Probability of combined events — Paper 4 · IGCSE Mathematics 0580

4 Ravi spins a biased 5-sided spinner, numbered 1 to 5. The probability of each number is shown in the table. Number 1 2 3 4 5 1 1 1 Probability x x 6 4 3 (a) Find the value of x. x = … [3] (b) Ravi spins the spinner once. Find the probability that the number is 2 or 3. … [2] (c) Ravi spins the spinner twice. Find the probability that (i) the number is 2 both times, … [2] (ii) the sum of the numbers is 3. … [3] (d) Ravi spins the spinner 72 times. Calculate how many times he expects the number 1. … [1]

11 marks

This question in 0580/42 Feb/March 2017

6 Each morning the probability that it rains is . 3 1 If it rains, the probability that Asha walks to school is . 7 4 If it does not rain, the probability that Asha walks to school is . 7 (a) Complete the tree diagram. 1 Walks 7 Rains 2 3 … Does not walk Walks … … Does not rain … Does not walk [2] (b) Find the probability that it rains and Asha walks to school. … [2] (c) (i) Find the probability that Asha does not walk to school. … [3] (ii) Find the expected number of days Asha does not walk to school in a term of 70 days. … [2] (d) Find the probability that it rains on exactly one morning in a school week of 5 days. … [2]

11 marks

Mark scheme: 6(a) 1 6 1 , correctly placed 3 7 4 3 1 , correctly placed 7 7 6(b) 2 2 2 1 oe M1 for × 21 3 7 6(c)(i) 15 3 2 6 1 3 oe M2 for × + × oe 21 3 7 3 7 2 6 1 3 or M1 for × oe or × oe seen 3 7 3 7 6(c)(ii) 50 2FT FT (70 × their (c)(i)) rounded up or down to integer M1 for 70 × their (c)(i) 2 6(d) 10 2 1 1 1 × × × oe M1 for × 1[×k ] oe nfww 243 3 3 3 3 3 where k is positive integer less than 5

This question in 0580/42 May/June 2017

5 (a) Haroon has 200 letters to post. The histogram shows information about the masses, m grams, of the letters. 8 7 6 5 Frequency density 4 3 2 1 0 m 0 10 20 30 40 50 Mass (grams) (i) Complete the frequency table for the 200 letters. Mass (m grams) 0 1 m G 10 10 1 m G 20 20 1 m G 25 25 1 m G 30 30 1 m G 50 Frequency 50 17 [3] (ii) Calculate an estimate of the mean mass. … g [4] (b) Haroon has 15 parcels to post. The table shows information about the sizes of these parcels. Size Small Large Frequency 9 6 Two parcels are selected at random. Find the probability that (i) both parcels are large, … [2] (ii) one parcel is small and the other is large. … [3] 3(c) The probability that a parcel arrives late is . 80 4000 parcels are posted. Calculate an estimate of the number of parcels expected to arrive late. … [1]

13 marks

Mark scheme: 5(a)(i) 80 33 20 1, 1, 1 5(a)(ii) 17.3 nfww 4 M1 for 5, 15, 22.5, 27.5, 40 soi M1 for ∑fx with their f’s and x in correct interval including both boundaries M1 (dep on 2nd M1) for ∑fx ÷ 200 5(b)(i) 30 2 6 5 oe M1 for × 210 15 14 36 If zero scored, SC1 for answer oe 225 5(b)(ii) 108 3 6 9 9 6 oe M2 for × + × oe 210 15 14 15 14 9 8 6 5 or 1 – × − × 15 14 15 14 6 9 9 6 or M1 for × or × 15 14 15 14 9 8 6 5 or × + × 15 14 15 14 108 If zero scored, SC1 for answer oe 225 5(c) 150 1

This question in 0580/43 May/June 2017

7 0 1 0 1 1 2 A B The diagram shows two fair dice. The numbers on dice A are 0, 0, 1, 1, 1, 3. The numbers on dice B are 1, 1, 2, 2, 2, 3. When a dice is rolled, the score is the number on the top face. (a) Dice A is rolled once. Find the probability that the score is not 3. … [1] (b) Dice A is rolled twice. Find the probability that the score is 0 both times. … [2] (c) Dice A is rolled 60 times. Calculate an estimate of the number of times the score is 0. … [1] (d) Dice A and dice B are each rolled once. The product of the scores is recorded. (i) Complete the possibility diagram. 3 0 0 2 0 0 2 0 0 Dice B 2 0 0 1 0 0 1 0 0 1 1 1 3 0 0 1 1 1 3 Dice A [2] (ii) Find the probability that the product of the scores is (a) 2, … [1] (b) greater than 3. … [1] (e) Eva keeps rolling dice B until 1 is scored. Find the probability that this happens on the 5th roll. … [2]

10 marks

Mark scheme: 7(a) 5 1 6 7(b) 4 2 2 2 oe M1 for × 36 6 6 7(c) 20 1 7(d)(i) Diagram completed correctly 2 B1 for 3 correct columns or for 4 correct rows x x 3 3 3 9 x x 2 2 2 6 x x 2 2 2 6 x x 2 2 2 6 x x 1 1 1 3 7(d)(ii)(a) 9 1FT FT their (d)(i) oe 36 7(d)(ii)(b) 4 1FT FT their (d)(i) oe 36 7(e) 512 2  4 k 2 oe M1 for oe k = 3, 4 or 5 only  × 7776  6  6

This question in 0580/42 Oct/Nov 2017

4 The table shows information about the time, t minutes, taken for each of 150 girls to complete an essay. Time (t minutes) 60 1 t G 65 65 1 t G 70 70 1 t G 80 80 1 t G 100 100 < t G 150 Frequency 10 26 34 58 22 (a) Write down the interval that contains the median time. … 1 t G … [1] (b) Calculate an estimate of the mean time. … min [4] (c) Rafay looks at the frequency table. (i) He says that it is not possible to work out the range of the times. Explain why he is correct. … … [1] (ii) He draws a pie chart to show this information. Calculate the sector angle for the interval 65 1 t G 70 minutes. … [2] (d) A girl is chosen at random. Work out the probability that she took more than 100 minutes to complete the essay. … [1] (e) Two girls are chosen at random. Work out the probability that, to complete the essay, (i) they both took 65 minutes or less, … [2] (ii) one took 65 minutes or less and the other took more than 100 minutes. … [3] (f) The information in the frequency table is shown in a histogram. The height of the block for the 60 1 t G 65 interval is 5 cm. Complete the table. Time (t minutes) 60 1 t G 65 65 1 t G 70 70 1 t G 80 80 1 t G 100 100 1 t G 150 Height of block 5 (cm) [3]

