Probabilities of combined outcomes | Year 8

01 / LEARN

Understand the idea.

Translate the event first

Translate the event before calculating. And requires both conditions; 'or' includes either condition or both unless stated otherwise. 'Exactly one' excludes the overlap, while 'at least one' excludes only the outcome where neither condition is satisfied.

For a die, even or greater than 4 includes ${2, 4, 5, 6}$ . The result 6 meets both conditions but is still one face, counted once. A complete outcome list helps make the intended event precise before using any probability rule.

Combine paths carefully

Multiply branch probabilities along a complete path, then add mutually exclusive successful paths. With independent stages, each stage keeps its original probability. Otherwise use the probability appropriate to the outcomes already observed, as shown on a correctly labelled tree.

Without replacement, remove the first token before listing second draws. The remaining total and colour counts change. Labelling physically different same-colour tokens can make the elementary outcomes visible and prevent mistaken assumptions that colour combinations are equally likely.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Worked example 1

Find P(at least one head) for two independent fair coins.

Worked solution

1
List all equally likely ordered results.
$H H, H T, T H, T T$
Two independent fair coins produce four complete outcomes of probability one quarter each.
2
Identify the only excluded result.
$No heads = {T T}$
At least one head permits one or two heads; only TT has none.
3
Use the complementary event.
$P (at least one head) = 1 - \frac{1}{4} = \frac{3}{4}$
The desired event contains HH, HT and TH, three of the four possible results.

Answer

3/4.

EXAMPLE 02

Worked example 2

A bag has B1, B2 and G. Draw two tokens at random without replacement. Find P(one of each colour).

Worked solution

1
Count ordered physical-token draws.
$3 \times 2 = 6$
There are three possible first tokens and two remaining choices after each; all six pairs are equally likely.
2
List the successful colour orders.
$\begin{array}{c} (B_{1}, G), (B_{2}, G), \\ (G, B_{1}), (G, B_{2}) \end{array}$
Each successful pair contains the green token and either blue token, in either order.
3
Divide the successful count by the full count.
$P (one of each colour) = \frac{4}{6} = \frac{2}{3}$
Four of the six equally likely physical pairs meet the event.
4
Check with changing branch probabilities.
$\frac{2}{3} \times \frac{1}{2} + \frac{1}{3} \times 1 = \frac{2}{3}$
Blue then green has a one-half second-stage chance; green then blue has a certain blue second draw.

Answer

2/3.

03 / YOUR TURN

Try it yourself.

Check your understanding

Find P(odd or multiple of 3) for a fair die.

Show answer & reasoning

2/3.

Odd faces are ${1, 3, 5}$ and multiples of 3 are ${3, 6}$ . Their union is ${1, 3, 5, 6}$ , counting 3 once, so the probability is $\frac{4}{6} = \frac{2}{3}$ .

Keep practising

6 worksheet questions · 14 marks

Open worksheet Worked solutions

Read the questions online

A fair coin and standard die are independent. Find $P (head and die greater than 4)$ .
2 marks
A bag has 2 red and 3 blue tokens. Draw at random and replace the token before a second random draw. Find $P (two red)$ .
2 marks
Find $P (two heads or two tails)$ for two independent fair coins.
2 marks
Two independent fair spinners each show 1,2,3. Find $P (sum 4)$ .
3 marks
Find $P (even or greater than 4)$ for a fair standard die.
2 marks
A bag has B1, B2 and G. Draw two tokens at random without replacement. Find $P (both blue)$ .
3 marks

Read the worked solutions

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

A fair coin and standard die are independent. Find $P (head and die greater than 4)$ .
2 marks
$\frac{1}{2} \times \frac{2}{6} = \frac{1}{6}$ .
$\frac{1}{6}$ .
A bag has 2 red and 3 blue tokens. Draw at random and replace the token before a second random draw. Find $P (two red)$ .
2 marks
$\frac{2}{5} \times \frac{2}{5} = \frac{4}{25}$ .
$\frac{4}{25}$ .
Find $P (two heads or two tails)$ for two independent fair coins.
2 marks
HH or TT gives 2 of 4 outcomes.
$\frac{1}{2}$ .
Two independent fair spinners each show 1,2,3. Find $P (sum 4)$ .
3 marks
Favourable pairs $(1, 3)$ , $(2, 2)$ , $(3, 1)$ give 3 of 9.
$\frac{1}{3}$ .
Find $P (even or greater than 4)$ for a fair standard die.
2 marks
Outcomes ${2, 4, 5, 6}$ give $\frac{4}{6}$ .
$\frac{2}{3}$ .
A bag has B1, B2 and G. Draw two tokens at random without replacement. Find $P (both blue)$ .
3 marks
Favourable pairs B1B2 and B2B1 give 2 of 6.
$\frac{1}{3}$ .

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Curriculum reference & lesson scope

Australian Curriculum v9 and SA Curriculum

Complementary events; two-stage sample spaces; tables, trees and Venn diagrams; probabilities of combined outcomes; experimental versus theoretical results; compound-event simulations and describing results. Formal conditional probability is Year 10.

Official curriculum source ↗

In this lesson

Understand the idea.

Translate the event first

Combine paths carefully

See the method in action.

Worked example 1

List all equally likely ordered results.

Identify the only excluded result.

Use the complementary event.

Worked example 2

Count ordered physical-token draws.

List the successful colour orders.

Divide the successful count by the full count.

Check with changing branch probabilities.

Try it yourself.

Check your understanding

Keep practising