AQA GCSE Mathematics 8300: Probability – Practice Questions

These are original questions written for Marlbridge, for revision and practice on this content. They are not reproduced past-paper questions, and they do not replicate the exam’s exact structure, question count or mark tariffs – examination boards hold copyright in their own papers. Use these alongside the official past papers from your board or school.

These questions 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). Questions and parts on P9, conditional probability, are labelled “Higher tier only”; everything else is for both tiers. Each question says whether it is calculator-free (Paper 1 style) or calculator allowed (Papers 2 and 3 style).

Learn the content first in the study guide and the revision notes. The course hub is AQA GCSE Mathematics and the printable checklist lists every statement.

Questions

1. (calculator-free) A biased six-sided dice is rolled. The table shows some of the probabilities.

Score	1	2	3	4	5	6
Probability	0.1	0.15	0.2	0.2	0.15

(a) Work out the probability of rolling a 6. [2] (b) The dice is rolled 300 times. Work out the number of 6s you would expect. [1]

2. (calculator allowed) Jas drops a drawing pin and records whether it lands point up.

Number of drops	20	100	400
Number of times point up	12	55	212

(a) Work out the relative frequency of “point up” after 400 drops. [1] (b) Explain which result gives the best estimate of the probability of “point up”. [1] (c) Estimate how many times the pin will land point up in 1000 drops. [1]

3. (calculator-free) 160 people are at a leisure centre. 5/8 of them are members and the rest are guests. 30% of the members use the pool. 1/4 of the guests use the pool.

(a) Draw and complete a frequency tree. [2] (b) A person is chosen at random. Work out the probability that they use the pool. [1] (c) (Higher tier only) A person who uses the pool is chosen at random. Work out the probability that they are a member. [2]

4. (calculator-free) Two fair six-sided dice are rolled. The score is the difference between the two numbers (larger minus smaller, or 0 if they are the same).

(a) Draw a sample space diagram showing all the scores. [2] (b) Work out P(score = 2). [1] (c) Work out P(score is 3 or more). [1]

5. (calculator-free) 60 people were asked whether they own a cat or a dog. 34 own a cat, 25 own a dog and 12 own neither.

(a) Draw a Venn diagram to show this information. [3] (b) A person is chosen at random. Work out the probability that they own exactly one of the two animals. [1] (c) (Higher tier only) A cat owner is chosen at random. Work out the probability that they also own a dog. [1]

6. (calculator allowed) The probability that Leo’s train is late is 0.12. The probability that his connecting bus is late is 0.25. The two events are independent.

(a) Calculate the probability that both are late. [1] (b) Calculate the probability that exactly one of them is late. [2]

7. (calculator-free) A box holds 4 lemon sweets and 6 orange sweets. Ana takes a sweet at random and eats it. She then takes a second sweet at random.

(a) Draw a tree diagram to show all the probabilities. [2] (b) Work out the probability that both sweets are orange. [2] (c) Work out the probability that at least one sweet is lemon. [2] (d) State one assumption you made in your calculations. [1]

8. (calculator allowed) A game costs 50p a go. A fair spinner has 8 equal sections. One section wins £2, two sections win 50p, and the other five win nothing. The game is played 200 times. Work out the expected profit for the person running the game. [4]

9. (calculator allowed) For two events A and B, P(A) = 0.45, P(B) = 0.3 and P(A and B) = 0.12.

(a) Work out P(A or B). [1] (b) Work out the probability that neither A nor B happens. [1] (c) Are A and B independent? Give a reason. [1]

10. (calculator allowed, Higher tier only) A factory has two machines. Machine X makes 60% of the items and machine Y makes the rest. 5% of the items from X are faulty. 8% of the items from Y are faulty.

(a) Draw a tree diagram to show this information. [2] (b) An item is chosen at random. Calculate the probability that it is faulty. [2] (c) A faulty item is chosen at random. Calculate the probability that it was made by machine Y. Give your answer as a fraction. [2] (d) The factory makes 2500 items in a day. Work out the expected number of faulty items. [1] (e) The manager says, “Most faulty items come from machine X, because it makes most of the items.” Use your answer to (c) to comment. [1]

11. (calculator-free, Higher tier only) 200 students each chose one option: Art, Drama or Music.

	Art	Drama	Music	Total
Year 10	40		40	110
Year 11		25	30	90
Total				200

(a) Complete the two-way table. [2] (b) A student is chosen at random. Work out the probability that they chose Drama. [1] (c) A Year 11 student is chosen at random. Work out the probability that they chose Music. [1] (d) A student who chose Art is chosen at random. Work out the probability that they are in Year 10. [2] (e) Two different students are chosen at random from those who chose Drama. Work out the probability that both are in Year 10. [2]

Answers

1. (a) 0.1 + 0.15 + 0.2 + 0.2 + 0.15 = 0.8 [1]; P(6) = 1 − 0.8 = 0.2 [1] (b) 0.2 × 300 = 60 [1] Examiner insight: The method mark is for using “sum to 1” (1 minus the total); if the total is added wrongly and no working is shown, there is nothing to award.

