Contingency Tables

Q: How do I test independence with a contingency table?

Calculate $P(A)$ and $P(B)$ from the totals, multiply them, and compare with $P(A \text{ and } B)$ from the cell.

A contingency table organises data about two features, such as grade and library use. It gives probabilities straight from the counts and is the standard way to test whether two events are independent.

Two-way contingency tables

Rows show one feature, columns another, and the margins show totals.

Example. 120 learners in Grades 10 and 11 were asked whether they use the library.

Grade 10: yes 24, no 36, total 60
Grade 11: yes 16, no 44, total 60
Totals: yes 40, no 80, grand total 120

Read probabilities by dividing by the grand total:

$P (Grade 11 and no) = \frac{44}{120} \approx 0, 367$
$P (yes) = \frac{40}{120} = \frac{1}{3}$

Fill in missing cells using row and column totals first.

Using tables to test independence

Events $A$ and $B$ are independent if $P (A and B) = P (A) \times P (B)$ .

Example. Using the table above:

$P (Grade 10) \times P (yes) = \frac{1}{2} \times \frac{1}{3} \approx 0, 167$

$P (Grade 10 and yes) = \frac{24}{120} = 0, 2$

These are not equal, so grade and library use are dependent in this sample.

Finding an unknown for independence. Set up $P (A and B) = P (A) \times P (B)$ with the unknown in the table and solve.

Key terms

Contingency table: A two-way table of counts for two categorical features.
Marginal total: A row or column total.
Grand total: The total of all the counts in the table.

Common mistakes

Dividing by a row total instead of the grand total.
Testing independence with counts rather than probabilities.
Rounding probabilities before comparing them, which can hide a difference.
Leaving out the conclusion after the calculation.

Practice questions

Reveal steps one at a time, then tick the marks you earned.

Question 1

4 marks total

routine

120 learners were asked whether they use the school library. 60 are in Grade 10 and 60 in Grade 11. 24 Grade 10 learners and 16 Grade 11 learners use the library.

(a) Complete a contingency table with all totals. (3)

(b) Calculate $P (Grade 11 and does not use the library)$ . (1)

Worked solution

Hidden - try the question first

Solution

(a) Grade 10: yes 24, no 36
Grade 11: yes 16, no 44
Totals: yes 40, no 80, total 120
(b) $\frac{44}{120} \approx 0, 367$

Question 2

5 marks total

routine

Use the table from Question 1. Are being in Grade 10 and using the library independent events? Show all calculations.

Worked solution

Hidden - try the question first

Solution

$P (Grade 10) = \frac{60}{120} = 0, 5$
$P (yes) = \frac{40}{120} = \frac{1}{3}$
$P (Grade 10) \times P (yes) \approx 0, 167$
$P (Grade 10 and yes) = \frac{24}{120} = 0, 2$
Not equal, so the events are dependent

Question 3

6 marks total

complex

A café records drink orders:

Male: coffee $a$ , tea 30
Female: coffee 40, tea 60

Determine the value of $a$ for which being male and ordering coffee are independent events.

Worked solution

Hidden - try the question first

Solution

Grand total $= a + 130$
$P (male) = \frac{a + 30}{a + 130}$ and $P (coffee) = \frac{a + 40}{a + 130}$
$P (male and coffee) = \frac{a}{a + 130}$
$\frac{a}{a + 130} = \frac{( a + 30 ) ( a + 40 )}{( a + 130 ) ^{2}} \Rightarrow a (a + 130) = (a + 30) (a + 40)$
$a^{2} + 130 a = a^{2} + 70 a + 1200$
$a = 20$

Question 4

8 marks total

problem-solving

An insurer studied 200 drivers. 30% of the drivers had an accident in the past year. 60 of the drivers are under 25, and 27 of them had an accident.

(a) Draw a contingency table. (3)

(b) Calculate $P (under 25 and had an accident)$ . (1)

(d) What would you advise the insurer? (1)

Worked solution

Hidden - try the question first

Solution

(a) Accidents: $= a + 130$ in total
Under 25: accident 27, no accident 33; 25 or older: accident 33, no accident 107
Totals: accident 60, no accident 140, total 200
(b) $\frac{27}{200} = 0, 135$
(c) $P (under 25) \times P (accident) = 0, 3 \times 0, 3 = 0, 09$
$P (under 25 and accident) = 0, 135$
Not equal, so the events are dependent
(d) Younger drivers have a higher accident rate, which the insurer can consider when setting premiums

FAQs

How do I test independence with a contingency table?

Calculate $P (A)$ and $P (B)$ from the totals, multiply them, and compare with $P (A and B)$ from the cell.

Which total do I divide by?

Divide by the grand total for probabilities of events in the whole group.

Do I need to write a conclusion?

Yes. State whether the events are independent or dependent, and why.

Two-way contingency tables

Using tables to test independence

Key terms

Common mistakes

Practice questions

FAQs

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