Sample size and theoretical probability - Edexcel GCSE Maths (Non-calculator) Exam Questions

Practise 18 exam-style questions on Sample size and theoretical probability for Edexcel GCSE Maths (Non-calculator). Write your answer as you would in the exam and get it marked in seconds against the mark scheme, point by point, with an AI tutor on hand if you get stuck.

18 questions
1–4 marks each
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18 questions, marked as you go

Sample questions with model answers

Here are 4 of the questions in this topic, each with a model answer, an explanation of why it earns the marks and the mark scheme an examiner would use.

Question 1

1 mark

A fair coin is flipped 6 times. It lands on heads every time.

Jen says,
"The next flip is more likely to be tails."

Explain why Jen is wrong.

Model answer

Each flip is independent of the previous flips - the coin does not remember what happened before. The probability of tails on the next flip is still \(\dfrac{1}{2}\).

Explanation

Relative frequency tends towards the theoretical probability over a large number of trials, but this is not because later results 'make up' for earlier ones. Every flip of a fair coin has the same probability of heads.

Mark scheme

1 mark
C1 for explaining that each flip is independent (the coin has no memory), so P(tails) on the next flip is still \(\dfrac{1}{2}\)
oe

Question 2

2 marks

Ben rolls a fair dice 60 times. He then carries on until he has rolled it 6000 times.
The table shows the relative frequency of each score after 60 rolls and after 6000 rolls.

Explain what the table shows about relative frequency and the number of rolls.

Table with columns "Relative frequency", "1", "2", "3", "4", "5", "6". Relative frequency: After 60 rolls; 1: 0.1; 2: 0.25; 3: 0.15; 4: 0.2; 5: 0.2; 6: 0.1 | Relative frequency: After 6000 rolls; 1: 0.168; 2: 0.165; 3: 0.17; 4: 0.164; 5: 0.166; 6: 0.167.

Model answer

After 60 rolls the relative frequencies are spread out, from 0.1 to 0.25.
After 6000 rolls they are all between 0.164 and 0.17, very close to \(\dfrac{1}{6} \approx 0.167\), the theoretical probability of each score on a fair dice.
As the number of rolls increases, the relative frequencies get closer to the theoretical probabilities.

Explanation

Each score on a fair dice has a theoretical probability of \(\dfrac{1}{6}\). A small sample varies a lot by chance, but a large sample gives results that follow the theoretical distribution closely.

Mark scheme

1 mark
C1 for saying that after 60 rolls the relative frequencies vary a lot (from 0.1 to 0.25)
oe
1 mark
C1 for saying that after 6000 rolls they are all very close to \(\dfrac{1}{6}\) (about 0.167), so more rolls give relative frequencies closer to the theoretical probability
oe

Question 3

3 marks

A bag contains 5 counters. Each counter is either red or blue.
Fay takes a counter at random, records its colour and puts it back.
After 20 trials, the relative frequency of red is 0.5
After 500 trials, the relative frequency of red is 0.39

Work out the most likely number of red counters in the bag. Give a reason for the result you used.

Model answer

Use 0.39, because 500 trials give a much more reliable estimate than 20 trials.
The possible probabilities are \(\dfrac{0}{5}, \dfrac{1}{5}, \dfrac{2}{5}, \ldots\) = 0, 0.2, 0.4, ...
0.39 × 5 = 1.95, which is closest to 2 (0.39 is closest to 0.4).
The most likely number of red counters is 2.

Explanation

The relative frequency from the larger number of trials should be close to the true probability, which must be a whole number of fifths. 0.39 is very close to \(\dfrac{2}{5}\), so there are probably 2 red counters.

Mark scheme

1 mark
C1 for using 0.39 because it is based on more trials (500), so it is more reliable
1 mark
M1 for 0.39 × 5 (= 1.95)
or comparing 0.39 with \(\dfrac{2}{5}\) = 0.4
1 mark
A1 for 2
cao

Question 4

4 marks

Maya spins a biased spinner.
The table shows the total number of times it landed on red after different numbers of spins.

Maya is going to spin the spinner another 1000 times.
Work out the best estimate for the number of times it will land on red. Give a reason for your choice of estimate.

Table with columns "Number of spins", "50", "100", "200", "400". Number of spins: Number of reds; 50: 22; 100: 38; 200: 72; 400: 140.

Model answer

The best estimate of P(red) comes from the most spins, 400, because it is the most reliable.
P(red) ≈ \(\dfrac{140}{400} = 0.35\)
Estimate for 1000 spins = 0.35 × 1000 = 350

Explanation

The numbers in the table are running totals, so the row for 400 spins includes all of Maya's results. A relative frequency from more trials is closer to the true probability, so use it to estimate the future number of reds.

Mark scheme

1 mark
C1 for choosing the result after 400 spins because it is based on the most trials
1 mark
M1 for \(\dfrac{140}{400}\) = 0.35
1 mark
M1 for their relative frequency × 1000
1 mark
A1 for 350
cao

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