1MA1 · P2–P3, P5ProbabilityFoundation & HigherGrades 2–5

Relative Frequency & Expected Outcomes

You cannot calculate the probability that a drawing pin lands point up, or that a particular spinner is fair, just by counting outcomes. Instead you experiment: repeat the trial many times and see how often the event happens. That proportion, the $\textbf{relative frequency}$ , estimates the probability — and the more trials, the better the estimate.

35 min Probability

The big picture

Theoretical probability predicts; relative frequency measures. When a coin is flipped 10 times, 7 heads is not unusual, but in 10 000 flips the proportion of heads settles very close to 0.5. That settling down is why larger samples give more reliable estimates, and why Edexcel asks which of several estimates is best.

Expected frequency turns a probability into a prediction: the number of times you would expect an event in a given number of trials. Comparing expected and observed frequencies is how you decide whether a dice or spinner might be biased.

What you'll learn

Calculate relative frequencies from experimental data
Explain why more trials give a more reliable estimate of probability
Calculate expected frequencies
Use relative frequencies to judge whether something is fair or biased

Relative frequency

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

It estimates the probability. Different experiments give different estimates, because results vary randomly.

\text{relative frequency} = \dfrac{\text{frequency}}{\text{number of trials}}

Experimental probability.

Example 1A spinner is spun 200 times and lands on red 46 times. Estimate the probability of red.

1Relative frequency

= \tfrac{46}{200}

Answer

0.23

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

A spinner lands on blue 81 times in 300 spins. Estimate the probability of blue.

Your answer

Show worked solution

Solution

1 $\frac{81}{300} = 0.27$ .

Answer: $0.27$

More trials, better estimates

As the number of trials increases, the relative frequency tends to get closer to the true probability.

So when several groups run the same experiment, the best estimate combines all the results, or uses the group with the most trials.

Relative frequency graphs typically fluctuate wildly at first, then level off.

Example 1Group A flips a drawing pin 20 times and it lands point up 12 times. Group B flips it 150 times and it lands point up 81 times. Which gives the better estimate, and what is the best estimate using all the data?

1Group B used more trials, so its estimate (

0.54

) is more reliable than A’s (

0.6

2All data:

\tfrac{12 + 81}{20 + 150} = \tfrac{93}{170}

AnswerGroup B; the best overall estimate is

\tfrac{93}{170} \approx 0.547

Why this works

This is the law of large numbers in action. Random variation does not disappear with more trials, but it becomes a smaller proportion of the total, so the estimate stabilises.

Expected outcomes

Expected frequency $=$ probability × number of trials.

It is what you would expect on average; the actual result will usually be close but not exact.

\text{expected frequency} = P(A) \times n

Expected number of times A happens.

Example 1Using the estimate

0.23

for red, how many times would you expect red in 500 spins?

0.23 \times 500

Answer115 times

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

Using the estimate $0.27$ , how many times would you expect blue in 800 spins?

Your answer

Show worked solution

Solution

1 $0.27 \times 800 = 216$ .

Answer: $216$

Fair or biased?

Compare the observed relative frequency with the theoretical probability. A large difference over many trials suggests bias.

Small differences over a few trials are expected by chance, so they are not strong evidence.

Example 1A dice is rolled 240 times and shows a six 60 times. Is it fair?

1Expected sixes if fair:

\tfrac16 \times 240 = 40

2Observed 60, relative frequency

0.25

compared with

0.167

AnswerFar more sixes than expected over many rolls, so the dice is probably biased.

Tip — Use cautious language: "probably biased" or "suggests it is biased". An experiment gives evidence, not certainty.

Think like an examiner

State that more trials give a more reliable estimate when comparing experiments.
Show expected frequency calculations.
Use cautious conclusions about bias.

Experimental probability

\tfrac{\text{frequency}}{\text{trials}}

Relative frequency.

P(A) \times n

Expected frequency.

Watch out for these

Choosing the estimate closest to the "expected" answer as the best.

The best estimate uses the most trials.

Claiming a coin is definitely biased after 10 flips with 7 heads.

Small samples vary; more trials are needed.

Dividing by the number of outcomes instead of trials.

Relative frequency divides by trials.

Expecting the exact expected frequency every time.

It is an average; actual results vary.

Stretch yourself

A biased spinner has four colours. After 400 spins the frequencies are red 90, blue 130, green 100, yellow 80. The spinner is spun 60 more times. Estimate how many times it lands on a colour other than blue, and explain how reliable your estimate is.

Hint — Estimate P(not blue) from the 400 spins, then multiply by 60.

Reveal solution

SolutionNot blue in the first 400 spins:

90 + 100 + 80 = 270

, so the relative frequency is

\tfrac{270}{400} = 0.675

. Expected in 60 spins:

0.675 \times 60 = 40.5

, so about 40 or 41 times. The estimate is fairly reliable because it is based on 400 spins, but the actual result will vary around it.

\blacksquare

Questions students ask

What is the difference between relative frequency and probability?

Probability is the true long-run chance; relative frequency is an estimate of it from an experiment.

Can expected frequency be a decimal?

Yes, as an average — though actual counts are whole numbers. Give the decimal or round sensibly as the question asks.

How many trials are enough?

There is no fixed number, but estimates improve steadily as trials increase.

Key takeaways

Relative frequency $=$ frequency ÷ trials estimates probability.
More trials give more reliable estimates.
Combine all data for the best estimate.
Expected frequency $=$ probability × number of trials.
Large, consistent differences from theory over many trials suggest bias.

Exam-style questions

Practise it the way it is examined

3 original questions · 7 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 coin is flipped 50 times and lands heads 18 times.

(a)Work out the relative frequency of heads.[1]
Show mark scheme (a)
- B1 $0.36$
Answer: $0.36$
(b)Using this, estimate the number of heads in 400 flips.[1]
Show mark scheme (b)
- B1 $144$
Answer: $144$

Question 2Grade 43 marks

Adam does 30 trials and gets 12 successes. Beth does 270 trials and gets 96 successes.

(a)Whose results give the better estimate? Give a reason.[1]
Show mark scheme (a)
- B1Beth, because she did more trials
(b)Combine the results to estimate the probability of success.[2]
Show mark scheme (b)
- M1 $\frac{108}{300}$
- A1 $0.36$
Answer: $0.36$

Question 3Grade 52 marks

A dice is rolled 600 times and shows a six 142 times. Is the dice fair? Give a reason.[2]
Show mark scheme
- M1Expected sixes for a fair dice: $100$
- C1Probably biased: 142 is much higher than 100 over a large number of rolls

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

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

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