Introduction to probability — 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

9 (a) A bag contains red beads and green beads. There are 80 beads altogether. The probability that a bead chosen at random is green is 0.35 . (i) Find the number of red beads in the bag. … [2] (ii) Marcos chooses a bead at random and replaces it in the bag. He does this 240 times. Find the number of times he would expect to choose a green bead. … [1] (b) A different bag contains 2 blue marbles, 3 yellow marbles and 4 white marbles. Huma chooses a marble at random, notes the colour, then replaces it in the bag. She does this three times. Find the probability that (i) all three marbles are yellow, … [2] (ii) all three marbles are different colours. … [3] (c) Another bag contains 2 green counters and 3 pink counters. Teresa chooses three counters at random without replacement. Find the probability that she chooses more pink counters than green counters. … [4]

12 marks

Mark scheme: 9(a)(i) 52 2 M1 for (1 – 0.35) × 80 oe 9(a)(ii) 84 1 9(b)(i) 27 2 3 3 3 oe M1 for × × 729 9 9 9 9(b)(ii) 144 3 2 3 4 oe M2 for × × × 6 oe 729 9 9 9 2 3 4 or M1 for × × oe isw 9 9 9 9(c) 42 4 3 2 1 3 2 2 oe M3 for × × + × × × 3 oe 60 5 4 3 5 4 3 3 2 2 or M2 for × × × 3 oe 5 4 3 3 2 1  3 2 2  or for × × +  × ×  [× 2] 5 4 3  5 4 3  3 2 1 3 2 2 or M1 for × × or × × oe isw 5 4 3 5 4 3 or for PPG, PGP, GPP and PPP selected soi

This question in 0580/41 Oct/Nov 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

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

4 (a) The test scores of 14 students are shown below. 21 21 23 26 25 21 22 20 21 23 23 27 24 21 (i) Find the range, mode, median and mean of the test scores. Range = … Mode = … Median = … Mean = … [6] (ii) A student is chosen at random. Find the probability that this student has a test score of more than 24. … [1] (b) Petra records the score in each test she takes. The mean of the first n scores is x. The mean of the first (n – 1) scores is (x + 1). Find the nth score in terms of n and x. Give your answer in its simplest form. … [3] (c) During one year the midday temperatures, t°C, in Zedford were recorded. The table shows the results. Temperature (t°C) 0 1 t G 10 10 1 t G 15 15 1 t G 20 20 1 t G 25 25 1 t G 35 Number of days 50 85 100 120 10 (i) Calculate an estimate of the mean. … °C [4] (ii) Complete the histogram to show the information in the table. 25 20 15 Frequency density 10 5 0 0 5 10 15 20 25 30 35 t Temperature (°C) [4]

18 marks

Mark scheme: 4(a)(i) range = 7 1 mode = 21 1 median = 22.5 2 M1 for evidence of middle value mean = 22.7 or 22.71… 2 M1 for use of Σ x ÷ 14 4(a)(ii) 3 1 oe 14 4(b) x − n + 1 final answer 3 M2 for nx − ( n − 1)( x + 1) or M1 for ( n − 1)( x + 1) 4(c)(i) 16.6 or 16.60 to 16.61 nfww 4 M1 for 5, 12.5, 17.5, 22.5, 30 soi M1 for Σfx where x is in correct interval, including boundaries M1 dep on second M1 for Σfx 50 + 85 + 100 + 120 + 10 4(c)(ii) Correct histogram 4 B1 for each correct block If 0 scored, SC1 for 5, 20, 24, 1 seen

This question in 0580/41 May/June 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 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

5 The cumulative frequency diagram shows information about the distance, d km, travelled by each of 60 male cyclists in one weekend. 60 50 40 Cumulative 30 frequency 20 10 0 d 0 20 40 60 80 100 120 Distance (km) (a) Use the cumulative frequency diagram to find an estimate of (i) the median, … km [1] (ii) the lower quartile, … km [1] (iii) the interquartile range. … km [1] (b) For the same weekend, the interquartile range for the distances travelled by a group of female cyclists is 40 km. Make one comment comparing the distribution of the distances travelled by the males with the distribution of the distances travelled by the females. … … [1] (c) A male cyclist is chosen at random. Find the probability that he travelled more than 50 km. … [2] (d) (i) Use the cumulative frequency diagram to complete this frequency table. Distance (d km) Number of male cyclists 0 1 d G 40 18 40 1 d G 50 9 50 1 d G 60 60 1 d G 70 70 1 d G 90 90 1 d G 120 2 [2] (ii) Calculate an estimate of the mean distance travelled. … km [4]