17 marks

Mark scheme: 4(a) 80 < t ⩽100 1 4(b) 86 nfww 4 M1 for midpoints soi M1 for use of Σfx with x in correct interval including both boundaries M1 (dep on 2nd M1) for Σfx ÷ 150 4(c)(i) Reference to not knowing the 1 individual values so we do not know the highest or the lowest values 4(c)(ii) 62.4 2 M1 for 26 ÷ 150 or 360 ÷ 150 soi 4(d) 22 1 oe 150 4(e)(i) 90 2 10 9 oe M1 for × 22350 150 149 100 After zero scored, SC1 for answer oe 22500 4(e)(ii) 440 3 10 22 22 10 oe M2 for × + × oe 22350 150 149 150 149 or 10 22 22 10 M1 for × or × oe 150 149 150 149 440 After zero scored, SC1 for answer oe 22500 4(f) 13, 8.5, 7.25, 1.1 3 B2 for 3 correct or B1 for 1 correct or for 3 correct FD.s 5.2, 3.4, 2.9, 0.44 oe

This question in 0580/43 Oct/Nov 2017

9 The probability that it will rain tomorrow is . 8 1 If it rains, the probability that Rafael walks to school is . 6 7 If it does not rain, the probability that Rafael walks to school is . 10 (a) Complete the tree diagram. Walks … Rains … … Does not walk Walks … … Does not rain … Does not walk [3] (b) Calculate the probability that it will rain tomorrow and Rafael walks to school. … [2] (c) Calculate the probability that Rafael does not walk to school. … [3]

8 marks

Mark scheme: 9(a) 5 3 3 B1 for each pair 8 8 1 5 6 6 7 3 10 10 9(b) 5 2 5 1 oe M1FT for their × their 48 8 6 9(c) 304 3 M2 for oe 5 5 3 3 480 their × their + their × their oe 8 6 8 10 or M1 for 5 5 3 3 their × their or their × their 8 6 8 10

This question in 0580/41 May/June 2018

4 (a) The diagram shows two sets of cards. Set A 1 1 2 2 2 Set B 0 1 1 1 2 (i) Jojo chooses two cards at random from Set A without replacement. Find the probability that the two cards have the same number. … [3] (ii) Jojo replaces the two cards. Kylie then chooses one card at random from Set A and one card at random from Set B. Find the probability that the two cards have the same number. … [3] (iii) Who is the most likely to choose two cards that have the same number? Show all your working. … [1] (b) Set C 4 4 5 5 5 Lena chooses three cards at random from Set C without replacement. Find the probability that the third card chosen is numbered 4. … [3]

10 marks

Mark scheme: 4(a)(i) 8 3 2 1 3 2 oe M2 for × + × 20 5 4 5 4 or M1 for one of these products OR M1 for probability tree identifying all 20 outcomes with the correct 8 identified OR M1 for completed possibility space / 2-way table identifying the 8 possible outcomes out of 20, oe 13 SC1 for with replacement 25 4(a)(ii) 9 3 2 3 3 1 oe M2 for × + × oe 25 5 5 5 5 or M1 for one of these products OR M1 for probability tree identifying all 25 outcomes with the correct 9 identified OR M1 for completed possibility space / 2-way table identifying the 9 possible outcomes out of 25, oe 4(a)(iii) 40 36 1 1FT their (i) and (ii) dep on being in range 0 to 1 Jojo and e.g. > 100 100 4(b) 24 3 2 3 1 3 2 1 3 2 2 oe M2 for × × + × × + × × oe 60 5 4 3 5 4 3 5 4 3 or M1 for any one correct product OR M1 for 4, 5, 4 and 5, 4, 4 and 5, 5, 4 clearly identified on a tree or in a list

This question in 0580/43 May/June 2018

7 Bag A Bag B Bag A contains 3 black balls and 2 white balls. Bag B contains 1 black ball and 3 white balls. (a) A ball is taken at random from each bag. (i) Show that a black ball is more likely to be taken from bag A than from bag B. [1] (ii) Find the probability that the two balls have different colours. … [3] (b) The balls are returned to their original bags. Three balls are taken at random from bag A, without replacement. Find the probability that (i) they are all black, … [2] (ii) they are all white. … [1] (c) The balls are returned to their original bags. A ball is taken at random from bag A and its colour is recorded. This ball is then placed in bag B. A ball is then taken at random from bag B. Find the probability that the ball taken from bag B has a different colour to the ball taken from bag A. … [3]

10 marks

Mark scheme: 7(a)(i) 3 1 12 k 5k 1 > oe or and 5 4 20 k 20 k or 0.6 and 0.25 or 60% and 25% 7(a)(ii) 11 3 3 3 2 1 oe M2 for × + × oe 20 5 4 5 4 3 1 2 3 or 1 – × – × oe 5 4 5 4 3 3 2 1 or M1 for × or × oe 5 4 5 4 (but not as part of a larger product) 7(b)(i) 6 2 3 2 1 oe M1 for × × oe 60 5 4 3 27 If 0 scored, SC1 for answer oe 125 7(b)(ii) 0 1 0 Accept 60 7(c) 11 3 3 3 2 1 oe M2 for × + × oe 25 5 5 5 5 3 2 2 4 or 1 – × – × oe 5 5 5 5 3 3 2 1 or M1 for × or × or for a correct tree 5 5 5 5 showing all 25 outcomes with the 11 correct outcomes identified

This question in 0580/43 Oct/Nov 2018

3 Sushila, Ravi and Talika each have a bag of balls. Each of the bags contains 10 red balls and 8 blue balls. (a) Sushila takes one ball at random from her bag. Find the probability that she takes a red ball. … [1] (b) Ravi takes two balls at random from his bag, without replacement. Find the probability that one ball is red and one ball is blue. … [3] (c) Talika takes three balls at random from her bag, without replacement. Calculate the probability that the three balls are the same colour. … [4]

8 marks

Mark scheme: 3(a) 5 1 oe 9 3(b) 80 3 10 8 oe M2 for 2 × × oe 153 18 17 10 8 or M1 for × oe 18 17 160 If 0 scored, SC1 for oe 324 3(c) 11 4 10 9 8 8 7 6 oe M3 for × × + × × oe 51 18 17 16 18 17 16 10 9 8 8 7 6 or M2 for × × oe or × × 18 17 16 18 17 16 oe 10 9 8 8 7 6 or M1 for , , or , , 18 17 16 18 17 16 1512 If 0 scored, SC1 for oe 5832

