Conditional versus unconditional probabilities | Year 10

01 / LEARN

Understand the idea.

Three different questions

An unconditional probability uses the full group. A joint probability also uses the full group but requires both characteristics. A conditional probability first restricts the reference group. Identify which question is being asked before choosing a denominator.

In the travel table, $P (girl) = \frac{30}{60}$ , $P (girl and walk) = \frac{18}{60}$ and $P (girl ∣ walk) = \frac{18}{40}$ . Each fraction is valid, but each describes a different selection rule.

Invented survey of 60 students; every student is in exactly one travel category.
Travel	Girl	Boy	Total
Cycle	12	8	20
Walk	18	22	40
Total	30	30	60

Use proportions, not raw counts

To test independence in a table, compare a conditional proportion with the corresponding overall proportion. Equal counts do not suffice when group sizes differ. If the proportion changes after conditioning, the events are dependent in the supplied data.

An empirical independence conclusion concerns this table. It does not automatically establish the same relationship throughout a larger population. Likewise, an association can arise from other factors or sampling choices, so a probability comparison alone cannot justify causation.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Unconditional versus conditional

Find P(girl) and P(girl|walk) in the travel table.

Invented survey of 60 students; every student is in exactly one travel category.
Travel	Girl	Boy	Total
Cycle	12	8	20
Walk	18	22	40
Total	30	30	60

Worked solution

1
Use the full group for the unconditional probability.
$P (girl) = \frac{30}{60} = \frac{1}{2}$
Thirty of all sixty students are girls.
2
Restrict to walkers for the conditional probability.
$P (girl ∣ walk) = \frac{18}{40} = \frac{9}{20}$
Only the forty walkers remain in the reference group.
3
Compare the two probabilities.
$\frac{1}{2} = \frac{10}{20} \neq \frac{9}{20}$
The girl proportion changes when the condition walk is supplied.

Answer

1/2 and 9/20; they differ.

EXAMPLE 02

A table with independence

A class has 20 girls and 20 boys. Ten girls and ten boys cycle. Are gender and cycling independent in this table?

Worked solution

1
Find the overall cycling proportion.
$P (cycle) = \frac{10 + 10}{20 + 20} = \frac{1}{2}$
Twenty of the forty students cycle.
2
Find the cycling proportion in each gender group.
$\begin{aligned} P (cycle ∣ girl) & = \frac{10}{20} = \frac{1}{2} \\ P (cycle ∣ boy) & = \frac{10}{20} = \frac{1}{2} \end{aligned}$
Both nonempty groups have the same cycling proportion as the whole class.
3
State the table-based conclusion.
Conditioning on either gender group does not change the cycling probability, so gender and cycling are independent in these supplied data.

Answer

Yes, in the supplied data.

03 / YOUR TURN

Try it yourself.

Check your understanding

In the travel table, compare P(walk) and P(walk|boy).

Show answer & reasoning

2/3 and 11/15.

Overall, $P (walk) = \frac{40}{60} = \frac{2}{3}$ . Among the thirty boys, twenty-two walk, so $P (walk ∣ boy) = \frac{22}{30} = \frac{11}{15}$ .

Keep practising

6 worksheet questions · 14 marks

Open worksheet Worked solutions

Read the questions online

Find $P (cycle)$ from the travel table.
2 marks
Invented survey of 60 students; every student is in exactly one travel category.
Travel Girl Boy Total
Cycle 12 8 20
Walk 18 22 40
Total 30 30 60
Find $P (boy and cycle)$ .
2 marks
Invented survey of 60 students; every student is in exactly one travel category.
Travel Girl Boy Total
Cycle 12 8 20
Walk 18 22 40
Total 30 30 60
Find $P (boy ∣ cycle)$ .
2 marks
Invented survey of 60 students; every student is in exactly one travel category.
Travel Girl Boy Total
Cycle 12 8 20
Walk 18 22 40
Total 30 30 60
Are being a boy and cycling independent in this table?
3 marks
Invented survey of 60 students; every student is in exactly one travel category.
Travel Girl Boy Total
Cycle 12 8 20
Walk 18 22 40
Total 30 30 60
$P (A) = 0.25$ and $P (A ∣ B) = 0.25$ , with $P (B) > 0$ . What can you conclude?
2 marks
A survey links extra tutoring with higher scores. Explain why a conditional proportion alone does not prove tutoring caused the increase.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Find $P (cycle)$ from the travel table.
2 marks
Invented survey of 60 students; every student is in exactly one travel category.
Travel Girl Boy Total
Cycle 12 8 20
Walk 18 22 40
Total 30 30 60
$\frac{20}{60} = \frac{1}{3}$ .
$\frac{1}{3}$ .
Find $P (boy and cycle)$ .
2 marks
Invented survey of 60 students; every student is in exactly one travel category.
Travel Girl Boy Total
Cycle 12 8 20
Walk 18 22 40
Total 30 30 60
$\frac{8}{60} = \frac{2}{15}$ .
$\frac{2}{15}$ .
Find $P (boy ∣ cycle)$ .
2 marks
Invented survey of 60 students; every student is in exactly one travel category.
Travel Girl Boy Total
Cycle 12 8 20
Walk 18 22 40
Total 30 30 60
$\frac{8}{20} = \frac{2}{5}$ .
$\frac{2}{5}$ .
Are being a boy and cycling independent in this table?
3 marks
Invented survey of 60 students; every student is in exactly one travel category.
Travel Girl Boy Total
Cycle 12 8 20
Walk 18 22 40
Total 30 30 60
$P (boy) = \frac{1}{2}$ , but $P (boy ∣ cycle) = \frac{2}{5}$ .
No.
$P (A) = 0.25$ and $P (A ∣ B) = 0.25$ , with $P (B) > 0$ . What can you conclude?
2 marks
The condition does not change the probability of A.
A and B are independent.
A survey links extra tutoring with higher scores. Explain why a conditional proportion alone does not prove tutoring caused the increase.
3 marks
Prior attainment, motivation and sampling can affect both groups.
A probability comparison alone does not isolate a cause.
Other differences may explain the association.

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Curriculum reference & lesson scope

Australian Curriculum v 9 / SA Curriculum

Compound events; systematic counting and the multiplication principle; factorials and simple arrangements; conditional language and restricted sample spaces; conditional probabilities from tables/trees; replacement and dependence; conditional versus unconditional probabilities; designing and interpreting simulations.

Official curriculum source ↗

In this lesson

Understand the idea.

Three different questions

Use proportions, not raw counts

See the method in action.

Unconditional versus conditional

Use the full group for the unconditional probability.

Restrict to walkers for the conditional probability.

Compare the two probabilities.

A table with independence

Find the overall cycling proportion.

Find the cycling proportion in each gender group.

State the table-based conclusion.

Try it yourself.

Check your understanding

Keep practising