OCR GCSE Mathematics: Probability scale and relative frequency (J560) – Revision Notes

These notes condense unit 11.01 (Basic probability and experiments, outcomes 11.01a to 11.01d) of the OCR GCSE (9-1) Mathematics J560 specification, Version 2.1 (August 2026). Nothing in the unit is marked Higher tier only, so everything here is for both tiers. For full explanations and worked examples, read the study guide. When you are ready to test yourself, use the practice questions, the course hub, the checklist and the free diagnostics.

Key facts at a glance

Ref	Idea	Key result
11.01a	Probability scale	0 impossible, 0.5 evens, 1 certain; never below 0 or above 1
11.01b	Relative frequency	number of times it happened / number of trials
11.01b	Frequency tree	numbers on branches add up to the number in the box they leave
11.01c	Estimating	relative frequency is an estimate of probability
11.01c	Many trials	relative frequency gets closer to the theoretical probability as trials increase
11.01d	Equally likely	P(event) = favourable outcomes / all outcomes
11.01d	Two dice	36 equally likely pairs
11.01d	Expected number	p × n

11.01a: Probability scale

0 ---------- 0.5 ---------- 1
impossible    evens        certain
unlikely <-- | --> likely

• Write probabilities as fractions, decimals or percentages. Ratios and “1 in 4” do not answer a question asking for a fraction or decimal.
• All outcomes of one experiment add up to 1.
• P(not A) = 1 - P(A).
• To place an event on a scale, decide: could it happen, must it happen, and is it nearer to evens or to an extreme?

Quick check: a fair dice shows a number below 7, probability 1. A fair dice shows 0, probability 0.

11.01d: Equally likely outcomes

Method in steps:

1. Say what the outcomes are and check that they are equally likely.
2. Count all the outcomes.
3. Count the outcomes that give the event.
4. Write favourable / total, then simplify.

Reminders:

• Coin: 2 outcomes. Dice: 6 outcomes with 3 odd, 3 even, 3 prime (2, 3, 5), 2 multiples of 3 (3, 6), 2 square numbers (1, 4).
• A bag with counters: add the numbers of each colour to find the total.
• Fair means each outcome is equally likely. Random means unpredictable outcome.
• Past results do not change the next result. A fair coin that has landed heads five times is still 1/2 for heads.

Must-know distinction: random and fair

Word	Meaning	Example
Random	Next outcome cannot be predicted	Any dice throw
Fair or unbiased	Every outcome equally likely	A normal dice
Biased	Some outcomes more likely than others	A dice that favours six

A biased dice is still random.

11.01d: Two dice

Both dice fair, scores added or compared. Sample space: 36 pairs.

Total	2	3	4	5	6	7	8	9	10	11	12
Ways	1	2	3	4	5	6	5	4	3	2	1

• The ways add up to 36. Use this as a check.
• Totals are not equally likely, but the 36 pairs are.
• P(doubles) = 6/36 = 1/6.
• P(total prime) = 15/36 = 5/12 (totals 2, 3, 5, 7, 11).
• Other combinations, such as the difference or the product of the two scores, need a table of all 36 pairs.

Worked reminder: P(total is a multiple of 5). Totals 5 and 10: 4 + 3 = 7 ways. P = 7/36. Check by hand with the table.

11.01d: Expected outcomes

expected number = probability × number of trials

• Use the theoretical probability if you have one, or the relative frequency if you do not.
• The answer is an average. The real result will often differ.
• It is fine for an expected number to be a decimal. Say “about” and round sensibly if the situation needs whole items.

Worked reminder: A fair dice rolled 90 times. Expected number of square numbers: 2/6 × 90 = 30.

Worked reminder: A machine makes parts with probability 0.04 of a defect. In a batch of 1500: 0.04 × 1500 = 60 defects expected.

11.01b: Relative frequency and frequency trees

Method for relative frequency:

1. Add up the trials to find the total.
2. Divide the count for the outcome by the total.
3. Give the answer as a fraction, decimal or percentage.
4. Check that all relative frequencies add up to 1.

Method for completing a frequency tree:

1. Put the total in the first box.
2. Put the numbers for each category on the branches.
3. Find missing numbers by subtracting: the branches from one box must add to that box.
4. Use the final numbers for relative frequencies, using the right total as the denominator (the category, not the whole group, when the question is about one category).

