OxfordAQA International GCSE Mathematics: Sample spaces, mutually exclusive events and Venn probability (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 are for OxfordAQA International GCSE Mathematics (9260), specification Version 5.1 (for exams May/June 2018 onwards), section 3.4.3 Probability, references S14 to S16. All of it is Core content, also assessable at Extension, so no question is Extension only. Start with the study guide if a topic is new. See also the course hub, the checklist and the free diagnostics.

Questions

1. The nine letters of the word COMMITTEE are written on separate cards. One card is chosen at random.

(a) Find the probability that the card shows T. [1] (b) Find the probability that the card shows a vowel (A, E, I, O or U). [1] (c) Explain why “the card shows T” and “the card shows a vowel” are mutually exclusive, and find the probability that the card shows T or a vowel. [2]

2. Four tiles are numbered 2, 5, 7 and 8. Two tiles are taken one after the other, without replacement. The first tile gives the tens digit and the second gives the units digit of a two-digit number.

(a) List all the possible two-digit numbers. [2] (b) Find the probability that the number is greater than 60. [1] (c) Find the probability that the number is a multiple of 5. [1]

3. A fair spinner has four equal sections coloured red (R), yellow (Y), blue (B) and green (G). It is spun twice.

(a) Complete the table of outcomes. The first row and first column are filled in. [2]

1st \ 2nd	R	Y	B	G
R	RR	RY	RB	RG
Y	YR
B	BR
G	GR

(b) Find the probability that both spins give the same colour. [1] (c) Find the probability that at least one spin gives red. [1] (d) Find the probability that one spin gives red and the other gives blue. [1]

4. A fair six-sided die is rolled. Event A is “the number is less than 3”. Event B is “the number is greater than 4”. Event C is “the number is odd”.

(a) For each pair of A, B and C, state whether the two events are mutually exclusive. Give a reason for each pair. [2] (b) Find P(A or B). [1] (c) Are A, B and C exhaustive? Give a reason. [1]

5. A four-sided spinner shows the numbers 1, 2, 3 and 4. The probabilities are shown below.

Number	1	2	3	4
Probability	x	x + 0.1	2x	0.3

(a) Work out the value of x. [3] (b) Find the probability that the spinner does not land on 3. [1]

6. A shop records what each customer buys. The probability that a customer buys bread is 0.55. The probability that a customer buys milk is 0.3. The probability that a customer buys neither is 0.25.

(a) Find the probability that a customer buys bread or milk (or both). [1] (b) Find the probability that a customer buys both. [2] (c) Find the probability that a customer buys milk but not bread. [1]

7. A year group of 36 pupils is asked about three clubs: art, band and chess. The results are:

15 pupils are in art, 12 in band and 11 in chess. Of these, 2 are in all three clubs, 3 are in art and band only, 4 are in art and chess only, and 1 is in band and chess only.

(a) Work out the number in art only, band only and chess only. [3] (b) Work out the probability that a randomly chosen pupil is in no club. [2] (c) Find the probability that a pupil is in exactly two clubs. [1] (d) Find the probability that a pupil is in art but not chess. [1]

8. The whole numbers from 1 to 36 are sorted using a Venn diagram. Set A is the square numbers. Set B is the multiples of 4.

(a) List the numbers in each of the four regions, or give how many there are in each. [3] (b) A number is chosen at random from 1 to 36. Find the probability that it is neither a square nor a multiple of 4. [1] (c) Find the probability that it is in exactly one of the sets. [1]

9. One spinner has three equal sections numbered 2, 3 and 4. A second has three equal sections numbered 1, 3 and 5. Both are spun and the score is the sum of the two numbers.

(a) Complete a table showing all the possible scores. [2] (b) Event L is “the score is 4 or less”. Event M is “the score is 5, 6 or 7”. Event N is “the score is 8 or more”. Find P(L), P(M) and P(N). [3] (c) Show that L, M and N are mutually exclusive and exhaustive. [1] (d) Use an addition of probabilities to find P(L or N). [2]

10. In a group of 50 students, 24 study French, 20 study Spanish and 9 study neither language. Let x be the number who study both.

(a) Write an equation in x and solve it. [3] (b) Complete a two-circle Venn diagram for the group. [1] (c) Find the probability that a student chosen at random studies French only. [1] (d) Find the probability that a student studies exactly one of the two languages. [2]

Answers

1. (a) There are 2 Ts among 9 cards, so P(T) = 2/9 [1] (b) The vowels are O, I, E, E: 4 cards, so P(vowel) = 4/9 [1] (c) No card is both T and a vowel, so the events cannot happen together [1]. P(T or vowel) = 2/9 + 4/9 = 6/9 = 2/3 [1] Examiner insight: Give a reason for “mutually exclusive”: no card can be both.

2. (a) The first tile has 4 options and the second 3, so there are 12 numbers: 25, 27, 28, 52, 57, 58, 72, 75, 78, 82, 85, 87 [2] (b) Greater than 60: 72, 75, 78, 82, 85, 87, which is 6 of 12, so 1/2 [1] (c) Multiples of 5: 25, 75, 85, which is 3 of 12, so 1/4 [1] Examiner insight: Without replacement, 22 or 55 cannot occur. Check that your list has 4 × 3 = 12 entries.

