Use and, inclusive or and exclusive or with observed data
Use the diagram or table as evidence. Distinguish exact selection from a recorded sample and an estimate for future observations
- And Both properties are required; use the overlap \(\displaystyle A\cap B\)
- Inclusive or At least one property, including both \(\displaystyle A\cup B\)
- Complements and only A′ means outside A and B′ means outside B. A only is in A but outside B. \(\displaystyle A\cap B'\)
- Exclusive or Exactly one property; add only-A and only-B. \(\displaystyle (A\cap B')\cup(A'\cap B)\)
- Relative frequency An observed event count divided by the number of observations estimates a future probability if the data are relevant. It does not guarantee the next result
Worked example
In 80 observed trials, A only occurred 18 times, B only 14, both 22 and neither 26. Estimate the probability of exactly one event
\[\text{Observed trials}=80\]
- \(\displaystyle 18+14=32\)Exclusive or excludes the overlap
- \(\displaystyle \frac{32}{80}=\frac{2}{5}\)The relative frequency estimates probability
- \(\displaystyle \Pr(\text{exactly one})\approx\frac{2}{5}\)Future results need not match the observed proportion
Common mistakes
Confusing the two meanings of or
✕\(\displaystyle n(\text{exactly one})=72\)✓\(\displaystyle \begin{gathered}24+18=42\\n(\text{exactly one})=42\end{gathered}\)
With 24 in A only, 18 in B only and 30 in both, exclusive or leaves out bothAdding overlapping totals
✕\(\displaystyle n(A\cup B)=54+48\)✓\(\displaystyle \begin{gathered}54+48-30=72\\n(A\cup B)=72\end{gathered}\)
With 54 in A, 48 in B and 30 in both, the shared 30 appears in both marginsTreating an estimate as a guarantee
✕\(\displaystyle \begin{gathered}\frac35\\\Rightarrow\text{guaranteed 3 of 5}\end{gathered}\)✓\(\displaystyle \text{Outcomes can vary}\)
Data provide an estimate under a comparable process, not a fixed scheduleDifficulty levels
introTranslate event language and calculate from overlapping or disjoint count regions, including exactly one
Questions about this topic
What is inclusive or?+
A or B or both: at least one event
What is exclusive or?+
A or B but not both: exactly one event
How is a probability estimated from data?+
Divide the observed event frequency by the total observations. Check whether the observations are relevant to the future process
Can two samples give different estimates?+
Yes. Random variation and sampling differences can change the relative frequency
What does mutually exclusive mean?+
Both events cannot occur together. Their overlap is empty; this is different from independence
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