Frequency and outcomes Revision Notes for Edexcel GCSE Maths

Frequency and outcomes

When dealing with probability problems, you often need to record all possible results from two or more events. There are two main tools to help you organise this information: frequency trees and sample space diagrams. These visual methods make complex probability calculations much more manageable and help prevent errors in your working.

Frequency trees

Frequency trees are visual diagrams that show the frequencies (how many times something happens) for each possible result of different events. They provide a systematic way to organise numerical data and make calculations involving multiple events much clearer.

The golden rule

The most important principle when working with frequency trees is a fundamental rule that governs how all the numbers relate to each other:

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Golden rule: In a frequency tree, each frequency equals the sum of its branches.

This means that any number at a junction point equals the total of the numbers on the branches coming from it.

How to use frequency trees

When completing a frequency tree, following a systematic approach will help you avoid mistakes and work efficiently:

Start with the information you know
Use the golden rule to calculate missing values
Work systematically through each branch
Check your answers by ensuring totals match

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Always double-check your frequency tree by verifying that each junction follows the golden rule. This simple check can help you catch calculation errors before moving to the final probability calculations.

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Worked Example: Reading habits

Let's say Wilfred owns 80 books total. He hasn't read 15 of them. Of the books he has read, 20 are hardbacks. He has 52 paperbacks in total, and the remaining books are hardbacks.

Step 1: Start with what you know
Write 15 in the "has not read" branch, since Wilfred hasn't read 15 books

Step 2: Use subtraction
If he owns 80 books total and hasn't read 15, then he has read $80 - 15 = 65$ books

Step 3: Calculate hardback totals
He has $80 - 52 = 28$ hardbacks in total. Since 20 hardbacks have been read, then $28 - 20 = 8$ hardbacks haven't been read

Step 4: Complete the tree
Use the golden rule to fill in remaining values

Answer: The probability that a randomly chosen book is a paperback he hasn't read would be $\frac{7}{80}$ .

Sample space diagrams

A sample space diagram displays all the possible results when two events happen together. These diagrams are particularly useful for showing combined outcomes and ensuring you don't miss any possible results when calculating probabilities.

Creating sample space diagrams

To create an effective sample space diagram that captures all possible outcomes:

List the outcomes of the first event along one axis
List the outcomes of the second event along the other axis
Fill in each cell with the combined result
Count the total possibilities to help calculate probabilities

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Sample space diagrams are especially powerful when dealing with independent events, as they make it easy to see that all outcomes are equally likely and help you count favourable outcomes accurately.

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Worked Example: Coin flipping

When flipping two coins, the sample space diagram shows four possible outcomes:

HH (heads, heads)
HT (heads, tails)
TH (tails, heads)
TT (tails, tails)

This makes it easy to see that there are four equally likely outcomes, each with probability $\frac{1}{4}$ .

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Worked Example: Counters in a bag

A bag contains 30 counters that are either black or white. A counter is chosen randomly, and the probability it's black is $\frac{1}{5}$ . How many white counters are there?

Solution:

$P(\text{White}) = 1 - P(\text{Black}) = 1 - \frac{1}{5} = \frac{4}{5}$
Number of white counters = $\frac{4}{5} \times 30 = 24$
Check: There are $30 - 24 = 6$ black counters, so $P(\text{Black}) = \frac{6}{30} = \frac{1}{5}$ ✓

Exam tips

Understanding these concepts is one thing, but performing well in exams requires specific strategies:

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Exam Success Strategies:

Show your working clearly - examiners want to see your method
Use the golden rule to check your frequency tree calculations
List all outcomes systematically in sample space diagrams
Check your final answers make sense in the context

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Key Points to Remember:

Frequency trees show how often different outcomes occur and use the golden rule where each frequency equals the sum of its branches
Sample space diagrams display all possible combinations when two events happen together
Always start with the information you know and work systematically through the problem
Check your work by ensuring totals add up correctly
Both tools help you calculate probabilities by organising information clearly

Frequency and outcomes (Edexcel GCSE Maths): Revision Notes