Introduction to probability — Paper 4 · IGCSE Mathematics 0580

4 (a) All 24 students in a class are asked whether they like football and whether they like basketball. Some of the results are shown in the Venn diagram below. F B 7 12 2 = {students in the class}. F = {students who like football}. B = {students who like basketball}. (i) How many students like both sports? [1] (ii) How many students do not like either sport? [1] (iii) Write down the value of n(F∪B). [1] (iv) Write down the value of n(F ′∩B). [1] (v) A student from the class is selected at random. What is the probability that this student likes basketball? [1] (vi) A student who likes football is selected at random. What is the probability that this student likes basketball? [1] (b) Two students are selected at random from a group of 10 boys and 12 girls. Find the probability that (i) they are both girls, [2] (ii) one is a boy and one is a girl. [3]

11 marks

Mark scheme: 4 (a) (i) 12 B1 (ii) 3 B1 (iii) 21 B1 (iv) 2 B1 (v) 14 o.e B1 Accept probabilities as fractions/decimals/% 24 (vi) 12 B1 o.e. 19 (b) (i) 12 × 11 M1 22 21 A1 2/7 in simplest form www2 132 o.e. (0.286) 462 (ii) 10 × 12 M1 22 21 M1 their 10 × 12 × 2 o.e. 22 21 A1 40/77 in simplest form www3 240 o.e.(0.519) 462 [11] IGCSE – NOVEMBER 2005 0580/0581 4

This question in 0580/41 Oct/Nov 2005

8 For Examiner's First Second Third Use Calculator Calculator Calculator p F F F p q NF NF F F q NF NF NF F = faulty NF = not faulty The tree diagram shows a testing procedure on calculators, taken from a large batch. 1 Each time a calculator is chosen at random, the probability that it is faulty (F) is . 20 (a) Write down the values of p and q. Answer(a) p = and q = [1] (b) Two calculators are chosen at random. Calculate the probability that (i) both are faulty, Answer(b)(i) [2] (ii) exactly one is faulty. Answer(b)(ii) [2] (c) If exactly one out of two calculators tested is faulty, then a third calculator is chosen at random. For Examiner's Calculate the probability that exactly one of the first two calculators is faulty and the third one Use is faulty. Answer(c) [2] (d) The whole batch of calculators is rejected either if the first two chosen are both faulty or if a third one needs to be chosen and it is faulty. Calculate the probability that the whole batch is rejected. Answer(d) [2] (e) In one month, 1000 batches of calculators are tested in this way. How many batches are expected to be rejected? Answer(e) [1]

10 marks

Mark scheme: 8 Throughout the question ratios score zero. If using decimals, 2 s.f. correct answers to parts (c) and (d) – penalty of 1 once Use of words e.g. 1 in 400 or 1 out of 400, Correct answers – penalty of one For method marks only accept probabilities p and q between 0 and 1 (a) p = 201 , q = 1920 o.e. B1 Could be on diagram (b) (i) 4001 o.e. c.a.o. B2 0.0025 allow M1 for (their p)2 o.e. (ii) 40038 o.e. c.a.o. B2 0.095 allow M1 for 2 (their p)( their q) o.e. (c) 800038 o.e. c.a.o. B2 0.00475 allow M1 for 2(their p)² (their q) o.e. including their (ii) × their p (d) their (b)(i) + their (c) M1 58 o.e. c.a.o. A1 0.00725 8000 (e) their (d) × 1000 = 7.25 o.e. ft B1 ft Accept 7 or 8 or an equivalent integer ft [10] IGCSE – May/June 2009 0580, 0581 04

This question in 0580/41 May/June 2009

3 For Examiner's Use 1 1 6 7 11 12 Six cards are numbered 1, 1, 6, 7, 11 and 12. In this question, give all probabilities as fractions. (a) One of the six cards is chosen at random. 1 (i) Which number has a probability of being chosen of ? 3 Answer(a)(i) [1] (ii) What is the probability of choosing a card with a number which is smaller than at least three of the other numbers? Answer(a)(ii) [1] (b) Two of the six cards are chosen at random, without replacement. Find the probability that (i) they are both numbered 1, Answer(b)(i) [2] (ii) the total of the two numbers is 18, Answer(b)(ii) [3] (iii) the first number is not a 1 and the second number is a 1. For Examiner's Use Answer(b)(iii) [2] (c) Cards are chosen, without replacement, until a card numbered 1 is chosen. Find the probability that this happens before the third card is chosen. Answer(c) [2] (d) A seventh card is added to the six cards shown in the diagram. The mean value of the seven numbers on the cards is 6. Find the number on the seventh card. Answer(d) [2]

