What is P5: Sampling and probability?

An empirical distribution is based on observed relative frequencies; a theoretical distribution is predicted by a probability model.

How many marks are P5 questions worth?

In verified past-paper appearances, P5 questions on Edexcel GCSE Maths 1MA1 have been worth 2–4 marks.

Which past exam papers has P5 appeared in?

P5 has appeared in 3 verified past-paper questions indexed on this page, with direct links to the official question paper and mark scheme for each.

Understand that empirical unbiased samples tend towards theoretical probability distributions, with increasing sample size

Sampling and probability

Worked answers, methods and verified real exam appearances for P5 on Edexcel GCSE Maths 1MA1.

Explanation

An empirical distribution is based on observed relative frequencies; a theoretical distribution is predicted by a probability model.
With unbiased, independent trials, a larger sample usually gives more stable relative frequencies that are closer to the theoretical probabilities.
This is a long-run tendency, not a promise that every larger sample will be closer.
Increasing sample size reduces random variation, but it does not remove bias caused by an unfair device or poor sampling method.

Worked example

A fair coin gives $162$ heads in $300$ tosses and $1038$ heads in $2000$ tosses. Compare the two relative frequencies with $0.5$ .

1.For $300$ tosses: $\dfrac{162}{300}=0.54$ .
2.For $2000$ tosses: $\dfrac{1038}{2000}=0.519$ .
3. $0.519$ is closer to the theoretical probability $0.5$ than $0.54$ .

Answer: The larger sample gives the closer relative frequency in this experiment.

Common mistakes

Don't fall into the trap of saying that a larger sample guarantees an exact theoretical distribution.
Don't fall into the trap of assuming more trials can correct a biased experiment.

Exam tip

When comparing experiments, calculate relative frequencies rather than comparing raw counts.

Worked practice

Tier 1 · Easy

A fair coin gives a relative frequency of heads of

0.64

after

25

tosses. Explain what is likely to happen to the relative frequency as many more unbiased tosses are made.

(1)

(Total for Question 1 is 1 mark)

Mark scheme

Mark scheme for question 1
Question	Answer	Mark	Mark scheme
1	It is likely to move closer to the theoretical probability $0.5$ , although it need not do so after every extra toss.	1	Use the long-run tendency: for unbiased coin tosses the empirical proportion tends towards the theoretical probability $1/2$ as the sample becomes large.

Tier 2 · Standard

A spinner is designed to land on green with probability

0.3

. It lands on green

8

times in

20

spins and

63

times in

200

spins. Compare the two empirical probabilities with the theoretical probability.

(3)

(Total for Question 2 is 3 marks)

Mark scheme

Mark scheme for question 2
Question	Answer	Mark	Mark scheme
2	For $20$ spins, the relative frequency is $0.4$ . For $200$ spins, the relative frequency is $0.315$ . The larger sample gives the value closer to $0.3$ .	3	Calculate $8/20=0.4$ and $63/200=0.315$ . Their distances from $0.3$ are $0.1$ and $0.015$ , so the empirical result from $200$ spins is closer to the theoretical probability.

Tier 3 · Hard

A fair die is tested. In the first

60

rolls, a six appears

16

times. After

600

rolls in total, a six has appeared

108

times. A student says the die must be biased because neither relative frequency equals

\dfrac{1}{6}

. Evaluate the student's claim.

(4)

(Total for Question 3 is 4 marks)

Mark scheme

Mark scheme for question 3
Question	Answer	Mark	Mark scheme
3	The relative frequencies are $\dfrac{16}{60}\approx0.267$ and $\dfrac{108}{600}=0.18$ . The theoretical probability is $\dfrac{1}{6}\approx0.167$ . The larger-sample result is closer to the theoretical value, and an exact match is not expected, so the evidence does not establish bias.	4	Compare each empirical probability with $1/6$ . The differences are about $0.100$ for the first $60$ rolls and $0.013$ after $600$ rolls. Random variation explains a non-zero difference, and the movement towards $1/6$ as the sample grows is consistent with a fair die.

Tier 1 · Easy

Kai tosses an unbiased coin many times and recalculates the relative frequency of heads after every

10

tosses. Give a reason why the first few relative frequencies are likely to fluctuate more than the later ones.

(1)

(Total for Question 4 is 1 mark)

Mark scheme

Mark scheme for question 4
Question	Answer	Mark	Mark scheme
4	The early relative frequencies use small samples, so each new result has a larger effect. With more tosses, the relative frequency tends to become more stable and approach $0.5$ .	1	For a small total, one extra head changes the fraction by a relatively large amount. As the denominator grows, one result changes the fraction less, so random variation has less effect.

Tier 2 · Standard

A spinner has theoretical probabilities

0.2

0.3

and

0.5

for red, yellow and blue. In

50

spins the frequencies are

8

19

and

23

. In

500

spins the frequencies are

102

146

and

252

. Work out the two empirical distributions and decide which is closer to the theoretical distribution.

(4)

(Total for Question 5 is 4 marks)

Mark scheme

Mark scheme for question 5
Question	Answer	Mark	Mark scheme
5	$50$ spins: red $0.16$ , yellow $0.38$ , blue $0.46$ . $500$ spins: red $0.204$ , yellow $0.292$ , blue $0.504$ . The $500$ -spin distribution is closer to the theoretical distribution.	4	Divide each frequency by its experiment total. For $50$ spins this gives $8/50=0.16$ , $19/50=0.38$ and $23/50=0.46$ . For $500$ spins it gives $102/500=0.204$ , $146/500=0.292$ and $252/500=0.504$ . Every value in the larger sample is closer to its theoretical probability.