This question in 0580/42 Feb/March 2019

6  = {students in a school} F = {students who play football} B = {students who play baseball} There are 240 students in the school. • 120 students play football • 40 students play baseball • 90 students play football but not baseball. (a) Complete the Venn diagram to show this information. F B … … … … [2] (b) Find n F l + B l . ^ h … [1] (c) A student in the school is chosen at random. Find the probability that this student plays baseball but not football. … [1] (d) Two students who play baseball are chosen at random. Find the probability that they both also play football. … [3]

7 marks

Mark scheme: 6(a) 2 B1 for any one correct 90 30 10 110 6(b) 110 1 FT their 110 in Venn diagram 6(c) 10 1 their10 oe FT 240 240 6(d) 870 3 their 30 their 30 − 1 oe M2 for × 1560 40 39 p p − 1 their 30 or M1 for × p < q or for q q − 1 40 soi

This question in 0580/41 May/June 2019

8 (a) Angelo has a bag containing 3 white counters and x black counters. He takes two counters at random from the bag, without replacement. (i) Complete the following statement. The probability that Angelo takes two black counters is x # . x + 3 [2] 7 (ii) The probability that Angelo takes two black counters is . 15 (a) Show that 4x2 - 25x - 21 = 0. [4] (b) Solve by factorisation. 4x2 - 25x - 21 = 0 x = … or x = … [3] (c) Write down the number of black counters in the bag. … [1] (b) Esme has a bag with 5 green counters and 4 red counters. She takes three counters at random from the bag without replacement. Work out the probability that the three counters are all the same colour. … [4]

14 marks

Mark scheme: 8(a)(i) x − 1 2 B1 for either numerator or denominator correct x + 2 8(a)(ii)(a) x x − 1 7 B1 7 × = FT their (a)(i) = x + 3 x + 2 15 15 15x(x – 1) = 7(x + 3)(x + 2) M1 Removes all algebraic fractions FT their equation if in comparable form 15x2 – 15x = 7x2 + 21x + 14x + 42 M1 Correctly expands all brackets FT their equation if in comparable form [8x2 – 50x – 42 = 0] A1 With no errors or omissions seen and one further 4x2 – 25x – 21 = 0 stage seen after final M1 8(a)(ii)(b) (4x + 3)(x – 7) [= 0] M2 M1 for 4x(x – 7) + 3(x – 7) or x (4x + 3) – 7(4x + 3) or for (4x + a)(x + b) where either ab = –21 or 4b + a = –25 If 0 scored, SC1 for 4x + 3 and x – 7 seen but not in factorised form 3 B1 7 and − 4 8(a)(ii)(c) 7 1 FT their positive solution 8(b) 1 4 5 4 3 4 3 2 oe M3 for × × + × × 6 9 8 7 9 8 7 5 4 3 4 3 2 or M2 for × × or × × 9 8 7 9 8 7 5 4 3 4 3 2 or M1 for , , seen or , , seen 9 8 7 9 8 7 53 + 4 3 If 0 scored, SC1 for oe 729

This question in 0580/43 May/June 2019

8 The diagram shows 5 cards. (a) Donald chooses a card at random. (i) Write down the probability that the number of dots on this card is an even number. … [1] (ii) Write down the probability that the number of dots on this card is a prime number. … [1] (b) Donald chooses two of the five cards at random, without replacement. He works out the total number of dots on these two cards. (i) Find the probability that the total number of dots is 5. … [3] (ii) Find the probability that the total number of dots is an odd number. … [3]

8 marks

Mark scheme: 8(a)(i) 4 1 oe 5 8(a)(ii) 4 1 oe 5 8(b)(i) 6 3 1 3 3 1 1 3 oe nfww M2 for × + × oe or 2 × × oe 20 5 4 5 4 5 4 1 3 3 1 or M1 for × alone or × alone or for 5 4 5 4 3 answer nfww 20 6 After 0 scored, SC1 for answer 25 8(b)(ii) 8 3 4 3 1 4 1 oe nfww M2 for 1 − × or × 1 + × oe or 20 5 4 5 5 4 1 2 × × 1 5 1 3 1 1 or 2 × × + 2 × × or 5 4 5 4 1 1 their (b)(i) + 2 × × 5 4 2 or 4 or 5 or 6 or 7 or M1 for answer oe nfww 20 8 After 0 scored, SC1 for answer 25

This question in 0580/41 Oct/Nov 2019

8 (a) A bag contains 4 red marbles and 2 yellow marbles. Behnaz picks two marbles at random without replacement. Find the probability that (i) the marbles are both red, … [2] (ii) the marbles are not both red. … [1] (b) Another bag contains 5 blue marbles and 2 green marbles. Bryn picks one marble at random without replacement. If this marble is not green, he picks another marble at random without replacement. He continues until he picks a green marble. Find the probability that he picks a green marble on his first, second or third attempt. … [4]

7 marks

Mark scheme: 8(a)(i) 2 2 4 3 oe M1 for × 5 6 5 8(a)(ii) 3 1 12 oe FT 1 – their oe 5 30 8(b) 5 4 2 5 2 5 4 2 oe nfww M3 for + × + × × oe 7 7 7 6 7 6 5 5 4 3 or for 1 − × × oe 7 6 5 5 2 5 4 2 or M1 for each of × and × × oe 7 6 7 6 5 or completed tree diagram with appropriate probabilities shown

This question in 0580/43 Oct/Nov 2019

9 This year, 40 students have each travelled by one or more of plane (P), train (T) or boat (B). 7 have travelled only by plane. 11 have travelled only by train. 9 have travelled only by boat. n ( P + T ) = 8 n ( B + T ) = 3 n ( B + P ) = 6 P T B (a) Complete the Venn diagram. [3] (b) Find n ( P , B ) l . ` j … [1] (c) Use set notation to complete the statement. ( P , T , B ) l = … [1] (d) Two students are chosen at random. Calculate the probability that they both have travelled only by plane. … [2] (e) Two students are chosen at random from those who have travelled by train. Calculate the probability that they both have also travelled by plane. … [2]

