P1, P3, P5ProbabilityFoundation & HigherGrades 3–5

Relative Frequency

There is no formula for the probability that a drawing pin lands point up. The only way to find out is to drop it many times and count. The fraction of trials with that outcome is the $\textbf{relative frequency}$ , and it is the best estimate of the probability when theory cannot help.

35 min AQA GCSE Maths

The big picture

Theoretical probability comes from equally likely outcomes, like the six faces of a fair dice. Experimental probability comes from data. The two are connected: as the number of trials grows, relative frequency settles closer to the true probability.

Once you have a probability — from theory or experiment — you can predict how often something should happen, and compare real results with those expectations to decide whether a dice, spinner or coin is fair.

What you'll learn

Calculate relative frequency from experimental data
Understand why more trials give a better estimate
Calculate expected outcomes
Use results to judge whether an experiment is fair

Calculating relative frequency

Relative frequency $= \dfrac{\text{number of times the outcome happens}}{\text{total number of trials}}$ .

A drawing pin dropped 300 times lands point up 186 times: relative frequency $= \dfrac{186}{300} = 0.62$ .

Example 1A spinner is spun 400 times with results A: 92, B: 108, C: 64, D: 136. Estimate the probability of landing on A or C.

1A or C:

92 + 64 = 156

times.

2Relative frequency:

\dfrac{156}{400}

Answer

0.39

Your turnSame method, new numbers — try it before moving on.

Using the same 400 spins (A: 92, B: 108, C: 64, D: 136), estimate the probability of landing on D.

Your answer

Show worked solution

Solution

1Relative frequency $= \frac{136}{400}$ .
2 $= 0.34$ .

Answer: $0.34$

Expected outcomes

Expected number $= \text{probability} \times \text{number of trials}$ .

Using the drawing pin estimate, in 800 drops we expect $0.62 \times 800 = 496$ to land point up.

Example 1The probability a train is late is 0.15. How many late trains would you expect in 240 journeys?

0.15 \times 240

Answer36 late trains

Your turnSame method, new numbers — try it before moving on.

The probability that an item from a factory is faulty is 0.08. How many faulty items would you expect in a batch of 750?

Your answer

Show worked solution

Solution

1 $0.08 \times 750 = 60$ .

Answer: $60$

Tip — An expected value is a prediction, not a promise. Real results vary around it.

Fairness and reliability

A fair dice should show each number about $\tfrac16$ of the time. Rolling it 60 times, you expect each number about 10 times.

Getting a six 18 times out of 60 (relative frequency 0.3) suggests bias, but 60 rolls is not many. More trials would give more reliable evidence.

Combining results from several groups gives a larger total and a better estimate.

Why this works

Small samples jump around. Ten coin flips can easily give 7 heads, but 7000 heads in 10 000 flips would be strong evidence of an unfair coin.

Think like an examiner

Refer to the number of trials when commenting on reliability.
Compare observed and expected results when judging fairness.
Give probabilities as fractions, decimals or percentages — never as ratios.

Remember these

\text{RF} = \dfrac{\text{frequency}}{\text{total trials}}

Relative frequency.

\text{expected} = P \times n

Expected outcomes.

\sum P = 1

Probabilities of all outcomes add to 1.

Watch out for these

Claiming a dice is definitely biased from a few rolls.

Say results suggest bias, and more trials are needed.

Using the result from the smallest sample as the best estimate.

Use the largest number of trials.

Expecting exactly the predicted number every time.

Real results vary around the expected value.

Writing a probability as "3 : 10".

Write

\tfrac{3}{10}

or 0.3.

Stretch yourself

Asha rolls a dice 50 times and gets a 4 on 13 of them. Ben rolls the same dice 250 times and gets a 4 on 42. (a) Whose results give the better estimate of P(4)? (b) Combine the results to estimate P(4). (c) Does the dice appear to be biased towards 4?

Hint — For (c), compare the combined estimate with $\tfrac16 \approx 0.167$ .

Reveal solution

Solution(a) Ben’s, because he did more trials. (b) Combined:

\dfrac{13 + 42}{300} = \dfrac{55}{300} = 0.183

(3 s.f.). (c) For a fair dice

P(4) = \tfrac16 = 0.167

, so we would expect 50 fours in 300 rolls; 55 is close to this. There is no strong evidence of bias.

\blacksquare

Questions students ask

Is relative frequency the same as probability?

It is an estimate of probability, which improves with more trials.

Can relative frequency be used for a fair coin?

Yes, and it should get close to 0.5 over many flips.

Should expected values be whole numbers?

Not always — 97.5 is a valid expected value, even though real counts are whole.

Key takeaways

Relative frequency estimates probability from experiments.
More trials give more reliable estimates.
Expected outcomes $= P \times n$ .
Compare observed and expected results to judge fairness.

Exam-style questions

Practise it the way it is examined

2 original questions · 5 marks, written to match the style and mark allocation of the real papers. Work on paper, then open the mark scheme and tick the marks you earned — method marks count even if the final answer slips.

Question 1Grade 32 marks

A dice is rolled 120 times. It lands on six 31 times.

(a)Work out the relative frequency of rolling a six.[1]
Show mark scheme (a)
- B1 $\frac{31}{120}$ or $0.258$
Answer: $\frac{31}{120}$
(b)Does the dice appear to be fair? Give a reason.[1]
Show mark scheme (b)
- B1A fair dice would give about 20 sixes; 31 is quite a lot more, so it may be biased (more trials would make this more certain)

Question 2Grade 53 marks

Ana flips a coin 20 times and gets 6 heads. Ben flips the same coin 500 times and gets 240 heads.

(a)Whose results give the more reliable estimate of the probability of heads? Explain why.[1]
Show mark scheme (a)
- B1Ben, because he did more trials
(b)Combine their results to estimate the probability of heads.[2]
Show mark scheme (b)
- M1 $\frac{6 + 240}{20 + 500}$
- A1 $\frac{246}{520}$ or $0.473$
Answer: $0.473$

Unlimited practice

Skill drill

A new question every time, marked instantly, with a full worked solution. Aim for a streak of five, then move up a level.

0 / 0 first-time correct Streak 0Best at this level: 0

Generating a question…

Test yourself

Answer a few questions, then review anything you missed.

Next step

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