Compound events | Year 10 | Ascent Library

01 / LEARN

The idea, step by step

Keep complete outcomes

A compound experiment has more than one stage. Record a complete outcome, such as (head,4) for a coin toss followed by a die roll. For a fair coin and fair die there are 2×6=12 equally likely complete outcomes.

“A and B” means both events occur. “A or B” normally includes outcomes in A, in B, or in both. If the events overlap, adding their probabilities counts the overlap twice, so subtract it once.

P (A \cup B) = P (A) + P (B) - P (A \cap B)

Equiprobable outcomes must be justified

For two fair dice, ordered pairs such as (1,6) and (6,1) are distinct outcomes. The 36 ordered pairs are equally likely; the 11 possible sums are not. Six pairs make a sum of 7 but only one pair makes 2.

Organise a sample space with a systematic list, grid or tree before counting favourable outcomes. Check that outcomes are neither omitted nor counted twice.

A common trap

Treating the sums of two dice as equally likely.

Instead: Count equally likely ordered pairs. Different sums occur in different numbers of ways.

02 / WORKED EXAMPLES

See the thinking

Example 1

Coin and die

A fair coin and fair six-sided die are used. Find the probability of a head and an even number.

There are 12 equally likely outcomes.
The favourable outcomes are (H,2), (H,4), (H,6).
3/12=1/4.

\frac{1}{4}

Example 2

An overlapping or event

A fair die is rolled. Find P(even or greater than 4).

Even={2,4,6}; greater than 4={5,6}.
The union is {2,4,5,6}; 6 must not be counted twice.

\frac{4}{6} = \frac{2}{3}

Check your understanding

For one fair die, what is P(even and greater than 3)?

Show answer

$\frac{1}{3}$ .

The favourable results are 4 and 6, so 2/6=1/3.

03 / WORKSHEET

Your turn

6 questions · 15 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

List the complete outcomes for two coin tosses.
2 marks
Two fair coins are tossed. Find P(exactly one head).
2 marks
A fair six-sided die is rolled. Find P(odd or prime).
2 marks
Two fair dice are rolled. Find P(sum 7).
3 marks
Find P(at least one 6) on two fair dice.
3 marks
A fair coin is tossed independently of a fair six-sided die roll. Find P(head or die shows 6).
3 marks

Read the worked solutions

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

List the complete outcomes for two coin tosses.
2 marks
Keep first and second toss positions distinct.
HH, HT, TH, TT.
Two fair coins are tossed. Find P(exactly one head).
2 marks
HT and TH are favourable: 2/4.
$\frac{1}{2}$ .
A fair six-sided die is rolled. Find P(odd or prime).
2 marks
Odd={1,3,5}, prime={2,3,5}.
Union={1,2,3,5}:4/6.
$\frac{2}{3}$ .
Two fair dice are rolled. Find P(sum 7).
3 marks
There are 36 ordered pairs.
Six pairs sum to 7, giving 6/36.
$\frac{1}{6}$ .
Find P(at least one 6) on two fair dice.
3 marks
The complement has neither die equal to 6.
1−(5/6)^2=11/36.
$\frac{11}{36}$ .
A fair coin is tossed independently of a fair six-sided die roll. Find P(head or die shows 6).
3 marks
Head contributes 6 outcomes, a 6 contributes 2, and (H,6) overlaps.
(6+2−1)/12=7/12.
$\frac{7}{12}$ .

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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 ↗

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