23.3 Frequency trees, tables and possibility spaces — Learn

Prerequisites: multiplication, fractions and systematic listing.

Objectives: organise data in a frequency tree or table; list outcomes systematically; calculate probabilities by counting outcomes.

A frequency tree splits a total into groups and subgroups. Each branch count tells how many observations are in that group. A two-way table displays counts for two categories. A possibility space lists every possible outcome of an experiment.

Example 1. A survey asks 30 students whether they walk to school. Of 18 who walk, 11 are girls. Of 12 who do not walk, 5 are girls. Complete the counts.

Walking girls: 11; walking boys: $18-11=7$.

Non-walking girls: 5; non-walking boys: $12-5=7$.

Girls total: $11+5=16$; boys total: $7+7=14$.

The table entries total $16+14=30$, which checks the survey total.

Example 2. Two fair coins are tossed. List the outcomes and find the probability of one head.

List the first coin then the second: HH, HT, TH, TT.

Two of four outcomes contain exactly one head: HT and TH.

Watch out. In a possibility space, HT and TH are different outcomes because the first and second tosses are distinct.

Progress check

A frequency tree of 50 spins has 28 red and 22 blue. If 16 reds come from the first 25 spins, work out the reds in the second 25. Check each pair of branches adds to its parent count.

You should now be able to

• read and complete frequency trees and two-way tables;
• list a possibility space systematically and use it to find probability.