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Complete and use a two-way probability table

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A displayed question mark is in a two-way table. What should you decide first?

\[\begin{array}{c|rr|r}&B&B^{\prime}&\text{Total}\\\hline A&17&9&?\\A^{\prime}&8&6&14\\\hline\text{Total}&25&15&40\end{array}\]
Year 11, VCE Units 1 & 2, Chapter 8, 4 difficulty levels, No calculatorBuild a worksheet

Two-way probability tables

A two-way table separates a sample by two binary properties. Complete the margins from the interior cells, then use the cells or margins that describe the requested event

\[\begin{gathered}\begin{array}{c|rr|r}&B&B^{\prime}&\text{Total}\\\hline A&17&9&26\\A^{\prime}&8&6&14\\\hline\text{Total}&25&15&40\end{array}\\[8pt]\begin{aligned}\Pr(A\cup B)&=\Pr(A)+\Pr(B)\\&\quad-\Pr(A\cap B)\end{aligned}\end{gathered}\]
  1. Read an interior cell An interior cell represents one combination of the two properties \(\displaystyle n(A\cap B)=17\)
  2. Add a row margin A row margin is the sum of the two interior cells in that row \(\displaystyle n(A)=17+9=26\)
  3. Add a column margin A column margin is the sum of the two interior cells in that column \(\displaystyle n(B)=17+8=25\)
  4. Remove the overlap once A union includes both margins, so subtract the shared cell once after adding the margins \(\displaystyle n(A\cup B)=26+25-17=34\)

Worked example

A synthetic classification contains 40 labelled cards. Event A means a card has a blue label and event B means it is placed in the first group. Select one card uniformly at random. Use the table to find the displayed probability

\[\begin{gathered}\begin{array}{c|rr|r}&B&B^{\prime}&\text{Total}\\\hline A&15&7&22\\A^{\prime}&10&8&18\\\hline\text{Total}&25&15&40\end{array}\\[6pt]\Pr(A\cup B)=?\end{gathered}\]
  1. \(\displaystyle n(A)=15+7=22\)Add the A and B cell to the A and not-B cell to find the A margin
  2. \(\displaystyle n(B)=15+10=25\)Add the A and B cell to the not-A and B cell to find the B margin
  3. \(\displaystyle n(A\cup B)=22+25-15=32\)Add the margins and subtract the shared cell once
  4. \(\displaystyle \Pr(A\cup B)=\frac{32}{40}=\frac{4}{5}=0.8\)Divide the union count by the grand total and reduce the fraction

Common mistakes

Copying a cell into a margin
✕\(\displaystyle n(A)=15\)✓\(\displaystyle n(A)=15+7=22\)
A row margin includes both interior cells, not just the shared cell
Counting the overlap twice
✕\(\displaystyle n(A\cup B)=22+25=47\)✓\(\displaystyle n(A\cup B)=22+25-15=32\)
The shared cell is included in both margins, so it must be subtracted once
Using the favourable count as the total
✕\(\displaystyle \Pr(A\cup B)=\frac{32}{32}=1\)✓\(\displaystyle \Pr(A\cup B)=\frac{32}{40}=\frac{4}{5}\)
The probability denominator is the grand total of the table, not the favourable count

Difficulty levels

introComplete a count table with one missing interior cell and use its margins
standardComplete a probability table with several pre-filled entries and calculate an event probability
challengeUse sparse givens, complement and margin reasoning before finding a compound-event probability
examSelect an internally consistent completion or calculate from a partially completed table

Questions about this topic

What is an interior cell?+

It is the count or probability for one combination of the two properties, such as being in A and in B

How is a margin found?+

Add the two interior cells in the relevant row or column. The row and column totals must agree with the grand total

Why is the overlap subtracted for a union?+

The overlap appears in both event margins, so adding the margins counts it twice. Subtracting it once leaves one copy

Can I use a calculator?+

No. This topic is practised without a calculator

Which year level and chapter is this?+

Year 11, VCE Units 1 & 2, Mathematical Methods, chapter 8: Probability and counting

In this chapter

Finite events, probability assignments and complementsRepresent a multi-stage finite sample spaceCompound events, the addition rule and mutual exclusivityComplete and use a two-way probability tableConditional probability from exact representationsDistribution of a finite random variableDesign and interpret a probability simulationCount restricted selections and calculate exact probabilitiesCount arrangements with product, sum and restriction rulesTest exact independenceExact probability of repeated independent successesMulti-stage probability, multiplication and total probabilityProbability from a uniform area modelProbabilities of sequences of independent events