AQA GCSE Mathematics 8300: Probability – Revision Notes

These revision notes cover Topic 5, Probability (P1 to P9), of the AQA GCSE Mathematics (8300) specification, for teaching from September 2015 with exams from May/June 2017 (version 1.0). P1 to P8 are for Foundation and Higher tier; P9 (conditional probability) is Higher tier only. Probability can appear on the non-calculator Paper 1 or on the calculator Papers 2 and 3, in the May/June and November series.

For full explanations and worked examples, read the Probability study guide. When you are ready, try the Probability practice questions. The AQA GCSE Mathematics hub, the printable checklist and the free diagnostics help you plan what to revise next.

Key words

Term	Meaning
Outcome	One possible result of an experiment
Event	One or more outcomes, e.g. “an even number”
Sample space	A list, table or grid of every possible outcome
Exhaustive	The outcomes cover everything that can happen
Mutually exclusive	The events cannot happen at the same time
Independent	One event does not change the probability of the other
Dependent	One event changes the probability of the other (e.g. no replacement)
Fair / unbiased	Every outcome is equally likely
Relative frequency	Times the outcome happened ÷ number of trials
Expected frequency	Probability × number of trials

Write probabilities as fractions, decimals or percentages. Never as a ratio.

Formulas you must know (P3, P4, P8, P9)

The “or” rule and the “and” rule are in the specification’s list of formulae you must know or be able to derive. None of these is printed on the exam paper.

Formula	Notes
P(not A) = 1 − P(A)	Exhaustive outcomes sum to 1 (P4)
P(A) = number of favourable outcomes ÷ total outcomes	Only for equally likely outcomes (P7)
Expected number = P(A) × n	n = number of trials (P2, P3)
P(A or B) = P(A) + P(B) − P(A and B)	If mutually exclusive, P(A and B) = 0
P(A and B) = P(A given B) × P(B)	If independent, this is P(A) × P(B)
P(A given B) = P(A and B) ÷ P(B)	Higher tier only (P9)

Relative frequency and sample size (P1, P3, P5)

• Relative frequency estimates a probability from an experiment.
• More trials give a more reliable estimate (P5). Always use the largest set of results.
• To test whether something is fair, compare the relative frequency after many trials with the theoretical probability.
• A frequency tree holds counts on its branches. Start with the total at the left, split it, then split each branch again.

Small reminder: a biased spinner lands on 3 in 45 out of 300 spins. The relative frequency is 45/300 = 0.15. If it is spun another 200 times, you would expect about 0.15 × 200 = 30 threes.

Expected outcomes and fairness (P2)

• Expected number = probability × number of trials. It is an average over many repeats, so it can be a decimal such as 12.5.
• If a question asks “how many times would you expect…”, give a number of times, not a probability.
• For a game, expected outcomes can be turned into money: expected number of wins × prize, compared with the total paid to play.
• “Fair” means equally likely outcomes. A single short run of results cannot prove a dice is biased; a large number of trials gives better evidence.

Listing combinations systematically (P6)

• Fix the first choice and run through every option for the second, then move the first choice on.
• For three items A, B, C in order there are 6 arrangements: ABC, ACB, BAC, BCA, CAB, CBA.
• For a meal of one of 3 starters and one of 4 mains, there are 3 × 4 = 12 combinations. This product rule for counting (spec N5) is Higher tier only; at Foundation, check by listing.

Method in steps: sample-space grid (P6, P7)

1. Write the outcomes of the first experiment down the side and the second across the top.
2. Fill every cell with the combined result (sum, product, difference or pair).
3. Count the total cells. This is the denominator.
4. Count the cells that match the event. This is the numerator.
5. Simplify the fraction.

Example: two fair dice give 36 cells. The sum 9 comes from (3,6), (4,5), (5,4), (6,3), so P(sum 9) = 4/36 = 1/9.

Method in steps: Venn diagram (P6, P9)

1. Put the “both” number in the overlap first.
2. Subtract it from each set total to get the “only” regions.
3. Subtract everything inside the circles from the grand total to get the region outside.
4. Check that all regions add up to the grand total.
5. For “given” questions, the denominator is the total of the circle you are told about (Higher tier only).

