Year 10: Probability with tables and diagrams

Tables and diagrams are useful probability tools for Year 10 students in Australia because they help organise information clearly. They are especially helpful in compound-event questions where it is easy to miss or double-count outcomes.

If the outcomes can be organised into rows and columns, a table often makes the favourable outcomes much easier to see.

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

Why use a table?

A table is a systematic way to list outcomes from two stages. It is often more compact than writing a long list of ordered pairs, and it makes patterns easier to spot.

Example 1: A coin is tossed and a die is rolled. A table could use the coin outcomes as rows and the die outcomes as columns. Every cell then represents one combined outcome.

Skill quiz: Organise outcomes in a table

Build a table with one experiment along each axis and one combined outcome in each cell.

Probability of compound events

Reading favourable regions

Once the table is built, mark the cells that satisfy the event. This is a strong method for “and” and “or” probability questions because you can literally see the region you are counting.

Example 2: A fair spinner labelled

1, 2, 3

is spun twice. Find the probability that the total is at least 5.

There are $3 \times 3 = 9$ table cells.
The totals at least 5 are from outcomes $(2, 3), (3, 2), (3, 3)$ .
So there are $3$ favourable outcomes.
$\Pr (total at least 5) = 3 / 9 = 1 / 3$ .

Skill quizzes: Identify favourable table cells

Mark the cells that satisfy the event, then practise reading probabilities from two-way tables.

Using tree diagrams

Tree diagrams are another way to organise probability. They are often useful when the stages are sequential or when the wording of the experiment naturally follows a “first, then” structure.

Example 3: Two fair coins are tossed. A tree diagram has two first branches

H

and

T

, then two more branches from each. The final outcomes are

HH, HT, TH, TT

Skill quiz: List outcomes along tree branches

Sketch a tree for the compound experiment and trace every complete path.

Probability of compound events

Connecting diagrams to probability

Whether you use a table or a tree, the next step is the same: count the favourable outcomes and divide by the total number of outcomes, provided the outcomes are equally likely.

Example 4: From the four outcomes

HH, HT, TH, TT

, the event “at least one head” includes

HH, HT, TH

. Therefore

\Pr (at least one head) = 3 / 4

Skill quizzes: Calculate from the diagram

Use the favourable and total equally likely outcomes to calculate simple and compound probabilities.

Common mistakes

Leaving blanks in the table and missing outcomes.
Marking the wrong cells because the event wording was not translated carefully.
Using a diagram but then counting favourable outcomes inaccurately.
Forgetting that the final probability still comes from favourable over total outcomes.

Skill quiz: Check cells, paths and event wording

Check that all outcomes are included and count only those satisfying the stated event.

Probability of compound events

Study routine

If a probability question has two stages, ask whether a table or tree diagram would make the outcomes clearer. In Year 10, neat organisation is often what separates a correct answer from a counting error.

Skill quiz: Choose and use a clear diagram

Choose a table or tree, list the outcomes systematically and use the diagram to support the calculation.

Probability of compound events