Estimate probabilities from experiments, decide whether a dice or spinner looks fair, and predict how many times an outcome will happen. It is on Foundation and Higher, aimed at grade 4. Free questions with answers are on this page. Worked examples show each line before you try the questions.
Objectives
Work out a relative frequency from a table of results.
Use it as an estimate of a probability and say why more trials help.
Find a missing relative frequency and an expected number.
Key points
Work out a relative frequency as the frequency of an outcome divided by the total number of trials.
Use a relative frequency as the best estimate of a probability, and know that more trials give a better estimate.
Compare a relative frequency with the probability for a fair dice, coin or spinner to judge whether it looks biased.
Find a missing relative frequency, because all the relative frequencies add up to 1.
Worked examples
Example 1
A coin is flipped 40 times and lands on heads 18 times. Write the relative frequency of heads as a decimal.
2 marks
The frequency of heads is 18 and the total number of trials is 40
Divide: 18 / 40 = 0.45
The relative frequency of heads is 0.45
Answer: 0.45
Common slips
Slip: The frequency of heads is 18 and the number of tails is 22 The total is all the flips, not just the tails.
Slip: Divide: 40 / 18 = 2.2 The frequency goes on top.
Slip: The relative frequency of heads is 18 A relative frequency is a fraction of the trials, not a count.
Example 2
A dice is rolled 300 times and shows a six 72 times. A fair dice would show about 50 sixes. Is the dice likely to be fair?
2 marks
The relative frequency of a six is 72 / 300 = 0.24
A fair dice has a probability of 1/6, which is about 0.17
0.24 is well above 0.17 over many rolls, so the dice is probably biased
Answer: probably biased
Common slips
Slip: The relative frequency of a six is 72 / 6 = 12 Divide by the number of rolls, not by 6.
Slip: A fair dice has a probability of 6 A probability is never above 1.
Slip: The dice is definitely biased Chance always causes some difference, so say probably.
Example 3
The relative frequencies of a spinner are red 0.3, blue 0.25, green x and yellow 0.15. Find x. The spinner is spun 80 times. How many greens are expected?
3 marks
The relative frequencies add up to 1, so x = 1 - 0.3 - 0.25 - 0.15
x = 1 - 0.7 = 0.3
The expected number of greens is 0.3 x 80
0.3 x 80 = 24
Answer: 24
Common slips
Slip: x = 0.3 + 0.25 + 0.15 The missing value is what is left to reach 1.
Slip: x = 0.7 0.7 is the sum of the known values.
Slip: The expected number of greens is 0.3 + 80 Multiply the probability by the number of trials.
Slip: 0.3 x 80 = 2.4 Check the place value: 3 tenths of 80 is 24.
Questions and answers
GCSE Maths relative frequency questions and answers. Try each one, then open the answer. Easy questions come first.
Question 1 Easy2 marks
A coin is flipped 50 times and lands on tails 20 times. Write the relative frequency of tails as a decimal.
Show answer
Answer: 0.4
Divide the frequency by the number of trials.
20 / 50 = 0.4.
M1 — Divide the frequency by the number of trials.
A1 — 20 / 50 = 0.4.
Question 2 Easy2 marks
A spinner is spun 80 times and lands on blue 20 times. Write the relative frequency of blue as a fraction in its simplest form.
Show answer
Answer: 1/4
Write 20 over 80.
Divide top and bottom by 20 to get 1/4.
M1 — Write 20 over 80.
A1 — Divide top and bottom by 20 to get 1/4.
Question 3 Medium2 marks
A train is late on 0.2 of its journeys. How many late trains do you expect in 150 journeys?
Show answer
Answer: 30
Multiply the probability by the number of trials.
0.2 x 150 = 30.
M1 — Multiply the probability by the number of trials.
A1 — 0.2 x 150 = 30.
Question 4 Medium2 marks
Relative frequencies of a spinner are 0.2, 0.35, 0.1 and x. Find x.
Show answer
Answer: 0.35
The relative frequencies add up to 1.
1 - 0.2 - 0.35 - 0.1 = 0.35.
M1 — The relative frequencies add up to 1.
A1 — 1 - 0.2 - 0.35 - 0.1 = 0.35.
Question 5 Hard2 marks
A biased dice is rolled 200 times and shows a six 50 times. The dice is rolled 600 more times. How many sixes should be expected?
Show answer
Answer: 150
The relative frequency of a six is 50 / 200 = 0.25.
0.25 x 600 = 150.
M1 — The relative frequency of a six is 50 / 200 = 0.25.
A1 — 0.25 x 600 = 150.
Question 6 Hard2 marks
Tom flips a coin 10 times and gets 7 heads. He says the coin must be biased. Give one reason his conclusion is not safe.
Show answer
Answer: there are too few trials
Ten flips is a small number of trials.
A fair coin can easily give 7 heads in 10 flips.
M1 — Ten flips is a small number of trials.
A1 — A fair coin can easily give 7 heads in 10 flips.
Question 7 Exam-style4 marks
Jo spins a spinner 150 times. It lands on 1 a total of 18 times, on 2 a total of 45 times, on 3 a total of 57 times and on 4 a total of 30 times. Estimate the probability that the spinner lands on 3. Jo spins it 400 more times. How many times should she expect it to land on 3?
Show answer
Answer: 0.38 and 152
57 / 150 = 0.38.
0.38 x 400 = 152.
M1 — Writes 57 over 150
A1 — 0.38
M1 — Multiplies 0.38 by 400
A1 — 152
Common mistakes
Most lost marks on this topic are one slip.
Dividing by the total of some of the outcomes
Divide by the total number of trials
Saying a coin is definitely biased after a few flips
Say it might be biased and that more trials are needed
Giving the sum of the known relative frequencies as the missing one
Take that sum away from 1
Treating the expected number as exactly what will happen
Say it is a prediction and the real number can differ
Spot the slip
A coin is flipped 40 times and shows heads 18 times. Ada says the relative frequency of heads is 18 out of 22. What did she divide by?
The number of tails
The number of trials
The number of heads
Show the slip
The number of tails She divided by the number of tails, 22. The total number of trials is 40, so the relative frequency is 18/40.
Exam tips
This topic can appear on a calculator or a non-calculator paper. Show enough working for a method mark.
If the question says “show that”, the result is given. The marks are for the steps that reach it.
The method is the same on AQA-style and Edexcel-style papers.
Printable worksheet
The same questions are in a PDF you can print. Answers start on a later page.