12 marks

Mark scheme: 5(a)(i) 52 1 5(a)(ii) 36 1 5(a)(iii) 26 1 FT 62 – their (a)(ii) evaluated correctly 5(b) Valid comment 1 Strict FT their (a)(iii), e.g. distances for females are more varied 5(c) 11 2 27 oe M1 for 27 written or answer of oe 20 60 5(d)(i) [18 9] 14 12 5 [2] 2 B1 for 1 correct value 5(d)(ii) 48.75 nfww 4 M1 for midpoints soi M1 for use of ∑fx with their frequencies M1 (dep on 2nd M1) for ∑fx ÷ (60 or by their ∑f )

This question in 0580/43 Oct/Nov 2019

6 Suleika has six cards numbered 1 to 6. 1 2 3 4 5 6 (a) She takes one card at random, records the number and replaces the card. (i) Write down the probability that the number is 5 or 6. … [1] (ii) Suleika does this 300 times. Find how many times she expects the number 5 or 6. … [1] (b) Suleika takes two cards at random, without replacement. (i) Find the probability that the sum of the numbers on the two cards is 5. … [3] (ii) Find the probability that at least one of the numbers on the cards is a square number. … [3]

8 marks

Mark scheme: 6(a)(i) 1 1 oe 3 6(a)(ii) 100 1 FT their (a)(i) × 300 to at least 3 sf or rounded to the nearest integer 6(b)(i) 2 3 1 1 oe M2 for 4 × × oe 15 6 5  1 1  or M1 for k  ×  oe  6 5  or list or indication of 4 correct pairs 6(b)(ii) 3 3 4 3 oe M2 for 1 – × 5 6 5  2 4  2 1 or 2  ×  + × oe  6 5  6 5 2  4 2  or +  ×  oe 6  6 5  4 3 2 4 or M1 for × oe seen or × [× 2 ] oe 6 5 6 5 seen 2 1 or × oe seen 6 5 or correct identification of 18 pairs or space diagram oe

This question in 0580/42 Feb/March 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 (a) A shop gives each of 1000 people a voucher. 28 people use their voucher. The shop now gives each of 16 500 people a voucher. Calculate how many of these 16 500 people are expected to use their voucher. … [1] (b) In a class activity, all the 15 students wear hats. 7 students wear red hats, 6 students wear green hats and 2 students wear white hats. (i) One of these students is picked at random. Find the probability that this student wears a red hat. … [1] (ii) Two of the 15 students are picked at random. 37 Show that the probability that these two students wear hats of the same colour is . 105 [3] (iii) Three of the 15 students are picked at random. Find the probability that at least two of these three students wear red hats. … [4]

9 marks

Mark scheme: 4(a) 462 1 4(b)(i) 7 1 oe 15 4(b)(ii) 7 6 6 5 2 1 3 M2 for addition of two of × + × + × 7 6 6 5 2 1 15 14 15 14 15 14 × + × + × 37 15 14 15 14 15 14 = or M1 for one of the products seen 105 4(b)(iii) 29 4 M3 for oe 7 6 5 7 6 6 7 6 2 65 × × + 3 × × × + 3 × × × oe 15 14 13 15 14 13 15 14 13  8 7 7   8 7 6  or 1 − 3  × ×  −  × ×  oe  15 14 13   15 14 13  or M2 for the sum of at least two of 7 6 5 7 6 6 7 6 2 × × , N × × × , N × × × 15 14 13 15 14 13 15 14 13 seen 7 6 13 or for × × 15 14 13 7 6 7 6 k or × + N × × × seen 15 14 15 14 13 or M1 for 7 6 5 7 6 6 7 6 2 × × or N × × × or N × × × 15 14 13 15 14 13 15 14 13 seen 1519 If 0 scored SC1 for oe 3375

This question in 0580/42 Feb/March 2021

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

8 (a) (i) Use set notation to describe the shaded region in the Venn diagram. B A … [1] (ii) Shade the correct region in each Venn diagram. K L Q P M Q , P ' ( K , L ) + M ' [2] (b) V E N N D I A G R A M The diagram shows 11 cards. (i) One of these cards is chosen at random. Write down the probability that the letter on the card is not A. … [1] (ii) A card is chosen at random from these 11 cards and then replaced. A second card is then chosen at random. Find the probability that exactly one card has the letter N. … [3] (c) E M … … … … 50 students are asked if they like English (E) and if they like mathematics (M). 3 say they do not like English and do not like mathematics. 33 say they like English. 42 say they like mathematics. (i) Complete the Venn diagram. [2] (ii) A student is chosen at random. Find the probability that this student likes English and likes mathematics. … [1] (iii) Two students are chosen at random. Find the probability that they both like mathematics. … [2] (iv) Two students who like English are chosen at random. Find the probability that they both also like mathematics. … [2]

