Frequency Trees

What Are Frequency Trees?

A frequency tree helps us organise information about how a group splits into different categories. It's especially useful when we're given totals and percentages, or when we want to calculate probabilities from combinations of options.

Each branch of a tree shows a choice, and each bubble (or number) tells us how many people or items fall into that group.

They're great for showing how two different characteristics are distributed across a group.

 

Key Features:

Start with the total in a circle.

Split it by the first category (e.g., Morning vs Afternoon).

Then split each branch again using the second category (e.g., Male vs Female).

Make sure each set of branches adds back up to its parent category

 

Example

A school surveyed 120 students. Each student either studies French or German.

70 study French.

Of the French students, 42 are girls.

Of the German students, 30 are boys.

(a) Draw a frequency tree to represent this information.

Let's fill in step by step:

Total: 120 students

French: 70 $\to$ German: 120 − 70 = 50

French girls: 42 $\to$ French boys: 70 − 42 = 28

German boys: 30 $\to$ German girls: 50 − 30 = 20

 

 

(b) Find the probability that a randomly selected student:

(i) Studies German and is a boy

We want P(German AND boy)

German boys = 30

Total students = 120

$P = \frac{30}{120} = \frac{1}{4}$

 

(ii) Studies French, given that the student is a girl

This is a conditional probability: P(French | Girl)

Total number of girls = 42 (French) + 20 (German) = 62 French girls = 42

$P(\text{French} \mid \text{Girl}) = \frac{42}{62} = \frac{21}{31}$