Tier 3 · Hard

A council wants to estimate the proportion of all teenagers who play chess online. It first asks

30

teenage chess-club members and

18

say yes. It later asks

900

teenage chess-club members at several tournaments and

540

say yes. Work out both empirical probabilities. Is the second result necessarily a reliable estimate for all teenagers because its sample is larger? Give a reason for your answer.

(4)

(Total for Question 6 is 4 marks)

Mark scheme

Mark scheme for question 6
Question	Answer	Mark	Mark scheme
6	No. Both empirical probabilities are $0.6$ . The second result is more stable for chess-club members, but the sample is biased towards teenagers interested in chess. A larger biased sample need not estimate the proportion for all teenagers reliably.	4	The first empirical probability is $18/30=0.6$ and the second is $540/900=0.6$ . The larger result has less random variation for the sampled group, but every person sampled is a chess-club member, so increasing the sample size does not remove this selection bias.

Tier 2 · Standard

A fair spinner has probability

0.25

of landing on blue. It lands on blue

5

times in its first

20

spins and

56

times after

200

spins in total. A student says a larger sample must always give a relative frequency closer to the theoretical probability. Work out both relative frequencies and decide whether these results support the student's claim. Give a reason for your answer.

(3)

(Total for Question 7 is 3 marks)

Mark scheme

Mark scheme for question 7
Question	Answer	Mark	Mark scheme
7	No, the results do not support the claim. The relative frequencies are $0.25$ and $0.28$ . The larger-sample value is farther from $0.25$ . A larger unbiased sample tends to be more stable, but it is not guaranteed to be closer every time.	3	Calculate $5/20=0.25$ and $56/200=0.28$ . The first value matches the theoretical probability, whereas the later value differs by $0.03$ . Long-run convergence is a tendency rather than a guarantee for every growing sample.

Tier 3 · Hard

A wheel has four sectors with theoretical probabilities

0.15

0.25

0.30

and

0.30

for red, blue, green and white. In

40

spins the frequencies are

9

8

13

and

10

. In

400

spins the frequencies are

58

104

126

and

112

. Work out both empirical distributions. For the

40

-spin distribution, write down the colour whose empirical probability differs most from its theoretical probability. State, with a reason, which of the two distributions you expect to be closer to the theoretical distribution.

(5)

(Total for Question 8 is 5 marks)

Mark scheme

Mark scheme for question 8
Question	Answer	Mark	Mark scheme
8	$40$ spins: $(0.225,0.20,0.325,0.25)$ ; $400$ spins: $(0.145,0.26,0.315,0.28)$ . Red differs most in the $40$ -spin distribution (difference $0.075$ ). The $400$ -spin distribution, because a larger number of trials gives relative frequencies closer to the theoretical probabilities.	5	Divide the first four frequencies by $40$ and the second four by $400$ . For $40$ spins the absolute differences are $0.075,0.05,0.025,0.05$ , so red differs most. For $400$ spins the differences are $0.005,0.01,0.015,0.02$ , all smaller, and the larger number of trials is expected to lie closer to the theoretical distribution.

Tier 3 · Hard

A spinner lands on red

34

times in its first

160

spins. It is then spun a further

80

times. After all

240

spins, the relative frequency of red is exactly

0.25

. A maintenance warning is issued if the relative frequency of red during the further

80

spins is greater than

0.30

. Decide whether a warning should be issued.

(4)

(Total for Question 9 is 4 marks)

Mark scheme

Mark scheme for question 9
Question	Answer	Mark	Mark scheme
9	$26$ of the further spins land on red. Yes. Their relative frequency is $26/80=0.325$ , which is greater than $0.30$ .	4	A relative frequency of $0.25$ after $240$ spins means there are $240\times0.25=60$ red results altogether. Therefore $60-34=26$ of the further $80$ spins land on red. Since $26/80=0.325>0.30$ , the maintenance warning should be issued.

Q10

Tier 3 · Hard

Two wheels are designed to land on red with probability

0.25

. Wheel A lands on red

17

times in its first

60

spins and

156

times after

600

spins in total. Wheel B lands on red

16

times in its first

60

spins and

204

times after

600

spins in total. Work out the four relative frequencies. Which wheel gives stronger evidence that it may be biased? Explain your answer.

(5)

(Total for Question 10 is 5 marks)

Mark scheme

Mark scheme for question 10
Question	Answer	Mark	Mark scheme
10	Wheel A: $0.283$ (to 3 d.p.) initially and $0.26$ overall. Wheel B: $0.267$ (to 3 d.p.) initially and $0.34$ overall. Wheel B gives stronger evidence that it may be biased, because its larger-sample relative frequency remains $0.09$ above $0.25$ , whereas wheel A's is only $0.01$ above it.	5	Divide each red frequency by its number of spins. The early results are both fairly close to $0.25$ , but the $600$ -spin results are more stable evidence. Wheel A moves closer to $0.25$ , while wheel B moves much farther away and remains there over the larger sample.

Verified exam appearances

Series	Paper	Question	Marks	Calculator	Tier	Links
2022-11	1H	Q10	4	Non-calculator	Higher	QP MS
2019-06	1F	Q17	2	Non-calculator	Foundation	QP MS
2023-06	2H	Q10	4	Allowed	Higher	QP MS

Other points in P Probability

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