2. (a) 212/400 = 0.53 [1] (b) The 400 drops, because it has the most trials, so it is the most reliable estimate [1] (c) 0.53 × 1000 = 530 [1] Examiner insight: For (b) the mark needs the reason “most trials”; just naming the 400 column is not enough.

3. (a) Members 100, guests 60 [1]; members: pool 30, no pool 70; guests: pool 15, no pool 45 [1] (b) (30 + 15)/160 = 45/160 = 9/32 [1] (c) Pool users = 45 [1]; P(member given pool) = 30/45 = 2/3 [1] Examiner insight: In (c) the denominator must be the 45 pool users; 30/160 scores zero because it answers a different question.

4. (a) 6 × 6 grid of differences [1], all 36 entries correct [1] (diagonal of 0s; 1s next to it; up to 5 in the corners) (b) 8 cells show 2, so P = 8/36 = 2/9 [1] (c) Scores 3, 4, 5 appear 6 + 4 + 2 = 12 times, so P = 12/36 = 1/3 [1] Examiner insight: An unsimplified correct fraction such as 8/36 usually earns the mark unless the question asks for simplest form; a wrong simplification loses it.

5. (a) Both = 34 + 25 + 12 − 60 = 11 [1]; cat only 34 − 11 = 23 and dog only 25 − 11 = 14 [1]; 12 outside the circles, all in a labelled rectangle [1] (b) (23 + 14)/60 = 37/60 [1] (c) 11 of the 34 cat owners also own a dog: 11/34 [1] Examiner insight: The overlap has to be found first; writing 34 and 25 inside the circles as the “only” values loses the method mark and every mark that follows.

6. (a) 0.12 × 0.25 = 0.03 [1] (b) 0.12 × 0.75 = 0.09 and 0.88 × 0.25 = 0.22 [1]; 0.09 + 0.22 = 0.31 [1] Examiner insight: “Exactly one” has two routes; the method mark needs at least one correct product using a “not late” probability, such as 0.88 or 0.75.

7. (a) First pick: L 4/10, O 6/10 [1]; second pick: after L: L 3/9, O 6/9; after O: L 4/9, O 5/9 [1] (b) 6/10 × 5/9 [1] = 30/90 = 1/3 [1] (c) 1 − 1/3 [1] = 2/3 [1] (or 12/90 + 24/90 + 24/90 = 60/90) (d) Each sweet left in the box is equally likely to be chosen (sweets taken at random) [1] Examiner insight: Denominators of 10 on the second branches show the “without replacement” idea was missed; the tree mark is lost and (b) can only earn follow-through credit.

8. Takings: 200 × 50p = £100 [1]; expected £2 wins = 200 × 1/8 = 25, paying £50 [1]; expected 50p wins = 200 × 2/8 = 50, paying £25 [1]; profit = 100 − 50 − 25 = £25 [1] Examiner insight: Mixing pence and pounds is the usual error; state units on the final answer, because “25” alone can be read as pence.

9. (a) 0.45 + 0.3 − 0.12 = 0.63 [1] (b) 1 − 0.63 = 0.37 [1] (c) 0.45 × 0.3 = 0.135, which is not 0.12, so not independent [1] Examiner insight: In (c) the mark needs the product 0.135 compared with 0.12; “no” with no calculation scores nothing.

10. (a) X 0.6, Y 0.4 [1]; faulty/not faulty branches 0.05, 0.95 and 0.08, 0.92 [1] (b) 0.6 × 0.05 + 0.4 × 0.08 [1] = 0.03 + 0.032 = 0.062 [1] (c) 0.032 ÷ 0.062 [1] = 16/31 [1] (d) 0.062 × 2500 = 155 [1] (e) 16/31 is more than a half, so more faulty items come from Y; the manager is wrong [1] Examiner insight: Part (e) needs a decision backed by the value from (c); “wrong because Y has a higher fault rate” without comparing 16/31 with 1/2 is not enough.

11. (a) Year 10 Drama 30; Year 11 Art 35 [1]; totals Art 75, Drama 55, Music 70 [1] (b) 55/200 = 11/40 [1] (c) 30/90 = 1/3 [1] (d) Art total = 75 [1]; 40/75 = 8/15 [1] (e) 30/55 × 29/54 [1] = 870/2970 = 29/99 [1] Examiner insight: In (e) the second fraction must drop to 29/54; squaring 30/55 treats the two picks as independent and scores no marks.

Where marks are usually lost

• Using the same denominator on both picks in a “without replacement” question.
• Adding along a tree path, or multiplying across different paths.
• Counting the overlap twice when two events can both happen.
• Using the grand total as the denominator in a “given” question.
• Giving a probability when the question asks “how many would you expect”.
• Leaving out units, or mixing pence and pounds, in expected-profit questions.
• Stating that events are or are not independent without a calculation.
• Writing probabilities as ratios or as “1 in 5”.
• Forgetting the “neither” region on a Venn diagram.

Next steps

• Probability revision notes
• Probability study guide
• AQA GCSE Mathematics course hub
• Printable checklist
• All free 10-minute diagnostics
• Book a free trial class

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.