Read, complete and connect Venn diagrams and tables
Use the diagram or table as evidence. Distinguish exact selection from a recorded sample and an estimate for future observations
- Both The overlap is in A and in B. \(\displaystyle A\cap B\)
- At least one Inclusive or includes A only, B only and both \(\displaystyle A\cup B\)
- Exactly one Exclusive or includes the two only regions and excludes the overlap
- Neither and total The universe includes the outside region. A′ means outside A; B′ means outside B. The notations n(A), N(A) and |A| all mean the number of items in A.
Worked example
Thirty counters are classified: A only 8, B only 6, both 4, neither 12. Choose one counter uniformly. Find the chance of A or B.
\[|S|=30\]
- \(\displaystyle 8+6+4=18\)Count all three regions in the union
- \(\displaystyle \Pr(A\cup B)=\frac{18}{30}\)Divide by the complete universe
- \(\displaystyle \Pr(A\cup B)=\frac{3}{5}\)Reduce the fraction
Common mistakes
A only is not all of A
✕\(\displaystyle n(A)=8\)✓\(\displaystyle n(A)=8+4=12\)
Include the shared region in the whole setInclusive or is not exactly one
✕\(\displaystyle n(A\cup B)=8+6\)✓\(\displaystyle n(A\cup B)=8+6+4=18\)
The union includes bothThe outside region belongs to the universe
✕\(\displaystyle |S|=18\)✓\(\displaystyle |S|=18+12=30\)
Neither is still an outcome in the universeDifficulty levels
introRead a region count or an exact selection probability
standardComplete a missing region before answering
challengeTransfer between a Venn diagram and a two-way table; formal notation is developed in 9C
Questions about this topic
What does both mean?+
Membership in A and B: the overlap
Does or include both?+
Inclusive or does; exclusive or means exactly one and excludes both
What does neither mean?+
Outside both sets but inside the universe
Why is the denominator the universe?+
Every item in the complete selection pool is equally likely here
What is taught in 9C?+
Formal set notation, membership and translation between words, symbols and diagrams
Can I use a calculator?+
A calculator is allowed unless a question is marked no 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