OxfordAQA International GCSE Mathematics: Probability scale, relative and expected frequency (9260) – 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 statements S9 to S13 in section 3.4.3 Probability of the OxfordAQA International GCSE Mathematics (9260) specification, Version 5.1. All of this content is Core, so it can appear on both the Core and the Extension papers, and no question here is Extension only.

If you need to learn the ideas first, read the study guide or the revision notes. The course hub is /boards/oxfordaqa/igcse/mathematics/ and the printable list of statements is at /checklists/oxfordaqa/igcse/mathematics/. The free 10-minute diagnostics show where you stand.

Questions

1. Choose the best word from: impossible, unlikely, evens, likely, certain.

(a) A fair die is rolled and shows 9. [1] (b) A fair coin is flipped and lands on heads. [1] (c) A day of the week is chosen at random and it is not Sunday. [1]

2. A fair spinner has 10 equal sectors. Four of the sectors say “win”. Write the probability of winning as a fraction in its simplest form, as a decimal and as a percentage. [3]

3. A bag holds 5 red counters, 3 green counters and 2 white counters. A counter is taken at random, its colour is noted and it is put back.

(a) Write the probability that the counter is green. [1] (b) The process is repeated 80 times. Work out the number of green counters you expect. [2]

4. A bus company records that its 8:15 bus was late on 7 of the last 40 days.

(a) Work out the relative frequency of the bus being late, as a decimal. [2] (b) Use your answer to estimate how many days the bus will be late in the next 200 days. [2]

5. A letter is chosen at random from the word STATISTICS.

(a) Find the probability that the letter is S. [1] (b) Find the probability that the letter is a vowel. [1] (c) Write the probability that the letter is Z, and the word that describes this event. [1]

6. Tomas flips a coin and keeps a running record of the number of heads.

Number of flips	10	40	100	500
Number of heads	6	21	52	255

(a) Work out the relative frequency of heads after each number of flips. [2] (b) State which relative frequency is the best estimate of the probability of heads for this coin, and why. [1] (c) The coin is fair. Work out the number of heads you would expect in 500 flips, and compare it with Tomas’s result. [2]

7. Asha rolls a die 15 times and gets four sixes. Ben rolls the same die 60 times and gets thirteen sixes.

(a) Work out each person’s relative frequency of a six, as a decimal to 3 decimal places. [2] (b) The die is fair, so the theoretical probability of a six is 1/6. Whose result is the better estimate, and why? [2]

8. A bag holds 40 counters. Some are yellow. The probability of taking a yellow counter at random is 0.15.

(a) How many yellow counters are in the bag? [2] (b) A counter is taken at random and replaced, 200 times. How many yellow counters do you expect? [2]

9. A factory makes 8000 phone cases in a week. Inspector A checks a sample of 50 cases and finds 12 are faulty. Inspector B checks a different sample of 400 cases and finds 90 are faulty.

(a) Work out the proportion of faulty cases found by each inspector. [2] (b) Use the more reliable sample to estimate the number of faulty cases made in the week. Say why you chose that sample. [2]

10. In a trial, 84 out of 120 seeds germinate.

(a) Work out the relative frequency of germination. [2] (b) A gardener plants 500 of these seeds. Estimate how many germinate. [2] (c) In fact 338 germinate. Give a reason why this is not exactly your answer to part (b). [1] (d) State how the trial could be changed to give a better estimate of the probability. [1]

11. A fair spinner has 5 equal sectors numbered 1 to 5. Reeta spins it 1500 times and gets a 5 on 330 spins.

(a) Write the theoretical probability of getting a 5. [1] (b) Work out the number of 5s you expect in 1500 spins. [2] (c) Work out the relative frequency of a 5 in Reeta’s experiment. [2] (d) Reeta says “The spinner must be biased.” Comment on her claim, using your answers. [2] (e) State one thing Reeta could do to get stronger evidence. [1]

12. A fairground game is advertised as “win with probability 1/12”. Leila plays it 300 times and wins 18 times.

(a) How many wins would you expect in 300 plays if the advert is correct? [2] (b) Work out the relative frequency of winning for Leila, as a decimal. [1] (c) Compare the observed and expected results. Does Leila’s data prove that the advert is wrong? [2] (d) Marcus has played 12 times and not won. He says “I am due a win now.” Comment. [2]

Answers

1. (a) A die cannot show 9, so the event is impossible [1] (b) The two outcomes are equally likely: evens [1] (c) Six of the seven days qualify, 6/7 is close to 1: likely [1] Examiner insight: A single word is enough here.

2. Favourable outcomes 4, total 10: 4/10 [1] Simplest fraction: 2/5 [1] Decimal 0.4 and percentage 40% [1] Examiner insight: When a question asks for three forms, check that all three describe the same number. Leaving a fraction unsimplified when “simplest form” is requested is an easy slip.

3. (a) Total counters 5 + 3 + 2 = 10, so P(green) = 3/10 [1] (b) Expected number = 3/10 × 80 [1] = 24 [1] Examiner insight: Show the multiplication of probability by number of trials, so that your method is visible even if an arithmetic slip creeps in.