9 marks

Mark scheme: 9(a) 3 B2 for 5 correct entries including ‘2’ correctly placed at the intersection of the P 3 sets T 6 7 11 or M1 for 2 k + 8 − k + 3 − k + 6 − k = 40 − (7 + 9 + 11) oe 4 1 or for k, 8 – k, 3 – k, 6 – k, seen correctly placed on diagram with 7, 11 and 9 9 correctly placed B 9(b) 11 1 9(c) Ø or { } 1 9(d) 7 2 7 6 oe M1 for × oe 260 40 39 9(e) 14 2 FT their Venn diagram oe 95 8 7 M1 for × 20 19

This question in 0580/42 Feb/March 2020

3 The speed, v km/h, of each of 200 cars passing a building is measured. The table shows the results. Speed (v km/h) 0 1 v G 20 20 1 v G 40 40 1 v G 45 45 1 v G 50 50 1 v G 60 60 1 v G 80 Frequency 16 34 62 58 26 4 (a) Calculate an estimate of the mean. … km/h [4] (b) (i) Use the frequency table to complete the cumulative frequency table. Speed (v km/h) v G 20 v G 40 v G 45 v G 50 v G 60 v G 80 Cumulative frequency 16 50 196 200 [1] (ii) On the grid, draw a cumulative frequency diagram. 200 180 160 140 120 Cumulative frequency 100 80 60 40 20 0 v 0 10 20 30 40 50 60 70 80 Speed (km/h) [3] (iii) Use your diagram to find an estimate of (a) the upper quartile, … km/h [1] (b) the number of cars with a speed greater than 35 km/h. … [2] (c) Two of the 200 cars are chosen at random. Find the probability that they both have a speed greater than 50 km/h. … [2] (d) A new frequency table is made by combining intervals. Speed (v km/h) 0 1 v G 40 40 1 v G 50 50 1 v G 80 Frequency 50 120 30 On the grid, draw a histogram to show the information in this table. 15 10 Frequency density 5 0 v 0 10 20 30 40 50 60 70 80 Speed (km/h) [3]

16 marks

Mark scheme: 3(a) 41.4 4 M1 for 10, 30, 42.5, 47.5, 55, 70 M1 for Σ fx where x lies in or on the boundary of each interval. Σfx M1 dep for dep on second M1 200 3(b)(i) 112, 170 1 3(b)(ii) Correct diagram 3 B1 for correct horizontal plot B1FT for correct vertical plots B1 FT dep on at least B1 earned for reasonable increasing curve or polygon through their 6 points If 0 scored SC1FT for 5 out of 6 points plotted correctly 3(b)(iii)(a) 48 1 3(b)(iii)(b) 160 2 M1 for 40 seen 3(c) 87 2 30 29 oe M1 for × oe 3980 200 199 3(d) Correct histogram 3 B1 for each column If 0 scored SC1 for correct frequency densities soi 1.25, 12, 1

This question in 0580/42 May/June 2020

7 Tanya plants some seeds. The probability that a seed will produce flowers is 0.8 . When a seed produces flowers, the probability that the flowers are red is 0.6 and the probability that the flowers are yellow is 0.3 . (a) Tanya has a seed that produces flowers. Find the probability that the flowers are not red and not yellow. … [1] (b) (i) Complete the tree diagram. Produces Colour flowers Red … … Yes Yellow 0.8 … Other colours No … [2] (ii) Find the probability that a seed chosen at random produces red flowers. … [2] (iii) Tanya chooses a seed at random. Find the probability that this seed does not produce red flowers and does not produce yellow flowers. … [3] (c) Two of the seeds are chosen at random. Find the probability that one produces flowers and one does not produce flowers. … [3]

11 marks

Mark scheme: 7(a) 0.1 1 7(b)(i) 0.2 oe 2 B1 for 0.2 0.6, 0.3, 0.1 oe B1 for 0.6, 0.3, 0.1 7(b)(ii) 0.48 oe 2 FT their 0.6 from tree diagram M1 for 0.8 × their 0.6 7(b)(iii) 0.28 oe 3 M2 for 0.2 + 0.8 × 0.1 oe or M1 for 0.2 or 0.8 × 0.1 or 0.8 × (0.6 + 0.3) 7(c) 0.32 oe 3 M2 for 0.8 × 0.2 + 0.2 × 0.8 oe M1 for one of these products

This question in 0580/42 May/June 2020

7 On any Saturday, the probability that Arun plays football is 4. On any Saturday, the probability that Bob plays football is 2. 5 (a) (i) Complete the tree diagram. Arun Bob … Plays Plays … Does not play … … Plays Does not … play Does not play … [2] (ii) Calculate the probability that, one Saturday, Arun and Bob both play football. … [2] (iii) Calculate the probability that, one Saturday, either Arun plays football or Bob plays football, but not both. … [3] (b) Calculate the probability that Bob plays football for 2 of the next 3 Saturdays. … [3] (c) When Arun plays football, the probability that he scores the winning goal is 1. 7 Calculate the probability that Arun scores the winning goal one Saturday. … [2]

12 marks

Mark scheme: 7(a)(i) 3 1 2 3 2 3 2 B1 for one correct pair , , , 4 4 5 5 5 5 7(a)(ii) 3 2 FT their tree diagram oe 3 2 10 M1 for × 4 5 7(a)(iii) 11 3 3 3 1 2 oe M2 for × + × 20 4 5 4 5 3 3 1 2 or M1 for × or × 4 5 4 5 7(b) 36 3  2 2 3 oe M2 for × × 3 oe   125  5  5  2 ×2 3 or M1 for    5  5 7(c) 3 2 3 1 oe M1 for × 28 4 7

This question in 0580/43 May/June 2020

9 (a) There are 32 students in a class. 5 do not study any languages. 15 study German (G). 18 study Spanish (S). G S (i) Complete the Venn diagram to show this information. [2] (ii) A student is chosen at random. Find the probability that the student studies Spanish but not German. … [1] (iii) A student who studies German is chosen at random. Find the probability that this student also studies Spanish. … [1] (b) A bag contains 54 red marbles and some blue marbles. 36% of the marbles in the bag are red. Find the number of blue marbles in the bag. … [2] (c) Another bag contains 15 red beads and 10 yellow beads. Ariana picks a bead at random, records its colour and replaces it in the bag. She then picks another bead at random. (i) Find the probability that she picks two red beads. … [2] (ii) Find the probability that she does not pick two red beads. … [1] (d) A box contains 15 red pencils, 8 yellow pencils and 2 green pencils. Two pencils are picked at random without replacement. Find the probability that at least one pencil is red. … [3]