14 marks

Mark scheme: 8(a)(i) A B 8(a)(ii) 2 B1 for each 8(b)(i) 9 11 1 8(b)(ii) 36 121 oe 3 M2 for 2 9 2 11 11   oe or M1 for 2 9 11 11  oe If 0 scored SC1 for 36 110 8(c)(i) 3, 5, 28, 14 correctly placed 2 B1 for 28 in the intersection 8(c)(ii) 28 50 oe 1 FT their 28 where their 28 50  8(c)(iii) 123 175 oe 2 M1 for 42 41 50 49  8(c)(iv) 63 88 oe 2 FT their 28 M1 for 28 28 1 33 32   their their

This question in 0580/41 May/June 2022

3 (a) The table shows information about the mass of each of 1000 eggs. Mass (m grams) 40 1 m G 50 50 1 m G 56 56 1 m G 64 64 1 m G 70 Frequency 126 520 154 200 (i) Calculate an estimate of the mean. … g [4] (ii) An egg is picked at random from the 1000 eggs. Find the probability that this egg has a mass greater than 56 g. Give your answer as a fraction in its simplest form. … [2] (b) One year, a farmer makes a profit of $24 730 selling eggs. Write this profit (i) correct to 2 significant figures $ … [1] (ii) in standard form. $ … [1] (c) On a farm, there are 500 hens, correct to the nearest 10. (i) In one year, the mean number of eggs laid per hen was 320 eggs, correct to the nearest 20. Calculate the upper bound for the total number of eggs all the hens lay in that year. … [3] (ii) Another farm has 800 hens, correct to the nearest 20. Calculate the lower bound for the difference between the number of hens on the two farms. … [2]

13 marks

Mark scheme: 3(a)(i) 55.87 4 M1 for midpoints soi M1 for use of where m is in the correct fm interval including boundaries M1 (dep on 2nd M1) for ÷1000 fm 3(a)(ii) 177 2 154  200 cao M1 for oe 500 1000 3(b)(i) 25000 1 3(b)(ii) 2.473  10 4 1 3(c)(i) 166 650 or 165816 nfww 3 M2 for (500 + 5) × ‘320 to 340’ or ‘500 to 510’ × (320 + 10) or M1 for 500  5 or 500  5 or 320 10 or 320 10 Alternative method M2 for 504 × ‘320 to 340’ or ‘500 to 510’ × 329 or M1 for 504 or 329 3(c)(ii) 285 or 286 nfww 2 M1 for 800 10

This question in 0580/41 May/June 2023

8 2 3 2 3 1 2 Dice A Dice B The diagram shows two fair dice. Dice A is numbered 1, 2, 2, 2, 3, 6. Dice B is numbered 2, 3, 3, 4, 4, 4. (a) (i) Dice A is rolled once. Write down the probability that it lands on the number 6. … [1] (ii) Dice A is rolled 150 times. Find the number of times it is expected to land on the number 6. … [1] (b) Dice A and Dice B are each rolled once. (i) Find the probability that the two numbers they land on have a total of 6. … [3] (ii) Find the probability that when the two numbers they land on have a total of 6, both numbers are 3. … [2] (c) Dice B is rolled n times. 32 The probability that on the nth roll it first lands on a number 3 is . 729 Find the value of n. n = … [2]

9 marks

Mark scheme: 8(a)(i) 1 1 oe 6 8(a)(ii) 25 1 FT their (a)(i) dep on 0 < (a) < 1 8(b)(i) 11 3 1 2 3 3 oe M2 for  +  oe or correct 36 6 6 6 6 possibility diagram with 11 outcomes identified 1 2 3 3 or M1 for  or  oe 6 6 6 6 or lists the 11 required outcomes or for possibility diagram but required outcomes not indicated 8(b)(ii) 2 2 2 p 11oe M1 for k or their 11 seen oe leading to answer 8(c) 6 2 k  4 2 32 = written oe M1 for    6  6 729 soi by one trial with k > 1 or 2 n −=1 32 or better or 3n = 729 or better

This question in 0580/42 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

12 A vase contains flowers that are red or pink or white. Ruth picks a flower at random from the vase. The probability that the flower is not red is 0.9 . The probability that the flower is not pink is 0.65 . Find the probability that the flower is white. … [2]

2 marks

Mark scheme: 12 0.55 oe 2 B1 for P(red) = 0.1 or P(pink) = 0.35 or M1 for 0.9 + 0.65 – 1 or 1 – (0.1 + 0.35)

This question in 0580/43 Oct/Nov 2025