4. (a) Relative frequency = 7/40 [1] = 0.175 [1] (b) Expected number of late days = 0.175 × 200 [1] = 35 [1] Examiner insight: The denominator is the total number of days recorded, not the number of days the bus was on time. Using 33 as the denominator is the usual slip on this kind of question.

5. (a) STATISTICS has 10 letters and S appears 3 times: 3/10 [1] (b) The vowels are A, I, I, so 3 letters: 3/10 [1] (c) There is no Z, so the probability is 0, and the event is impossible [1] Examiner insight: Count letters one at a time and write down the total before dividing. Repeated letters count each time they occur, so counting S once is a common slip.

6. (a) 6/10 = 0.6, 21/40 = 0.525, 52/100 = 0.52, 255/500 = 0.51 [1] for any two correct, [1] for all four (b) 0.51 (the result from 500 flips), because it is based on the largest number of trials [1] (c) Expected = 1/2 × 500 = 250 [1]. Tomas got 255, which is close to the expected number (a difference of 5) [1] Examiner insight: “Why” needs a reason that refers to the data. Saying “it is the last one” is not a reason, whereas “it comes from the most trials” does. In a comparison give both numbers, not just the word “close”.

7. (a) Asha: 4/15 = 0.267 [1]; Ben: 13/60 = 0.217 [1] (b) 1/6 ≈ 0.167, and Ben’s value (0.217) is closer than Asha’s (0.267) [1]; Ben’s sample is larger, so his is the better estimate [1] Examiner insight: Rounding to the stated 3 decimal places matters. Quote the number of trials in your reason, since “larger sample” backed by 60 against 15 is clearer than a general statement.

8. (a) Number of yellow = 0.15 × 40 [1] = 6 [1] (b) Expected yellow = 0.15 × 200 [1] = 30 [1] Examiner insight: Part (a) must be a whole number of counters. In part (b) use the number of draws, 200, not the number of counters.

9. (a) A: 12/50 = 0.24 [1]; B: 90/400 = 0.225 [1] (b) Use B’s sample because it is much larger (400 against 50) [1]. Estimate = 0.225 × 8000 = 1800 [1] Examiner insight: Choose the sample and justify it with sample size. Choosing the right sample without a reason leaves the answer incomplete.

10. (a) 84/120 [1] = 0.7 [1] (b) 0.7 × 500 [1] = 350 [1] (c) Repeating an experiment usually gives different results, because of chance, so the number need not be exactly 350 [1] (d) Use more seeds (a larger sample) [1] Examiner insight: For part (c), a reference to chance or variation in repeated experiments is the idea being tested. “The gardener made a mistake” does not answer the question, and neither does “the probability changed”.

11. (a) 1/5 [1] (b) 1/5 × 1500 [1] = 300 [1] (c) 330/1500 [1] = 0.22 [1] (d) The relative frequency, 0.22, is close to the theoretical 0.2, and 330 is only 30 more than the expected 300 out of 1500 spins [1]. This is not strong evidence of bias, as the difference could be due to chance, so her claim is not supported [1] (e) Spin the spinner many more times and see whether the relative frequency stays near 0.22 or moves towards 0.2 [1] Examiner insight: A conclusion about bias needs the numbers from earlier parts, not just an opinion. Say what the data shows and what it does not prove. Answers that leap to “definitely biased” or “definitely fair” go beyond what the data shows in part (d).

12. (a) 1/12 × 300 [1] = 25 [1] (b) 18/300 = 0.06 [1] (c) Leila won 18 times against an expected 25, and 0.06 is a little below 1/12 ≈ 0.083 [1]. One experiment cannot prove the advert wrong, because results vary by chance, but a larger sample would give stronger evidence [1] (d) Each play has the same probability of winning (1/12) however many earlier plays were lost, so he is not “due” a win [1]. Twelve plays is also a small number, so no wins in 12 is not unusual [1] Examiner insight: Part (d) needs a reason, not just “no”. The earlier plays do not change the probability of the next one. Use the probability value 1/12 in your sentence to show you are applying it.

Where marks are usually lost

• Writing a probability as a ratio, or with a value above 1.
• Dividing by the number of unfavourable outcomes instead of the total.
• Using 1/2 for any two-outcome event, even when the outcomes are not equally likely.
• Forgetting that relative frequency uses the total number of trials as the denominator.
• Multiplying by the wrong quantity in an expected frequency question, such as the number of counters instead of the number of draws.
• Calling a die or spinner biased from one small experiment, or giving a reason for a better estimate without mentioning sample size.
• Saying that earlier results make the next result more or less likely.

Next steps

• Read the revision notes and the study guide again for any weak spots.
• Return to the course hub and tick off S9 to S13 on the checklist.
• Try the free 10-minute diagnostics to find the next topic to revise.
• Book a free trial class: /trial/.

Official syllabus

These questions 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.