OxfordAQA International GCSE Mathematics: Probability scale, relative and expected frequency (9260) – Revision Notes

These revision notes condense statements S9 to S13 of section 3.4.3 Probability in the OxfordAQA International GCSE Mathematics (9260) specification, Version 5.1. All of this content is Core, so it can be assessed on both the Core and the Extension papers. The study guide has the full explanations. For testing yourself, try the practice questions.

Related links: the course hub at /boards/oxfordaqa/igcse/mathematics/, the printable checklist, and the free 10-minute diagnostics.

Key words (S9)

Word	Meaning
Experiment	An activity with an unpredictable result
Trial	One go at the experiment
Outcome	One possible result
Event	An outcome, or group of outcomes, you are interested in
Fair or unbiased	All outcomes equally likely
Biased	Some outcomes favoured
At random	Each possible choice equally likely
Population	The whole group being studied
Sample	The part of the population you actually observe

The probability scale (S9)

Value	Word
0	impossible
between 0 and 0.5	unlikely
0.5 (1/2, 50%)	evens, even chance
between 0.5 and 1	likely
1 (100%)	certain

• Probabilities are fractions, decimals or percentages, never ratios.
• Never below 0 and never above 1.
• Convert freely: 4/25 = 0.16 = 16%.
• To mark an event on a scale, work out the number first, then place it.

Formulas (S10)

Name	Formula	Use when
Theoretical probability	favourable outcomes / total outcomes	outcomes equally likely
Relative frequency	times the event happened / total trials	from an experiment or survey
Expected frequency	probability × number of trials	to predict a count

Method: theoretical probability

1. Check the outcomes are equally likely (look for “fair”, “at random”).
2. Count the favourable outcomes.
3. Count all the outcomes.
4. Divide, and simplify if asked.

Method: relative frequency

1. Find the total number of trials. This is often the sum of a frequency column.
2. Divide the frequency of the event by that total.
3. State it as an estimate of the probability.

Method: expected frequency

1. Write the probability (exact if possible).
2. Multiply by the number of trials.
3. Give a sensible answer. A non-whole number is allowed unless the question asks for a whole number of items.

Small reminders

• A fair spinner has 8 equal sectors, 3 shaded. P(shaded) = 3/8. In 240 spins, expected shaded = 3/8 × 240 = 90.
• A pin lands point up 74 times in 200 drops. Relative frequency = 0.37. In 1500 drops, expected = 0.37 × 1500 = 555.
• A sample survey has 18 yes answers out of 60. Estimated proportion = 18/60 = 0.3. For a population of 400, expect 0.3 × 400 = 120.
• Working backwards: expected frequency 45 in 150 trials gives an estimated probability 45/150 = 0.3.
• A bag with 12 counters and P(blue) = 1/4 has 1/4 × 12 = 3 blue counters.

Comparing experiment and theory (S11)

1. Work out the theoretical probability from the set-up.
2. Work out the relative frequency from the data.
3. Compare them, using the expected frequency if that helps: observed count against expected count.
4. Comment on the number of trials. A small number means a gap could easily be chance.
5. Suggest more trials before deciding.

Example: a die rolled 120 times gives 27 sixes. Expected 120 × 1/6 = 20. Relative frequency 27/120 = 0.225 against theoretical 1/6 ≈ 0.167. The gap is 7 sixes, but 120 rolls is a small sample, so it does not prove bias.

Repeating an experiment (S12)

• Repeated experiments usually give different results.
• The differences are due to chance, not mistakes.
• A result that differs from the expected frequency does not mean the model is wrong.
• Past results do not change the probability of the next trial. After a run of misses, the next trial still has the same probability.

Sample size (S13)

• More trials generally give a relative frequency closer to the true probability.
• “Generally” matters: a larger sample makes a better estimate likely, not certain.
• To estimate a population characteristic, use the larger sample if both are fairly chosen.
• A large sample taken in a biased way is still a poor sample.
• For a population estimate: sample proportion × population size.

Worked layouts to copy

From a frequency table

A box of 25 raffle-style tokens is sampled. Colours drawn (with replacement): green 14, yellow 9, red 7.