3. (a) The completed table [2]:

1st \ 2nd	R	Y	B	G
R	RR	RY	RB	RG
Y	YR	YY	YB	YG
B	BR	BY	BB	BG
G	GR	GY	GB	GG

(b) The matching pairs RR, YY, BB, GG are 4 of 16, so 1/4 [1] (c) Row R has 4 cells and column R adds 3 more, so 7 of 16: 7/16 [1] (d) RB and BR are 2 of 16, so 1/8 [1] Examiner insight: Count each cell once: the RR cell must not be counted twice in part (c).

4. (a) A and B: mutually exclusive, because no number is both less than 3 and greater than 4 [1]. A and C: not mutually exclusive, because 1 is in both. B and C: not mutually exclusive, because 5 is in both [1] (b) P(A or B) = 2/6 + 2/6 = 4/6 = 2/3 [1] (c) Not exhaustive: the number 4 is in none of A, B or C [1] Examiner insight: Naming a number that sits in both events is a precise reason for “not mutually exclusive”.

5. (a) The probabilities sum to 1: x + (x + 0.1) + 2x + 0.3 = 1 [1]. This gives 4x + 0.4 = 1 [1], so 4x = 0.6 and x = 0.15 [1] (b) P(3) = 2 × 0.15 = 0.3, so P(not 3) = 1 − 0.3 = 0.7 [1] Examiner insight: Say why the equation holds (“the probabilities sum to 1”), then check by adding.

6. (a) P(bread or milk) = 1 − 0.25 = 0.75 [1] (b) P(both) = 0.55 + 0.3 − 0.75 [1] = 0.1 [1] (c) Milk but not bread = 0.3 − 0.1 = 0.2 [1] Examiner insight: Adding 0.55 + 0.3 to get 0.85 would leave the regions not summing to 1, so check the regions total 1.

7. (a) Art only = 15 − 2 − 3 − 4 = 6 [1]. Band only = 12 − 2 − 3 − 1 = 6 [1]. Chess only = 11 − 2 − 4 − 1 = 4 [1] (b) The seven regions total 6 + 6 + 4 + 3 + 4 + 1 + 2 = 26 [1]. So 36 − 26 = 10 are in no club and P = 10/36 = 5/18 [1] (c) Exactly two clubs: 3 + 4 + 1 = 8, so P = 8/36 = 2/9 [1] (d) Art but not chess: art only (6) plus art and band only (3) = 9, so P = 9/36 = 1/4 [1] Examiner insight: Each “only” value needs three subtractions. Totalling the regions first helps you catch a slip.

8. (a) Squares: 1, 4, 9, 16, 25, 36. Multiples of 4: 4, 8, …, 36 (9 numbers). In both: 4, 16, 36 [1]. Square only: 1, 9, 25 [1]. Multiple of 4 only: 8, 12, 20, 24, 28, 32 [1] (b) In at least one set: 3 + 3 + 6 = 12, so neither = 24 and P = 24/36 = 2/3 [1] (c) Exactly one set: 3 + 6 = 9, so P = 9/36 = 1/4 [1] Examiner insight: List each set’s members first. The number 1 is a square and is easy to overlook.

9. (a) The completed table [2]:

1st \ 2nd	1	3	5
2	3	5	7
3	4	6	8
4	5	7	9

(b) There are 9 outcomes [1]. Scores of 4 or less: 3, 4, so P(L) = 2/9. Scores of 5 to 7: 5, 7, 6, 5, 7, so P(M) = 5/9 [1]. Scores of 8 or more: 8, 9, so P(N) = 2/9 [1] (c) Every score is in exactly one group, so the events cannot overlap and cover everything. Also 2/9 + 5/9 + 2/9 = 1 [1] (d) L and N are mutually exclusive, so P(L or N) = P(L) + P(N) [1] = 2/9 + 2/9 = 4/9 [1] Examiner insight: In a “show that” part, give both reasons: no overlap, and no score left out. The sum to 1 supports the claim but does not replace it.

10. (a) The regions are French only (24 − x), both (x), Spanish only (20 − x) and neither (9), totalling 50 [1]. So (24 − x) + x + (20 − x) + 9 = 50, which gives 53 − x = 50 [1], so x = 3 [1] (b) French only 21, both 3, Spanish only 17, neither 9 [1] (c) P(French only) = 21/50 [1] (d) Exactly one language: 21 + 17 = 38 [1], so P = 38/50 = 19/25 [1] Examiner insight: Define the unknown on the diagram, then write the equation from the total.

Where marks are usually lost

• Listing outcomes that are not equally likely, such as one outcome per colour.
• Losing one of a pair like (3, 5) and (5, 3), or including an impossible pair.
• Adding probabilities for events that overlap, or claiming “mutually exclusive” with no reason.
• Putting circle totals into the circles instead of the “only” and “both” values.
• Leaving out the “neither” region, or not checking that the regions total the group size.
• Giving a count (such as 21) when the question asks for a probability (such as 21/50).

Next steps

• Revise with the revision notes.
• Re-read the study guide.
• Go to the course hub and the checklist.
• Take one of the free 10-minute diagnostics.
• Book a free trial class.

Official syllabus

OxfordAQA International GCSE Mathematics (9260), Version 5.1 (for exams May/June 2018 onwards), published by OxfordAQA, section 3.4.3 Probability, references S14 to S16.