Complete and interpret two-way count tables
Use the diagram or table as evidence. Distinguish exact selection from a recorded sample and an estimate for future observations
- Interior cells Each item belongs to exactly one combination of the two properties
- Margins Add the interior cells in that row or column. Margins repeat the same items; do not add them to the interior counts
- Complete a blank Subtract the other interior count from a known row or column total
- Probability For uniform selection from recorded items, divide the favourable count by the grand total
Worked example
A count table has both 7, A only 9, B only 5 and neither 11. Choose one of the 32 items uniformly. Find the probability of exactly one property
\[|S|=32\]
- \(\displaystyle 9+5=14\)Choose the two only cells
- \(\displaystyle \Pr(\text{exactly one})=\frac{14}{32}\)Use the grand total as denominator
- \(\displaystyle \Pr(\text{exactly one})=\frac{7}{16}\)Simplify
Common mistakes
Copying a cell into a margin
✕\(\displaystyle n(A)=7\)✓\(\displaystyle n(A)=7+9=16\)
A contains the overlap and A onlyDouble-counting the overlap
✕\(\displaystyle n(A\cup B)=16+12\)✓\(\displaystyle n(A\cup B)=16+12-7=21\)
Subtract the shared seven onceUsing the favourable count as denominator
✕\(\displaystyle \Pr(A\text{ only})=\frac{9}{9}\)✓\(\displaystyle \Pr(A\text{ only})=\frac{9}{32}\)
All 32 recorded items are selectableDifficulty levels
introComplete a missing count and find a union, intersection, only-A or complement probability
Questions about this topic
What is a margin?+
A row or column total, including both of its interior cells
Why do the row and column totals agree?+
Both count the same complete group, organised in different ways
Can I use either direction to find a blank?+
Yes. A valid completion satisfies both its row and its column
Does A only include both?+
No. It is A together with Not B.
Is a recorded proportion always a future probability?+
No. It is exact for uniform selection from that recorded group, but only an estimate for future cases under a comparable process
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 9 Mathematics, chapter 9: Probability and statistics
In this chapter
Finite events, probability assignments and complementsDescribe and order probability values9B Venn diagrams and tables: Venn diagrams, two-way tables and set notation for two eventsComplete and use a two-way probability tableCompound events, the addition rule and mutual exclusivity9C Set notation: Event notation and shaded regionsFinite sets and set operationsBuild two-step sample spaces with and without replacementUse two-step probability treesExperimental probability from observed frequenciesDesign and interpret a probability simulationFind the mean and the median, and choose between themWork backwards from a mean or medianMode, bimodal and no-mode data, and the effect of an outlierConstruct an ordered stem plot from raw dataConstruct and read back-to-back stem-and-leaf plotsDescribe the shape of a distributionGroup numerical data into class intervalsConstruct a histogram from a frequency tableRead values from a histogramFind the range, the quartiles and the IQRProduce a five-number summary and construct a boxplotRead values and percentages from a boxplotCompare a numerical variable across groupsSurveys and statistical investigations