Total trials = 14 + 9 + 7 = 30
Relative frequency of red = 7/30 ≈ 0.233
Expected reds in 300 draws = 7/30 × 300 = 70

Write the total first. Most errors here come from using the wrong denominator.

Choosing the model

Fair die, P(multiple of 3) = 2/6 = 1/3      theoretical
Biased die, 15 threes in 60 rolls: 15/60 = 0.25   relative frequency

If the question says the die is biased, or gives you results, use the data. If it says fair, use the equally likely model.

Finding how many from a probability

P(winning) = 0.15, 40 tickets
Winning tickets = 0.15 × 40 = 6

The answer to a “how many” question about a fixed set (a bag, a pack) should be a whole number. If you get 6.4 you have the wrong probability or the wrong number.

Wording for written answers (S11 to S13)

Use short, specific sentences that refer to the numbers.

• “The relative frequency, 0.225, is close to the theoretical probability, 0.167, but the sample is small.”
• “The observed number, 27, is above the expected number, 20, which could be due to chance.”
• “More trials would give a better estimate.”
• “Different samples will usually give different results.”
• “The larger sample gives the better estimate, provided it is chosen fairly.”

Avoid vague phrases such as “it is not fair” with no numbers, or “it is wrong” when you mean “it is different”.

Quick checks before you move on

• Is the answer between 0 and 1 (or 0% and 100%)?
• Does an expected frequency come out smaller than the number of trials?
• Did you use the total number of trials, not just one frequency, as the denominator?
• Does your wording say “estimate” when the probability came from data?
• If asked to compare, did you give both numbers and mention sample size?

Must-know distinctions

Idea A	Idea B
Theoretical probability: from the model	Relative frequency: from data
Expected frequency: a prediction	Observed frequency: what actually happened
Probability of one trial	Frequency over many trials
Sample: the part you observe	Population: the whole group

Quick self-test

1. A bag has 3 red counters and 7 blue counters, picked at random. State P(red) as a fraction, a decimal and a percentage.
2. Which probability word describes an event with probability 0.5?
3. A fair spinner has 5 equal sectors numbered 1 to 5. It is spun 400 times. How many 4s do you expect?
4. A coin is flipped 80 times and lands heads 46 times. Write the relative frequency of heads as a decimal.
5. A factory tests 500 items and finds 35 faulty. Estimate the probability that an item is faulty.
6. Using your answer to question 5, how many faulty items do you expect in 2000 items?
7. Can a probability be 1.2? Give a reason.
8. A fair die is rolled 90 times. How many even numbers do you expect?
9. A survey of 10 people and a survey of 400 people give different estimates of the same proportion. Which should you trust more, and why?
10. After 5 tails in a row, is heads more likely on the next flip of a fair coin?

Answers

1. 3/10 = 0.3 = 30%.
2. Evens (an even chance).
3. P(4) = 1/5, so 1/5 × 400 = 80.
4. 46/80 = 0.575.
5. 35/500 = 0.07.
6. 0.07 × 2000 = 140.
7. No. A probability cannot be more than 1.
8. P(even) = 3/6 = 1/2, so 1/2 × 90 = 45.
9. The 400-person survey, because larger samples generally give better estimates (assuming both samples are chosen fairly).
10. No. The coin has no memory, so the probability is still 1/2.

Where marks are usually lost

• Giving a probability as a ratio, or writing “3 in 10” when a fraction, decimal or percentage is asked for.
• Dividing by the wrong total, such as favourable over unfavourable.
• Using theoretical probability when the question gives experimental data, or the reverse.
• Forgetting to add up the frequencies to get the total number of trials.
• Multiplying the probability by the wrong number of trials.
• Rounding a fraction early and carrying an inaccurate decimal forward.
• Claiming a die or coin is biased from a small experiment, or claiming it is fair because the results are not identical to the expected values.
• Not mentioning sample size when a question asks which estimate is better.
• Writing “it will happen 90 times” when the question is about an expected frequency.

Official syllabus

These notes follow the OxfordAQA International GCSE Mathematics (9260) specification, Version 5.1, for exams May/June 2018 onwards, published by OxfordAQA, section 3.4.3 Probability, statements S9 to S13.