12 marks

Mark scheme: 9(a)(i) 2 B1 for two correct values 5 Or 9 6 12 B1 5 outside and total in G = 15 and total in G S S = 18 9(a)(ii) 3 1 their 12 oe FT 8 32 9(a)(iii) 2 1 their 6 oe FT 5 15 9(b) 96 2 36 54 54 M1 for = oe or 36 = × 100 64 x ( 54 + b ) oe If 0 scored SC1 for answer 150 9(c)(i) 9 2 15 15 oe M1 for × oe 25 25 25 9(c)(ii) 16 1 FT 1 – their (c)(i) oe 25 9(d) 17 3 10 9 oe M2 for 1 − × oe 20 25 24 15 14 15 8 15 2 8 15 or for × + × + × + × 25 24 25 24 25 24 25 24 2 15 + × oe 25 24 or M1 for one correct relevant product

This question in 0580/41 Oct/Nov 2020

6 4 6 3 4 2 3 Red Yellow Blue Blue Yellow Blue The diagram shows six discs. Each disc has a colour and a number. (a) One disc is picked at random. Write down the probability that (i) the disc has the number 4, … [1] (ii) the disc is red and has the number 3, … [1] (iii) the disc is blue and has the number 4. … [1] (b) Two of the six discs are picked at random without replacement. Find the probability that (i) both discs have the number 3, … [2] (ii) both discs have the same colour. … [3] (c) Two of the six discs are picked at random with replacement. Find the probability that both discs have the same colour. … [3]

11 marks

Mark scheme: 6(a)(i) 1 1 oe 3 6(a)(ii) 0 1 6(a)(iii) 1 1 oe 6 6(b)(i) 1 2 2 1 oe M1 for × or equivalent method 15 6 5 6(b)(ii) 4 3 2 1 3 2 oe M2 for × + × or equivalent method 15 6 5 6 5 2 1 3 2 or M1 for × oe seen or × oe seen 6 5 6 5 6(c) 7 3  1  2  2  2  3  2 oe M2 for + + oe       18  6   6   6  or M1 for one correct product seen or sample space with 14 correct pairs identified

This question in 0580/42 Oct/Nov 2020

4 P O S S I B I L I T Y Morgan picks two of these letters, at random, without replacement. (a) Find the probability that he picks (i) the letter Y first, … [1] (ii) the letter B then the letter Y, … [2] (iii) two letters that are the same. … [3] (b) Morgan now picks a third letter at random. Find the probability that (i) all three letters are the same, … [2] (ii) exactly two of the three letters are the same, … [5] (iii) all three letters are different. … [2]

15 marks

Mark scheme: 4(a)(i) 1 1 oe 11 4(a)(ii) 1 2 1 1 oe M1 for × oe 110 11 10 4(a)(iii) 4 3  2 1   3 2  oe M2 for  ×  +  ×  oe 55  11 10   11 10   2 1   3 2  or M1 for  ×  or  ×  seen  11 10   11 10  oe 4(b)(i) 1 2 3 2 1 oe M1 for × × oe 165 11 10 9 4(b)(ii) 1 5  2 1 9   3 2 8  × × × oe M4 for 3   + 3   ×  5 10 10 9 9  11   11  oe  3 2 8  or M3 for 3  × ×   11 10 9   2 1 9  or M2 for 3  × ×   or  11 10 9  3 2 8 × × oe 11 10 9 2 1 k or M1 for × × where k is 3, 6  11 10 9 or 9 4(b)(iii) 131 2 M1 for 1 – (their (b)(i) + their (b)(ii)) oe oe 165

This question in 0580/43 Oct/Nov 2020

6 In a class of 24 students, 18 students like homework (H ), 15 students like tests (T ) and 1 student does not like homework and does not like tests. (a) Complete the Venn diagram to show this information. H T … … … 1 … [2] (b) Write down the number of students who like both homework and tests. … [1] (c) Find n (H l + T ) . … [1] (d) A student is picked at random from the class. Write down the probability that this student likes tests but does not like homework. … [1] (e) Two students are picked at random from the class. Find the probability that both students do not like homework and do not like tests. … [1] (f) Two of the students who like homework are picked at random. Find the probability that both students also like tests. … [3]

9 marks

Mark scheme: 6(a) 2 i.e. 8, 10 and 5 correctly placed T H B1 for 10 correctly placed or M1 for 18 – x , x and 15 – x correctly 8 10 5 placed on diagram and x = 10 seen 1 6(b) 10 1 FT their Venn diagram 6(c) 5 1 FT their Venn diagram 6(d) 5 1 FT their 5 on the Venn diagram oe 24 6(e) 0 1 6(f) 5 3 their10 their 9 oe M2 for × 17 18 17 their10 their 9 or B1FT for or seen 18 17 25 After 0 scored, SC1 for answer oe 81

This question in 0580/43 May/June 2021

10 (a) Sarah spins a fair four-sided spinner numbered 0, 1, 1 and 3. (i) What number is the spinner most likely to land on? … [1] (ii) Sarah spins the spinner twice. Find the probability that it lands on the number 1 both times. … [2] (iii) Sarah spins the spinner until it lands on the number 3. 729 The probability that this happens on the nth spin is . 16384 Find the value of n. n = … [2] (b) Scott takes an examination. The examination is in two parts, a theory test and a practical test. Both parts must be passed to pass the examination. The probability that Scott passes the theory test is 0.9 . The probability that Scott passes the practical test is 0.8 . Find the probability that (i) Scott passes the examination, … [2] (ii) Scott passes the theory test or the practical test but not both. … [3]

10 marks

Mark scheme: 10(a)(i) 1 1 10(a)(ii) 1 2 2 2 oe nfww M1 for × oe 4 4 4 10(a)(iii) 7 2 k M1 for trials with  3 × 1 soi    4  4 10(b)(i) 0.72 oe 2 M1 for 0.9 × 0.8 10(b)(ii) 0.26 oe 3 M2 for 0.9 × 0.2 + 0.1 × 0.8 or 1 – their (b)(i) – 0.1 × 0.2 or M1 for 0.9 × 0.2 or 0.1 × 0.8 or 1 – their (b)(i) or 1 – 0.1 × 0.2