Method in steps: tree diagram (P8)

1. Write probabilities on each branch. Each pair from one point sums to 1.
2. Without replacement, take 1 off the total for the second pick, and take 1 off the colour already chosen.
3. Multiply along the branches to get each outcome.
4. Add the outcomes that fit the event.
5. For “at least one”, use 1 − P(none).

Small reminder: a bag holds 3 green and 2 white beads. Two are taken without replacement. P(both white) = 2/5 × 1/4 = 2/20 = 1/10.

Conditional probability (P9) – Higher tier only

• “Given” cuts the sample space down to the group you know about.
• Two-way table: use the row or column total as the denominator, not the grand total.
• Tree diagram: P(first was A given the outcome) = (probability of the path you want) ÷ (sum of all paths that give that outcome).
• Thinking in expected frequencies helps: turn probabilities into counts out of, say, 100 or 1000, then read off the fraction.

Small reminder: 60% of pupils take the bus and 25% of bus users arrive late. Overall, 20% of pupils arrive late. Out of 100 pupils, 60 take the bus and 15 of those are late, while 20 pupils in total are late. So P(took the bus given late) = 15/20 = 0.75.

Must-know distinctions

This	Is not the same as
Mutually exclusive: cannot both happen, so P(A and B) = 0	Independent: can both happen, and P(A and B) = P(A) × P(B)
With replacement: branches repeat	Without replacement: denominators drop by 1
Relative frequency (from an experiment)	Theoretical probability (from equally likely outcomes)
Expected number (an average, may be a decimal)	What will actually happen
P(A and B) (out of everyone)	P(A given B) (out of B only)

Quick self-test

1. P(A) = 0.37. Find P(not A).
2. A fair spinner has five equal sectors, two of them red. It is spun 80 times. How many reds do you expect?
3. A coin lands heads 58 times in 100 throws. Write the relative frequency of heads as a decimal.
4. A fair coin is thrown and a fair dice is rolled. How many outcomes are in the sample space?
5. Two fair dice are rolled. Find P(the sum is 9).
6. A and B are mutually exclusive. P(A) = 0.2 and P(B) = 0.45. Find P(A or B).
7. A and B are independent. P(A) = 0.4 and P(B) = 0.3. Find P(A and B).
8. A bag holds 3 green and 2 white beads. Two are taken without replacement. Find P(both white).
9. P(A) = 0.5, P(B) = 0.4 and P(A and B) = 0.15. Find P(A or B).
10. (Higher tier only) P(A and B) = 0.18 and P(B) = 0.6. Find P(A given B).
11. (Higher tier only) In a class, 12 girls and 9 boys wear glasses. A pupil who wears glasses is chosen at random. Find the probability that it is a girl.

Answers

1. 1 − 0.37 = 0.63
2. 2/5 × 80 = 32
3. 58/100 = 0.58
4. 2 × 6 = 12
5. 4/36 = 1/9
6. 0.2 + 0.45 = 0.65
7. 0.4 × 0.3 = 0.12
8. 2/5 × 1/4 = 1/10
9. 0.5 + 0.4 − 0.15 = 0.75
10. 0.18 ÷ 0.6 = 0.3
11. 12/(12 + 9) = 12/21 = 4/7

Where marks are usually lost

• Adding along the branches of a tree diagram instead of multiplying.
• Keeping the same denominator on the second pick when there is no replacement.
• Using P(A) + P(B) for events that overlap, which counts P(A and B) twice.
• Giving an expected frequency as a probability (writing 0.15 instead of 9 wins).
• Using a relative frequency from a small number of trials when a larger set is given.
• Leaving an unfinished calculation, such as 20/56 + 6/56, without adding and simplifying.
• Writing a probability as a ratio or as “3 in 8”.
• In a Venn diagram, forgetting the region outside the circles.
• (Higher tier only) Dividing by the grand total instead of the “given” total in a conditional question.

Official syllabus

AQA GCSE Mathematics (8300) specification, for teaching from September 2015, exams from May/June 2017, version 1.0, published by AQA – section 3.5 Probability (P1 to P9) and Appendix: mathematical formulae.