Relative frequency | Pearson Edexcel GCSE Maths Notes

Visual model

Relative frequency estimates probability

successestrials

18

60

\text{estimate}=\frac{18}{60}=0.3

Gold-standard guide

20 mins

What you will learn

Estimate probabilities using experimental results.

Use a clear step-by-step method for relative frequency.

Check your answer and avoid the most common exam mistake.

Useful before you start

Core number skillsEarlier probability skillsShowing clear working

Core knowledge

Know the rule, then use it

These are the short notes. Read each one, then check you can use it in the worked example below.

Method

Relative frequency = frequency of outcome / total trials

Step 1

Calculate the relative frequency

Relative frequency = $26\frac{0}{4}00$ = 0.65

Step 2

Use it as the estimated probability

P(heads) ≈ 0.65

Step 3

Predict for 600 flips

Expected heads = 0.65 × 600 = 390

Watch out

Students confuse relative frequency with theoretical probability

Estimate

$relative frequency = successe\frac{s}{trials}.$

Expected number

$expected number = probability \times number of trials.$

Worked example

A biased coin is flipped 400 times and lands heads 260 times. Estimate the probability of heads and predict the number of heads in 600 further flips.

Calculate the relative frequency: Relative frequency = $26\frac{0}{4}00$ = 0.65.

Use it as the estimated probability: P(heads) ≈ 0.65.

Predict for 600 flips: Expected heads = 0.65 × 600 = 390.

Final answer

P(heads) ≈ 0.65; expected heads in 600 flips = 390

Question ladder

Build up to the hardest questions

Do them in order. If you miss a step, read the solution, then redo the question without looking.

Workedreasoning

A biased coin is flipped 400 times and lands heads 260 times. Estimate the probability of heads and predict the number of heads in 600 further flips.

3 marks4 minsrelative-frequency-worked

Show solution

Worked solution

1.Calculate the relative frequency: Relative frequency = $26\frac{0}{4}00$ = 0.65.
2.Use it as the estimated probability: P(heads) ≈ 0.65.
3.Predict for 600 flips: Expected heads = 0.65 × 600 = 390.

Final answer

P(heads) ≈ 0.65; expected heads in 600 flips = 390

Mark points

M1: calculate the relative frequency
M1: use it as the estimated probability
M1: predict for 600 flips
A1: P(heads) ≈ 0.65; expected heads in 600 flips = 390

Watch out

Students confuse relative frequency with theoretical probability.Relative frequency is an estimate based on experimental results — it may differ from the theoretical value, especially with few trials.State 'estimate' in your answer.

Diagnosticrecall

A drawing $pin$ lands point-up 72 times in 120 throws. Estimate P(point-up).

1 mark2 minsrelative-frequency-q1

Show solution

Worked solution

1.Spot the skill: Relative frequency = frequency of outcome / total trials.
2.Use the calculate the relative frequency stage first, then use it as the estimated probability.
3.Keep the final answer visible: 0.6.

Final answer

0.6

Mark points

M1: use the correct relative frequency = frequency of outcome / total trials.this estimates the theoretical probability. the more trials, the closer the estimate to the true probability.expected frequency = relative frequency × number of trials.
A1: 0.6

Watch out

Easyprocedure

A spinner is spun 500 times. Sector A comes up 175 times. Estimate P(A).

2 marks3 minsrelative-frequency-q2

Show solution

Worked solution

1.Spot the skill: Relative frequency = frequency of outcome / total trials.
2.Use the use it as the estimated probability stage first, then predict for 600 flips.
3.Keep the final answer visible: 0.35.

Final answer

0.35

Mark points

M1: use the correct relative frequency = frequency of outcome / total trials.this estimates the theoretical probability. the more trials, the closer the estimate to the true probability.expected frequency = relative frequency × number of trials.
A1: 0.35

Watch out

Mediumreasoning

Expected frequency of an outcome with probability 0.4 in 250 trials.