This question in 0580/43 Oct/Nov 2021

11 (a) The probability that Shalini is late for school on any day is 6. (i) Complete the tree diagram for Monday and Tuesday. Monday Tuesday Late … Late … … Not late Late … … Not late … Not late [2] (ii) Calculate the probability that Shalini is late on Monday but is not late on Tuesday. … [2] (b) The Venn diagram shows the number of students in a group of 50 students who wear glasses (G), who wear trainers (T ) and who have a mobile phone (M ). G T 0 2 3 2 14 19 1 9 M (i) Use set notation to describe the region that contains only one student. … [1] (ii) Find n T l + G , M b ` jl. … [1] (iii) One student is picked at random from the 50 students. Find the probability that this student wears trainers but does not wear glasses. … [1] (iv) Two students are picked at random from those wearing trainers. Find the probability that both students have mobile phones. … [3]

10 marks

Mark scheme: 11(a)(i) 1 2 1 5 oe on all late branches B1 for one correct vertical pair oe and oe 6 6 6 5 oe on all not late branches 6 11(a)(ii) 5 2 FT their tree oe 1 5 36 M1 for their × their 6 6 11(b)(i) 1 ( G ∪ T ∪ M )′ oe 11(b)(ii) 28 1 11(b)(iii) 17 1 oe 50 11(b)(iv) 4 3 16 15 oe M2 for × 7 21 20 n n − 1 16 15 or M1 for × or for and seen 21 20 21 20 256 If 0 scored SC1 for answer oe 441

This question in 0580/42 Feb/March 2022

6 (a) At a festival, 380 people out of 500 people questioned say that they are camping. There are 55 300 people at the festival. Calculate an estimate of the total number of people camping at the festival. … [2] (b) 12 friends travel to the festival. 5 travel by car, 4 travel by bus and 3 travel by train. Two people are chosen at random from the 12 friends. Calculate the probability that they travel by different types of transport. … [4] (c) Arno buys a student ticket for $43.68 . This is a saving of 16% on the full price of a ticket. Calculate the full price of a ticket. $ … [2] (d) At a football match, there are 29 800 people, correct to the nearest 100. (i) At the end of the football match, the people leave at a rate of 400 people per minute, correct to the nearest 50 people. Calculate the lower bound for the number of minutes it takes for all the people to leave. … min [3] (ii) At a cricket match there are 27 500 people, correct to the nearest 100. Calculate the upper bound for the difference between the number of people at the football match and at the cricket match. … [2]

13 marks

Mark scheme: 6(a) 42 028 2 380 M1 for oe soi isw 500 6(b) 47 4 0.712[1…] oe 66  5 4   4 3   5 3  M3 for 2   2   2         12 11   12 11   12 11  oe  5 4 4 3 3 2  or 1 –      oe    12 11 12 11 12 11  or M2 for sum of 3 or more correct product pairs and no incorrect pairs 5 4 4 3 3 2 or for      and no other 12 11 12 11 12 11 pairs k j or M1 for  seen 12 11 94 If 0 scored SC1 for answer oe 144 6(c) 52 2 100  16 M1 for x   43.68 oe or better 100 6(d)(i) 70 or 70.16[5…] or 70.17 or 70.2 3 29750 to 29800 29750 to 29800 M2 for or or 400  25 400  24 29800  50 400to425 or B1 for 29 750 or 29 850 or 29 849 or 375 or 425 or 424 seen 6(d)(ii) 2399 2 B1 for 27 450 or 27 550 or 27 549 or 29 850 or or 2400 nfww 29 849 seen

This question in 0580/42 May/June 2022

3 Kai and Ann carry out a survey on the distances travelled, in kilometres, by 200 cars. Kai completes this frequency table for the data collected. Distance (d km) 80 1 d G 100 100 1 d G 150 150 1 d G 200 200 1 d G 300 300 1 d G 400 Frequency 7 33 76 52 32 (a) (i) Calculate an estimate of the mean. … km [4] (ii) Ann uses this frequency table for the same data. There is a different interval for the final group. Distance (d km) 80 1 d G 100 100 1 d G 150 150 1 d G 200 200 1 d G 300 300 1 d G 360 Frequency 7 33 76 52 32 Without calculating an estimate of the mean for this data, find the difference between Ann’s and Kai’s estimate of the mean. You must show all your working. … km [2] (iii) A histogram is drawn showing the information in Kai’s frequency table. The height of the block for the interval 200 1 d G 300 is 2.6 cm. Calculate the height of the block for each of the following intervals. 80 1 d G 100 … cm 150 1 d G 200 … cm 300 1 d G 400 … cm [3] (b) One car is picked at random. Find the probability that the car has travelled more than 300 km. … [1] (c) Two of the 200 cars are picked at random. Find the probability that (i) both cars have travelled 150 km or less, … [2] (ii) one car has travelled more than 200 km and the other car has travelled 100 km or less. … [3]

15 marks

Mark scheme: 3(a)(i) 211.275 4 M1 for mid-points soi (90, 125, 175, 250, 350) M1 for use of fm with m in correct interval including both boundaries M1 for (dep on 2nd M1) for fm  200 3(a)(ii) 32  350 – 32  330 oe or better, or the reverse of this M1 3.2 or – 3.2 final answer B1 3(a)(iii) 1.75 3 B2 for two correct heights or B1 for one correct height or 3 correct frequency densities 7.6 1.6 or M1 for scale factor of 5 or 0.2 3(b) 4 1 oe 25 3(c)(i) 39 2 40 39 oe M1 for  oe 995 200 199 3(c)(ii) 147 3 oe 4975 84 7 M2 for [2]  oe 200 199 84 7 84 7 or B1 for and or and oe 200 199 199 200 147 If 0 scored, SC1 for answer oe 5000

This question in 0580/42 Oct/Nov 2022

7 Regan is playing a game with these six number cards. – 3 – 2 2 3 5 7 (a) She takes two cards at random, without replacement, and multiplies the two numbers to give a score. Find the probability that (i) the score is 35 … [3] (ii) the score is a positive number. … [3] (b) Regan now takes three cards at random from the six cards, without replacement, and adds the three numbers to give a total. Find the probability that her total is 5. … [4]

