Year 10: Probability of compound events

A compound event involves more than one stage or more than one condition. In Year 10 Mathematics in Australia, these questions often involve two spins, two tosses, rolling two dice, or choosing items under more than one condition.

For compound events, start by listing the sample space clearly. Tables, tree diagrams, and ordered pairs are especially useful.

Work through the sections in order, then try the linked Year 10 skill quizzes after each section.

Two-stage experiments

A compound event question often combines two smaller experiments. To solve it well, show every possible combined outcome and then count the favourable ones.

Example 1: A fair coin is tossed and a fair die is rolled. Find the probability of getting a head and an even number.

There are $2 \times 6 = 12$ total outcomes.
Favourable outcomes are $(H, 2), (H, 4), (H, 6)$ .
So there are $3$ favourable outcomes.
$\Pr (H and even) = 3 / 12 = 1 / 4$ .

Skill quiz: List combined outcomes

Record all outcomes from both stages before counting those that satisfy the event.

Probability of compound events

Using ordered pairs and tables

Ordered pairs are a neat way to record compound outcomes. Tables are also useful, especially when the two stages come from different categories such as colour and shape.

Example 2: Two fair coins are tossed. Find the probability of getting exactly one head.

The sample space is ${HH, HT, TH, TT}$ .
The favourable outcomes are $HT$ and $TH$ .
$\Pr (exactly one head) = 2 / 4 = 1 / 2$ .

Skill quiz: Keep the two stages in order

Use a table or ordered pairs so that distinct outcomes are not combined or missed.

Probability of compound events

Worded compound-event questions

In school assessments, the event may be described in words. Translate the wording into outcomes before you calculate anything. That step usually prevents the biggest mistakes.

Example 3: A spinner has four equal sectors labelled

1, 2, 3, 4

. It is spun twice. Find the probability that the total is 5.

There are $4 \times 4 = 16$ ordered outcomes.
The favourable outcomes are $(1, 4), (2, 3), (3, 2), (4, 1)$ .
So there are $4$ favourable outcomes.
$\Pr (total 5) = 4 / 16 = 1 / 4$ .

Skill quiz: Translate the event into outcomes

Underline words such as and, or and exactly before identifying the favourable outcomes.

Probability of compound events

Common mistakes

Not listing the full two-stage sample space.
Ignoring order when order does matter.
Counting one favourable outcome twice or missing it entirely.
Confusing “and” with “or” in the event wording.

Skill quiz: Check the full sample space

Check order, avoid double-counting and verify the interpretation of the event wording.

Probability of compound events

Study routine

For a compound-event question, write the total number of possible combined outcomes first. Then mark the favourable outcomes one by one. This is slower than guessing, but much more reliable.

Skill quiz: Count before forming the probability

Write the total number of combined outcomes, mark the favourable ones and simplify the probability.

Probability of compound events