Worked reminder: 160 pupils surveyed, 90 own a bicycle. Of those with a bicycle, 63 cycle to school. Of those without, 14 cycle to school.

Bicycle 90: cycle 63, do not 27
No bicycle 70: cycle 14, do not 56
Pupils who cycle: 63 + 14 = 77 out of 160, so 77/160 = 0.48125
Relative frequency of cycling among bicycle owners: 63/90 = 0.7

11.01c: Relative frequency as an estimate

• A relative frequency is an estimate of the probability. It is not exact.
• More trials give a more reliable estimate. Use the largest sample.
• The estimate gets closer to the theoretical value as the number of trials increases, but it can still be a little different.
• To judge whether something is fair, compare the relative frequency from many trials with the theoretical probability. If they are far apart, the experiment may be biased. If you are unsure, say more trials are needed.
• Do not claim that a few trials prove or disprove fairness.

Describing and analysing results

When a question says “describe” or “analyse” experimental results, write a sentence that uses the numbers and says what they mean.

• Compare relative frequencies, not raw counts, when the groups have different sizes. A group with 30 successes out of 60 (0.5) did better than a group with 36 successes out of 90 (0.4), even though 36 is bigger than 30.
• Say which result is based on more trials and so is more reliable.
• Link back to the theoretical probability when you have one: “The relative frequency of 0.52 is close to 1/2, so there is no strong evidence that the coin is biased.”
• If the results disagree with the theory, suggest repeating the experiment with more trials.

Worked reminder: a bag holds 4 red and 6 green counters, and a counter is taken, noted and replaced 50 times, giving 17 reds. The theoretical P(red) = 4/10 = 0.4. The relative frequency is 17/50 = 0.34, which is a little lower. With only 50 trials this is not surprising, and a larger number of trials would be expected to give a value nearer 0.4.

Quick self-test

1. What is the probability of an impossible event, and of a certain event?
2. The chance that a train runs on time is 0.35. Find the chance that it does not run on time.
3. A bag has 3 red, 5 blue and 2 yellow counters. Find P(blue).
4. A fair dice is rolled. Find P(even number greater than 2).
5. Two fair dice are rolled and the scores added. Find P(total is 9).
6. A coin lands heads 38 times in 80 flips. Find the relative frequency of heads.
7. A spinner is expected to land on green with probability 0.2. How many greens are expected in 350 spins?
8. A fair dice is rolled twice and lands on six both times. What is the probability of a six on the third roll?
9. After 20 spins a spinner has landed on red 9 times and after 2000 spins 506 times. Which gives the better estimate of P(red), and what is it?
10. A frequency tree starts with 120 customers: 75 pay by card and the rest pay by cash. 18 of the cash customers use a coupon. How many cash customers do not use a coupon?

Answers

1. 0 and 1.
2. 1 - 0.35 = 0.65.
3. Total 10 counters, so 5/10 = 1/2.
4. Even numbers greater than 2 are 4 and 6, so 2/6 = 1/3.
5. Pairs (3,6) (4,5) (5,4) (6,3): 4/36 = 1/9.
6. 38/80 = 0.475.
7. 0.2 × 350 = 70.
8. 1/6. Earlier rolls do not change the next one.
9. 2000 spins is the larger sample: 506/2000 = 0.253.
10. Cash customers 120 - 75 = 45; without a coupon 45 - 18 = 27.

Where marks are usually lost

• Giving 11 outcomes for two dice instead of 36.
• Writing a probability as a ratio or as “1 in 6” when asked for a fraction or decimal.
• Leaving an unsimplified fraction when the question says “simplify”, or simplifying wrongly.
• Forgetting that P(not A) = 1 - P(A), and adding instead.
• Using the whole group as the denominator when the question is about one branch of a frequency tree.
• Saying “it is biased” from a small sample, rather than saying more trials are needed.
• Saying a coin is “due” to land on a different side after a run of the same result.
• Writing a probability above 1, or a negative one.
• Showing an expected number without the working p × n, so your method is clear even if the final number is wrong.
• Not giving a reason when the question says “explain” or “justify”.

Official syllabus

OCR GCSE (9-1) Mathematics J560, Version 2.1 (August 2026), for first assessment in 2017, published by OCR. Unit 11.01: Basic probability and experiments.