10 marks

Mark scheme: 7(a)(i) 1 3 1 1 oe M2 for 2   oe 15 6 5 1 1 or M1 for  oe 6 5 or list or indication of 2 correct pairs 1 If 0 scored, SC1 for answer oe 18 7(a)(ii) 7 3  4 3   1 1   1 1  oe M2 for    + 2    oe or 14    oe 15  6 5   6 5   6 5   2 4  or 1 – 2     6 5   4 3   1 1   2 4  or M1 for    or 2    oe or 2     6 5   6 5   6 5  or correct identification of 14 pairs 5 If 0 scored, SC1 for answer 9 7(b) 1 4  1 1 1   1 1 1  oe nfww M3 for 6     + 6     oe 10  6 5 4   6 5 4   1 1 1   1 1 1  or M2 for 6     oe or 2     oe  6 5 4   6 5 4   1 1 1  or M1 for k     where k is an integer and 1 ⩽ k ⩽ 12 but  6 5 4  not k = 2 or k = 6 or identifies –2, 2 and 5 or –3, 3 and 5 as the 3 cards needed 1 If 0 scored, SC1 for answer 18

This question in 0580/43 Oct/Nov 2022

7 = {students in a class} P = {students who study Physics} C = {students who study Chemistry} n() = 24 n ( P) = 17 n ( C ) = 14 n ( P k C ) = 9 (a) Complete the Venn diagram. P C 9 … … … [2] (b) (i) Find n ( P k C l) . … [1] (ii) Find n ( P j C l) . … [1] (c) Two students are picked from the class at random. Find the probability that one student studies both subjects and one student studies Chemistry but not Physics. … [3] (d) Two of the students who study Physics are picked at random. Find the probability that they both study Chemistry. … [2]

9 marks

Mark scheme: 7(a) Completed Venn diagram. 2 B1 for two correct values Ɛ C P 8 [9] 5 2 7(b)(i) 8 1 FT their (a) their 8 dep < 24 7(b)(ii) 19 1 FT their (a) 24 – their 5 dep on positive answer 7(c) 15 3 oe 92 9 their 5 M2 for [2]24  23 oe 9 their 5 their 5 9 or M1 for and or and 24 23 24 23 5 If 0 scored SC1 for answer oe 32 7(d) 9 2 9 oe B1 for seen 34 17

This question in 0580/42 Feb/March 2023

4 (a) Lucia has two fair spinners. Spinner A is five-sided and is numbered 1, 2, 3, 4, 5. Spinner B is nine-sided and is numbered 3, 3, 3, 4, 4, 4, 4, 5, 5. Lucia spins the two spinners and records whether they land on a prime number. (i) Complete the tree diagram. Spinner A Spinner B prime … prime 3 5 not … prime prime … not … prime not … prime [2] (ii) Find the probability that (a) the two numbers are both prime … [2] (b) the two numbers are not both prime. … [1] (b) Lucia spins Spinner A 120 times. Find the expected number of times the spinner lands on a prime number. … [1] (c) Lucia spins Spinner B twice. Find the probability that the two numbers it lands on add up to 9 or more. … [3] (d) Lucia keeps spinning Spinner B until it lands on a 4. Find an expression, in terms of n, for the probability that this happens on the nth spin. … [2]

11 marks

Mark scheme: 4(a)(i) 2 5 4 5 4 2 2 , , B1 for and a pair of probabilities for spinner B 5 9 9 9 9 5 that sum to 1 4a(ii)(a) 1 2 FT dep their tree diagram oe 3 5 3 M1 for  their 5 9 4a(ii)(b) 2 1 1 oe FT dep 1 – their 3 3 4(b) 72 1 4(c) 20 3 2 4 2 2 oe M2 for  [ 2] +  oe 81 9 9 9 9 2 4 2 2 or M1 for  or  oe 9 9 9 9 4(d) n−1 2 n−1  5  4  5  [] oe final answer M1 for seen      9  9  9 

This question in 0580/41 Oct/Nov 2023

5 Indira records the time taken for workers in her company to travel to work. The table and the histogram each show part of this information. Time (t minutes) 0 1 t G 10 10 1 t G 25 25 1 t G 40 40 1 t G 60 60 1 t G 80 Frequency 57 38 12 4 3 Frequency 2 density 1 0 t 0 10 20 30 40 50 60 70 80 Time (minutes) (a) Complete the table and the histogram. [5] (b) Calculate an estimate of the mean time. … min [4] (c) Rashid says: ‘The longest time that any of these workers take to travel to work is 80 minutes.’ Give a reason why Rashid may be wrong. … … [1] (d) Indira picks three workers at random from those who take longer than 25 minutes to travel to work. Calculate the probability that one worker takes 60 minutes or less and the other two each take more than 60 minutes. … [4]

14 marks

Mark scheme: 5(a) 28 and 45 on table B2 B1 for each Histogram correctly completed B3 B1 for each correct bar If 0 scored, SC1 for two of FD’s 3.8, 1.9 or 0.6 oe soi 5(b) 30.7 or 30.66 to 30.67 4 M1 for midpoints soi M1 for use of ∑fh with h in correct interval including both boundaries M1 (dep on 2nd M1) for ∑fh ÷ (their 28 + their 45 + 57 + 38 + 12) 5(c) Exact values are not known oe 1 5(d) 1254 4 M3 for oe 39 697  38 + 57 12 11  N     oe  57 + 38 + 12 56 + 38 + 12 56 + 38 + 11  where N = 1, 2 or 3 38 + 57 12 or M2 for and 57 + 38 + 12 56 + 38 + 12 12 11 or and oe seen 57 + 38 + 12 57 + 38 + 11 38 + 57 12 or M1 for or oe seen 57 + 38 + 12 57 + 38 + 12 41040 If 0 scored SC1 for answer or 0.0335… 1225043

This question in 0580/43 Oct/Nov 2023

9 N A M I B I A The diagram shows 7 cards. (a) Amir picks a card at random. Find the probability that the card shows (i) the letter H … [1] (ii) the letter B. … [1] (b) Fumika picks one of the 7 cards at random. She replaces it and picks a second card at random. Find the probability that both cards show the letter I. … [2] (c) Marcos picks two of the 7 cards at random, without replacement. (i) Find the probability that one card shows the letter I and the other card shows the letter N. … [3] (ii) Find the probability that the two cards show different letters. … [3] (d) Nina picks one of the 7 cards at random without replacement. She continues picking cards at random without replacement until she picks a card that shows the letter A. 4 The probability that this occurs when she picks the nth card is . 21 Find the value of n. n = … [2]

12 marks

Mark scheme: 9(a)(i) 0 1 9(a)(ii) 1 1 oe 7 9(b) 4 2 2 2 oe M1 for  49 7 7 9(c)(i) 2 3 2 1 1 2 oe M2 for  +  oe 21 7 6 7 6 2 1 1 2 or M1 for  or  oe seen 7 6 7 6 4 If 0 scored SC1 for 49 9(c)(ii) 19 3  2 1   2 1  oe M2 for 1         oe 21  7 6   7 6   2 1   2 1  or M1 for        oe  7 6   7 6  ALTERNATIVE 1 2 5 M2 for [ 1]  3 +   2 7 7 6 2 5 1 or M1 for  or [ 1]  3 7 6 7 38 If 0 scored SC1 for 49 9(d) 3 2 5 4 2or3 M1 for   7 6 5

