What is P3: Relative frequency?

Probabilities lie on a scale from 0 to 1: 0 is impossible, 1 is certain, and 0.5 means equally likely to happen or not happen.

How many marks are P3 questions worth?

In verified past-paper appearances, P3 questions on Edexcel GCSE Maths 1MA1 have been worth 2–6 marks.

Which past exam papers has P3 appeared in?

P3 has appeared in 19 verified past-paper questions indexed on this page, with direct links to the official question paper and mark scheme for each.

Relate relative expected frequencies to theoretical probability, using appropriate language and the 0-1 probability scale

Relative frequency

Worked answers, methods and verified real exam appearances for P3 on Edexcel GCSE Maths 1MA1.

Explanation

Probabilities lie on a scale from $0$ to $1$ : $0$ is impossible, $1$ is certain, and $0.5$ means equally likely to happen or not happen.
Theoretical probability comes from a model, while relative frequency comes from results: $\text{relative frequency}=\dfrac{\text{frequency}}{\text{number of trials}}$ .
In many unbiased trials, relative frequency is expected to settle near the theoretical probability, but random variation means the two values need not be exactly equal.

Worked example

A coin lands heads $156$ times in $300$ tosses. Find the relative frequency of heads and compare it with the theoretical probability.

1.Relative frequency $=\dfrac{156}{300}=0.52$ .
2.For a fair coin, the theoretical probability is $0.5$ .
3. $0.52$ is close to $0.5$ ; the difference can be due to random variation.

Answer: The relative frequency is $0.52$ , close to the theoretical value $0.5$ .

Common mistakes

Don't fall into the trap of dividing by the wrong total number of trials.
Don't fall into the trap of claiming a small experimental difference proves that the model is wrong.

Exam tip

Use probability language such as impossible, unlikely, even chance, likely and certain accurately.

Worked practice

Tier 1 · Easy

Place these events in order from least likely to most likely: event A has probability

0.72

, event B has probability

\dfrac{1}{4}

, and event C has probability

0.5

(1)

(Total for Question 1 is 1 mark)

Mark scheme

Mark scheme for question 1
Question	Answer	Mark	Mark scheme
1	B, C, A	1	Write every probability as a decimal: $\dfrac{1}{4}=0.25$ , then compare $0.25<0.5<0.72$ .

Tier 2 · Standard

An outcome occurs

84

times in

140

trials. Its theoretical probability is

\dfrac{5}{8}

. Work out the relative frequency and the expected frequency in

560

further trials. Comment on the experimental result.

(3)

(Total for Question 2 is 3 marks)

Mark scheme

Mark scheme for question 2
Question	Answer	Mark	Mark scheme
2	Relative frequency $=0.6$ . Expected frequency $=350$ . $0.6$ is reasonably close to the theoretical probability $0.625$ .	3	The relative frequency is $84/140=0.6$ . The theoretical probability is $5/8=0.625$ , so the expected frequency is $560\times5/8=350$ . The observed value differs by only $0.025$ , which is plausible experimental variation.

Tier 3 · Hard

A model says that a device flashes with probability

0.5

on each trial. In a short run it flashes

41

times in

80

trials. After

500

trials in total it has flashed

247

times. Calculate both relative frequencies and decide which result gives stronger evidence about the model.

(4)

(Total for Question 3 is 4 marks)

Mark scheme

Mark scheme for question 3
Question	Answer	Mark	Mark scheme
3	Short-run relative frequency $=0.5125$ . Overall relative frequency $=0.494$ . The $500$ -trial result gives stronger evidence and is very close to $0.5$ , so it supports the model.	4	For the short run, calculate $41/80=0.5125$ . For all trials, calculate $247/500=0.494$ . Both are close to $0.5$ , but the larger sample is less affected by random variation, so the overall relative frequency is the stronger comparison.

Tier 1 · Easy

A card is chosen at random from cards labelled CAT, DOG and OWL. Write down the probability that the chosen word has three letters and the probability that it has four letters.

(2)

(Total for Question 4 is 2 marks)

Mark scheme

Mark scheme for question 4
Question	Answer	Mark	Mark scheme
4	Three letters: $1$ ; four letters: $0$ .	2	Every word has three letters, so that event is certain and has probability $1$ . No word has four letters, so that event is impossible and has probability $0$ .

Tier 2 · Standard

An event occurs

18

times in

80

trials. Its theoretical probability is

0.2

. Work out the relative frequency and choose the best description of the theoretical probability: impossible, unlikely, even chance, likely or certain.

(2)

(Total for Question 5 is 2 marks)

Mark scheme

Mark scheme for question 5
Question	Answer	Mark	Mark scheme
5	Relative frequency $=0.225$ ; the event is unlikely.	2	The relative frequency is $18\div80=0.225$ . A probability of $0.2$ lies between $0$ and $0.5$ , so 'unlikely' is the best description.

Tier 3 · Hard

A spinner is designed to land on blue with probability

0.35

. In a test it lands on blue

52

times in

160

spins. Use the relative frequency to estimate the number of blues in the next

600

spins. Compare this with the number predicted by the theoretical probability.