3 marks4 minsrelative-frequency-q3

Show solution

Worked solution

1.Spot the skill: Relative frequency = frequency of outcome / total trials.
2.Use the predict for 600 flips stage first, then calculate the relative frequency.
3.Keep the final answer visible: 100.

Final answer

100

Mark points

M1: use the correct relative frequency = frequency of outcome / total trials.this estimates the theoretical probability. the more trials, the closer the estimate to the true probability.expected frequency = relative frequency × number of trials.
A1: 100

Watch out

Hardproblem solving

Relative frequency of an event after n trials is 0.45. After 20 more trials with 8 successes, what is the new relative frequency? (original n = 100)

3 marks5 minsrelative-frequency-q4

Show solution

Worked solution

1.Spot the skill: Relative frequency = frequency of outcome / total trials.
2.Use the calculate the relative frequency stage first, then use it as the estimated probability.
3.Keep the final answer visible: (45 + 8)/120 = $5\frac{3}{1}20$ ≈ 0.44.

Final answer

(45 + 8)/120 = $5\frac{3}{1}20$ ≈ 0.44

Mark points

M1: use the correct relative frequency = frequency of outcome / total trials.this estimates the theoretical probability. the more trials, the closer the estimate to the true probability.expected frequency = relative frequency × number of trials.
A1: (45 + 8)/120 = $5\frac{3}{1}20$ ≈ 0.44

Watch out

Exam-stylemulti-step

Why does relative frequency become more reliable as n increases?

4 marks6 minsrelative-frequency-q5

Show solution

Worked solution

1.Spot the skill: Relative frequency = frequency of outcome / total trials.
2.Use the use it as the estimated probability stage first, then predict for 600 flips.
3.Keep the final answer visible: Law of large numbers: more data means experimental estimates converge to the true probability.

Final answer

Law of large numbers: more data means experimental estimates converge to the true probability

Mark points

M1: use the correct relative frequency = frequency of outcome / total trials.this estimates the theoretical probability. the more trials, the closer the estimate to the true probability.expected frequency = relative frequency × number of trials.
A1: Law of large numbers: more data means experimental estimates converge to the true probability

Watch out

Grade 9 stretchproblem solving

A biased spinner lands blue 84 times in 120 spins. Estimate how many blue results to expect in 350 further spins.

4 marks7 minsrelative-frequency-g9

Show solution

Worked solution

1.Estimate P(blue).
2.Multiply the probability by 350.

Final answer

245

Mark points

M1: use $8\frac{4}{1}20$ = 0.7
A1: 245

Watch out

Do not rush straight into arithmetic. Select the relevant method and show a complete chain of working.

Timed checkpoint

12 mins - 9 marks

Switch between skills

Set a timer and attempt all four questions before opening any answers. This is closer to the way skills appear in a real paper.

1Relative frequency - 2 marksA drawing

pin

lands point-up 72 times in 120 throws. Estimate P(point-up).Mark answer

Answer

0.6

2Probability basics - 2 marksP(A) = 0.35. Find P(not A).Mark answer

Answer

0.65

3Sample spaces and frequency trees - 2 marksA bag has 4 red, 2 blue balls. Two are drawn with replacement. Find P(both red).Mark answer

Answer

$\frac{4}{6}$ × $\frac{4}{6}$ = $\frac{4}{9}$

4Venn diagrams and set notation - 3 marksIn a Venn diagram, the 'only A' region has 9, 'only B' has 7, 'both' has 4. Find P(A).Mark answer

Answer

P(A) = $1\frac{3}{2}0$

Mastery check

I can explain the method for relative frequency.
I can show clear working without skipping key steps.
I can avoid this mistake: Students confuse relative frequency with theoretical probability.Relative frequency is an estimate based on experimental results — it may differ from the theoretical value, especially with few trials.State 'estimate' in your answer.

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