This question in 0580/43 May/June 2024

5 A box contains 3 blue pens and 5 red pens. (a) Mia picks a pen from the box at random. Find the probability that she picks a red pen. … [1] (b) Mia puts the pen back into the box. She then picks a pen at random and replaces it. She then picks a second pen at random. (i) Complete the tree diagram. First Pen Second Pen Blue … Blue … … Red Blue … … Red … Red [2] (ii) Find the probability that Mia picks two pens that have the same colour. … [3] (c) Mia now picks 3 of the 8 pens in the box at random without replacement. Find the probability that she picks 2 blue pens and 1 red pen. … [3]

9 marks

Mark scheme: 5(a) 5 1 oe 8 5(b)(i) Tree diagram correct probabilities on 3 pairs 2 B1FT for one pair of branches of first of branches stage or second stage correct 3 8 5 8 5(b)(ii) 17 3 oe 32  3 3   5 5  M2FT for their    +    oe  8 8   8 8  or M1FT for one correct product seen 5(c) 15 3 oe 56 3 2 5 M2 FT for    k where k is 1, 2 or 8 7 6 3 3 2 5 or M1FT for and and seen oe 8 7 6 or for showing the 3 possible combinations 135 If 0 scored, SC1 for answer oe 512

This question in 0580/41 Oct/Nov 2024

8 (a) A bag contains 24 coloured beads. Some are red, some are blue and 10 are yellow. One bead is picked at random from the bag. Find the probability that (i) the bead is yellow … [1] (ii) the bead is not yellow. … [1] (b) Another bag contains 5 green marbles, 6 white marbles and 4 black marbles. Meera picks 2 marbles at random from the bag, without replacement. Find the probability that (i) the first marble is black and the second marble is white … [2] (ii) both marbles have different colours. … [4]

8 marks

Mark scheme: 8(a)(i) 5 1 oe 12 8(a)(ii) 7 1 FT 1 – their (a)(i) oe 12 8(b)(i) 4 2 4 6 oe M1 for  35 15 14 8(b)(ii) 74 4  5 4 6 5 4 3  oe M3 for 1 –   +  +   oe 105  15 14 15 14 15 14  5 4 6 5 4 3 or M2 for  +  +  oe 15 14 15 14 15 14 k k − 1 or M1 for  where k is 4, 5 or 6 oe 15 14 148 If 0 scored, SC1 for 225 ALTERNATIVE 1 5 10 6 9 4 11 M3 for  +  +  oe 15 14 15 14 15 14 or M2 for two of these products added oe k 15 − k or M1 for  where k is 4, 5 or 6 oe 15 14 148 If 0 scored, SC1 for 225 ALTERNATIVE 2 5 6 5 4 6 4 M3 for   2 +   2 +   2 oe 15 14 15 14 15 14 or M2 for at least two of these different products added oe or M1 for one correct product 148 If 0 scored, SC1 for 225

This question in 0580/42 Oct/Nov 2024

9 A bag contains 5 white balls and 3 black balls. (a) (i) Marwan picks a ball from the bag at random and then replaces it. Find the probability that the ball is white. … [1] (ii) Naomi picks a ball from the bag at random and then replaces it. She repeats this 120 times. Find the number of times the ball is expected to be white. … [1] (b) Oscar picks a ball from the bag at random. He replaces it and then picks a second ball from the bag at random. (i) Find the probability that the balls are the same colour. … [3] (ii) Find the probability that the balls are not the same colour. … [1] (c) Priya picks 3 of the 8 balls from the bag at random without replacement. Find the probability that she picks two white balls and one black ball. … [3]

9 marks

Mark scheme: 9(a)(i) 5 1 oe 8 9(a)(ii) 75 1 FT their (a)(i) 9(b)(i) 17 3 5 5 3 3 oe M2 for  +  32 8 8 8 8 5 5 3 3 or M1 for  or  8 8 8 8 9(b)(ii) 15 1 FT their (b)(i) oe 32 9(c) 15 3 5 4 3 oe M2 for   k , k = 1 or 2 or 3 oe 28 8 7 6 5 4 3 M1 for and and or showing the 8 7 6 three possible combinations oe 225 If 0 scored SC1 for oe 512

This question in 0580/43 Oct/Nov 2024

16 On any day, the probability that the weather will be sunny is 0.7 . (a) Find the probability that on any day the weather will not be sunny. … [1] (b) When the weather is sunny, the probability that Rohit goes for a walk is 0.9 . When the weather is not sunny, the probability that Rohit goes for a walk is 0.2 . Find the probability that on any day Rohit goes for a walk. … [3]

4 marks

Mark scheme: 16(a) 0.3 oe 1 16(b) 0.69 oe 3 M2 for 0.7 × 0.9 + (their 0.3) × 0.2 oe or M1 for 0.7 × 0.9 oe or (their 0.3) × 0.2 oe

This question in 0580/42 May/June 2025

16 The histogram shows information about the masses of some coconuts. The masses are classified into four categories A, B, C and D. 200 150 Frequency density 100 C 50 B D A 0 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 Mass (kg) (a) Show that there are 10 coconuts in category D. [1] (b) Two of the coconuts from those in category C and category D are chosen at random. Find the probability that both are from category D. … [3] (c) Calculate an estimate of the mean mass of the coconuts. … kg [4]

8 marks

Mark scheme: 16(a) (1.6 – 1.35) oe × 40 [= 10] 1 16(b) 5 3 10 9 oe M2 for  oe 39 10 + 170  0.1 9 + 170  0.1 10 9 9 10 or M1 for or or or oe 10 + 17 10 + 16 10 + 17 10 + 16 seen 10 9 or  oe with k > 10 and an integer k k − 1 m m − 1 or  oe with 0 < m < 27 and an integer 10 + 17 10 + 16 16(c) 1.26 or 1.259… nfww 4 M1 for frequencies 6, 15, 17 soi M1 for midpoints soi (1, 1.175, 1.3, 1.475) M1 for use of their fm  their f with m in correct interval including both boundaries

This question in 0580/42 Oct/Nov 2025