GCSE Maths Probability

Probability Distribution

Relative Frequency

# Relative Frequency

Here we will learn about relative frequency, including how to calculate it.

There are also relative frequency worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

## What is relative frequency?

Relative frequency is the number of times an event happens divided by the total number of outcomes that took place in an experiment, known as the number of trials.

To calculate the relative frequency we can use the formula,

\text{Relative frequency = }\frac{\text{frequency of event occurring}}{\text{total number of trials of the experiment}}

For example,

Jo made a four-sided spinner out of a piece of card and a pencil. She spun the spinner 50 times and the spinner landed on a three 15 times. Calculate the relative frequency of the spinner landing on a three.

Relative frequency of landing on a three = \frac{15}{50} or \frac{3}{10} or 0.3 or 30\%.

The more times an experiment is carried out, the more reliable the relative frequency will be and closer to the theoretical probability.

### What is relative frequency? ## How to calculate relative frequency

In order to calculate relative frequency:

1. Find the number of times the event occurs.
2. Find the number of trials of the experiment.
3. Use the formula

\textbf{Relative frequency = }\frac{\textbf{frequency of event occurring}}{\textbf{total number of trials of the experiment}}

### Explain how to calculate relative frequency ## Relative frequency examples

### Example 1: calculating relative frequency

A coin is flipped 50 times and shows heads on 22 occasions. Calculate the relative frequency of the coin showing heads.

1. Find the number of times the event occurs.

The coin showed heads 22 times.

2Find the number of trials of the experiment.

The coin was flipped 50 times.

3Use the formula

\textbf{Relative frequency = }\frac{\textbf{frequency of event occurring}}{\textbf{total number of trials of the experiment}}

Relative frequency =\frac{22}{50}=0.44

### Example 2: calculating relative frequency

A coin shows heads 15 times having been flipped 40 times. Calculate the relative frequency of the coin showing tails.

If the coin showed heads 15 times, it would have showed tails 25 times.

The coin was flipped 40 times.

Relative frequency =\frac{25}{40}=0.625

### Example 3: calculating the number of trials if relative frequency is known

An experiment rolling a normal 6 -sided dice obtained the number 4 on 32 occasions.

The relative frequency of rolling a 4 was calculated as 0.2. Find the number of times the dice was rolled.

The number 4 was rolled 32 times.

This is what we need to calculate using step 3.

As we already know the relative frequency, substitute this and the frequency of rolling a 4 into the formula.

0.2=\frac{32}{\text{number of times the die was rolled}}

Rearrange the formula.

Number of times the dice was rolled =\frac{32}{0.2}=160 .

### Common misconceptions

• Assuming relative frequency is equal to the theoretical probability

The relative frequency of an event is determined by the number of trials or observations. If an experiment only has a small number of trials it is possible for the relative frequency to be very different from the theoretical probability of the same event. For example – if a coin is flipped only 5 times, it is very possible for it to show heads on all 5 occasions.

• Assuming a relative frequency with few trials is a good estimate for the theoretical probability

The more times an experiment is carried out, the more reliable the relative frequency will be and closer to the theoretical probability.

### Practice relative frequency questions

1. A coin is flipped 80 times and shows heads 50 times. Find the relative frequency for the coin showing heads, giving your answer as a fraction in its simplest form.

\frac{50}{80} \frac{5}{8} \frac{3}{8} \frac{1}{2} \frac{50}{80} simplifies to \frac{5}{8} .

2. A coin is flipped 60 times and shows tails 24 times. Find the relative frequency for the coin showing tails, giving your answer as a decimal.

\frac{24}{60} 0.5 0.4 0.6 \frac{24}{60}=0.4

3. A coin is flipped 50 times and shows heads 31 times. Find the relative frequency for the coin showing tails, giving your answer as a fraction.

\frac{19}{50} \frac{31}{50} \frac{25}{50} \frac{29}{50} If the coin shows heads 31 times it must show tails 19 times. 50-31 = 19.

4. A 3 sided spinner, numbered 1, \ 2 and 3, is spun 200 times. It lands on 1, \ 60 times, and lands on 2, \ 85 times. Calculate the relative frequency of the spinner landing on 3. Give your answer as a fraction in its simplest form.

\frac{3}{10} \frac{17}{40} \frac{55}{200} \frac{11}{40} 200-60-85=55

\frac{55}{200}=\frac{11}{40}

5. A normal 6- sided dice is rolled 150 times. The relative frequency of the die landing on 2 is 0.36. How many times did the die land on 2?

54 36 25 150 0.36 \times 150 = 54

6. A 4- sided spinner numbered 1 to 4 is spun n times in an experiment. The spinner lands on 3, 76 times. The relative frequency of the spinner landing on 3 was calculated as \frac{19}{75}. Find the value of n.

75 150 76 300 76\div \frac{19}{75}=300

### Relative frequency GCSE questions

1. A school recorded the number of times different fruits were bought by students for their lunch. Find the relative frequency of

(a) a student buying an apple.

(b) a student not buying grapes.

(4 marks)

(a)

Evidence of a fraction with denominator 150 .

(1)

\frac{53}{150} or 0.353… or equivalent.

(1)

(b)

Evidence of a fraction with denominator 150 .

(1)

\frac{121}{150} or 0.8066… or equivalent.

(1)

2. Jenny is doing an experiment to test the fairness of a 6- sided dice. She rolled the dice 12 times and got the following results. Jenny claims that the dice is not fair based on her results.

(a) Explain why Jenny may not be correct.

Simon repeats the experiment with the same die but rolls the die 36 times. His results are below. (b) Using all the information available find the relative frequency of rolling a 4 on the dice.

(4 marks)

(a)

“Jenny has not rolled the die many times” or “if she rolled the die more times her results would be more accurate”

(1)

(b)

7 + 2 = 9

(1)

12 + 36 = 48

(1)

\frac{9}{48} or 0.1875 or equivalent.

(1)

3. A coin is flipped 500 times and shows tails 324 times.

Find the relative frequency of the coin showing heads.

(2 marks)

500-324 = 176

(1)

0.352 or equivalent.

(1)

## Learning checklist

You have now learned how to:

• Calculate the relative frequency of an event occurring from a probability experiment

## The next lessons are

• Theoretical probability
• Sample space
• Experimental probability
• Expected frequency
• Probability distributions
• Histograms

## Still stuck?

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#### FREE GCSE Maths Practice Papers - 2022 Topics

Practice paper packs based on the advanced information for the Summer 2022 exam series from Edexcel, AQA and OCR.

Designed to help your GCSE students revise some of the topics that will come up in the Summer exams.