(4)

(Total for Question 6 is 4 marks)

Mark scheme

Mark scheme for question 6
Question	Answer	Mark	Mark scheme
6	Relative frequency $=0.325$ ; estimate from the test $=195$ blues. The theoretical prediction is $210$ blues, which is $15$ more than the test-based estimate.	4	The observed relative frequency is $52/160=0.325$ . This gives $600\times0.325=195$ blues. The theoretical model gives $600\times0.35=210$ , and $210-195=15$ .

Tier 2 · Standard

The theoretical probability of event A is

0.18

. In an experiment, event A occurs

23

times in

100

trials. Event B is the complement of event A in the theoretical model. Work out the experimental relative frequency of A and the theoretical probability of B. Put these two values and

0.18

in increasing order, then describe A and B as likely or unlikely using the theoretical model.

(3)

(Total for Question 7 is 3 marks)

Mark scheme

Mark scheme for question 7
Question	Answer	Mark	Mark scheme
7	Experimental relative frequency of A $=0.23$ ; theoretical $P(B)=0.82$ . $0.18<0.23<0.82$ ; A is unlikely and B is likely in the theoretical model.	3	The experimental relative frequency is $23/100=0.23$ . Since B is the complement of A, its theoretical probability is $1-0.18=0.82$ . Order the decimals directly. A theoretical probability below $0.5$ is described as unlikely, while one above $0.5$ is described as likely.

Tier 3 · Hard

A probability model predicts that an event will occur

144

times in

360

trials. In an experiment, the event occurs

93

times in

240

trials. Work out the theoretical probability and the experimental relative frequency. For

800

further trials, work out the expected frequency from each value and compare the predictions.

(4)

(Total for Question 8 is 4 marks)

Mark scheme

Mark scheme for question 8
Question	Answer	Mark	Mark scheme
8	Theoretical probability $=0.4$ ; experimental relative frequency $=0.3875$ . Model prediction: $320$ ; experiment-based prediction: $310$ ; the model predicts $10$ more occurrences.	4	The model probability is $144/360=0.4$ . The experimental relative frequency is $93/240=0.3875$ . Multiplying each by $800$ gives $320$ from the model and $310$ from the experiment, a difference of $10$ .

Tier 3 · Hard

A probability model gives an event a theoretical probability of

0.4

. The event occurs

57

times in the first

150

trials and

105

times in the next

250

trials. Work out the relative frequency for each block of trials and for all

400

trials. Which result is closest to the theoretical probability, and which result gives the strongest overall evidence about the model?

(4)

(Total for Question 9 is 4 marks)

Mark scheme

Mark scheme for question 9
Question	Answer	Mark	Mark scheme
9	First block: $0.38$ ; second block: $0.42$ ; all $400$ trials: $0.405$ . The combined result is closest to $0.4$ and gives the strongest overall evidence because it uses all $400$ trials.	4	Calculate $57/150=0.38$ and $105/250=0.42$ . Across both blocks the event occurs $162$ times, so the combined relative frequency is $162/400=0.405$ . Its distance from $0.4$ is $0.005$ , compared with $0.02$ for either block, and it uses the largest sample.

Q10

Tier 3 · Hard

An event occurs

44

times in the first

110

trials. It then occurs

32

more times. The relative frequency after all the trials is

0.38

. Work out how many further trials were carried out. The theoretical probability of the event is

0.375

. Compare the first and final relative frequencies with this value.

(4)

(Total for Question 10 is 4 marks)

Mark scheme

Mark scheme for question 10
Question	Answer	Mark	Mark scheme
10	$90$ further trials were carried out. First relative frequency: $0.4$ ; final relative frequency: $0.38$ . The final value is closer to $0.375$ .	4	The event occurs $44+32=76$ times altogether. Since $76/N=0.38$ , the total number of trials is $N=200$ , so there were $200-110=90$ further trials. The first relative frequency is $44/110=0.4$ , which is $0.025$ from $0.375$ ; the final value is only $0.005$ away.

Verified exam appearances

Series	Paper	Question	Marks	Calculator	Tier	Links
2019-11	2F	Q9	6	Allowed	Foundation	QP MS
2023-06	2F	Q6	2	Allowed	Foundation	QP MS
2024-11	2F	Q7	5	Allowed	Foundation	QP MS
2021-11	2F	Q16	5	Allowed	Foundation	QP MS
2022-11	1F	Q7	3	Non-calculator	Foundation	QP MS
2019-06	1H	Q1	4	Non-calculator	Higher	QP MS
2023-06	3F	Q7	3	Allowed	Foundation	QP MS
2022-06	3F	Q10	4	Allowed	Foundation	QP MS
2019-11	1F	Q15	4	Non-calculator	Foundation	QP MS
2023-11	3F	Q15	5	Allowed	Foundation	QP MS
2024-06	1H	Q4	3	Non-calculator	Higher	QP MS
2024-06	2F	Q23	4	Allowed	Foundation	QP MS
2019-11	1H	Q10	6	Non-calculator	Higher	QP MS
2019-06	1F	Q22	4	Non-calculator	Foundation	QP MS
2022-06	1F	Q13	2	Non-calculator	Foundation	QP MS
2022-06	1F	Q17	5	Non-calculator	Foundation	QP MS
2023-06	2H	Q10	4	Allowed	Higher	QP MS
2021-11	3F	Q26	4	Allowed	Foundation	QP MS
2024-11	1F	Q19	4	Non-calculator	Foundation	QP MS

Other points in P Probability

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