known neighbouring cell; continue until the requested entries are determined
- Keep the event under transposition Swapping rows and columns moves a cell but does not change which outcomes it represents
Worked example
A collection has 60 items. In rows A and not A and columns B and not B, the four counts are 12, 18, 9 and 21. Select one item uniformly. Find the probability of A or B.
- \(\displaystyle N(A)=12+18=30\)Add row A.
- \(\displaystyle N(B)=12+9=21\)Add column B.
- \(\displaystyle \begin{aligned}N(A\cup B)&=30+21-12\\&=39\end{aligned}\)Remove one copy of the overlap
- \(\displaystyle \Pr(A\cup B)=\frac{39}{60}=\frac{13}{20}\)Divide by the whole collection
Common mistakes
Using a cell as a whole event
✕\(\displaystyle N(A)=12\)✓\(\displaystyle N(A)=12+18=30\)
A contains both cells in its rowCounting a union twice
✕\(\displaystyle \begin{aligned}N(A\cup B)&=30+21\\&=51\end{aligned}\)✓\(\displaystyle \begin{aligned}N(A\cup B)&=30+21-12\\&=39\end{aligned}\)
The overlap is included in both marginsNormalising each row in a joint table
✕\(\displaystyle \Pr(A\cap B)=\frac{12}{30}\)✓\(\displaystyle \Pr(A\cap B)=\frac{12}{60}=\frac15\)
The denominator for a joint probability is the grand total; a restricted denominator asks a different questionDifficulty levels
introComplete a count table and calculate an event probability
standardComplete diagonal or opposite-diagonal joint probabilities
challengeRecover cells and margins from sparse exact givens
examRecognise a missing entry, complete other entries and find an event probability
Questions about this topic
What is a margin?+
A row or column total that summarises its interior cells
Can a joint probability be zero?+
Yes. It means no outcome in the stated model lies in that joint category
Must each row total one?+
No. Only the four joint probabilities together must total one
Can I use decimals?+
Use exact fractions or terminating decimals when an exact answer is required
Does a table show independence automatically?+
No. Its layout alone establishes neither independence nor mutual exclusivity
Does transposing the table change an answer?+
No, provided the event is read from the same category labels
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