13 marks

Mark scheme: 3 (a) (i) 1 B1 Penalty of –1 in question if any answers given as decimals or percentages (to 3sf) alone, but isw cancelling/conversion after correct answer (ii) 3 B1 oe 6 (b) (i) 2 oe www2 B2 M1 for 2 × 1 30 6 5 (ii) 6–12 and 12–6 and 7–11 and 11–7 soi M1 Evidence of all pairs adding up to 18 but no extras e.g. 4/6 × 1/6 k × 1 × 1 for k = integer M1 Without seeing the first M, 4 × 1 oe scores 6 5 6 5 M2, 2 × 1 oe scores M1 4 6 5 oe www3 A1 30 (iii) 4 × 2 M1 6 5 8 oe www2 A1 30 (c) 2 + 4 × 2 oe M1 2 + their (b) (iii) 6 6 5 6 18 oe cao www2 A1 30 (d) 4 B2 M1 for (1 + 1 + 6 + 7 + 11 + 12 + x) ÷ 7 = 6 or better [13] IGCSE – October/November 2009 0580 04

This question in 0580/41 Oct/Nov 2009

4 Examiner's Use A B Box A contains 3 black balls and 1 white ball. Box B contains 3 black balls and 2 white balls. (a) A ball can be chosen at random from either box. Complete the following statement. There is a greater probability of choosing a white ball from Box . Explain your answer. Answer(a) [1] (b) Abdul chooses a box and then chooses a ball from this box at random. The probability that he chooses box A is 23 . (i) Complete the tree diagram by writing the four probabilities in the empty spaces. BOX COLOUR 1 white 4 A 2 3 black white B black [4] For (ii) Find the probability that Abdul chooses box A and a black ball. Examiner's Use Answer(b)(ii) [2] (iii) Find the probability that Abdul chooses a black ball. Answer(b)(iii) [2] (c) Tatiana chooses a box and then chooses two balls from this box at random (without replacement). The probability that she chooses box A is 23 . Find the probability that Tatiana chooses two white balls. Answer(c) [2]

11 marks

Mark scheme: 4 In (b) and (c) isw any cancelling or changing to other forms, after correct answer seen. Penalty of – 1 for 2 sf decimals or percentages. Do not accept ratio or worded forms. 2 1 (a) B and , oe 1 Allow any reasonable explanation, e.g. 2 out of 5 4 5 greater than 1 out of 4. 1 3 2 3 (b) (i) , , , 4 B1 B1 B1 B1 3 4 5 5 6 1 2 3 (ii) oe cao www 2 2 , 0.5 etc M1 for × their i.e. product of 12 2 3 4 correct branches on their tree 42 7 (iii) oe cao www2 2 , 0.7 etc 60 10 1 3 M1 for their (ii) + their × their from their 3 5 tree 2 1 (c) oe cao www2 2 , 0.0333(3……..) etc 60 30  2 1  1 2 1 M1 for  × × 0  + × ×  3 4  3 5 4

This question in 0580/41 May/June 2010

3 Examiner's Use 2 1 1 10 2 1 The diagram shows a spinner with six numbered sections. Some of the sections are shaded. Each time the spinner is spun it stops on one of the six sections. It is equally likely that it stops on any one of the sections. (a) The spinner is spun once. Find the probability that it stops on (i) a shaded section, Answer(a)(i) [1] (ii) a section numbered 1, Answer(a)(ii) [1] (iii) a shaded section numbered 1, Answer(a)(iii) [1] (iv) a shaded section or a section numbered 1. Answer(a)(iv) [1] For (b) The spinner is now spun twice. Examiner's Use Find the probability that the total of the two numbers is (i) 20, Answer(b)(i) [2] (ii) 11. Answer(b)(ii) [2] (c) (i) The spinner stops on a shaded section. Find the probability that this section is numbered 2. Answer(c)(i) [1] (ii) The spinner stops on a section numbered 2. Find the probability that this section is shaded. Answer(c)(ii) [1] (d) The spinner is now spun until it stops on a section numbered 2. 16 The probability that this happens on the nth spin is . 243 Find the value of n. Answer(d) n = [2]

12 marks

Mark scheme: 3 Throughout this question isw any cancelling or changing to other forms, after correct answer seen. Penalty of –1 for 2 sf decimals or percentages. Do not accept ratio or worded forms. 4 (a) (i) oe (0.667) 1 Allow 0.6666 – 0.6667 6 3 (ii) oe 1 6 2 (iii) oe (0.333) 1 Allow 0.3333… 6 5 (iv) oe (0.833) 1 Allow 0.8333… 6 1 1 1 (b) (i) oe (0.0278) 2 Allow 0.02777 – 0.02778, M1 for × 36 6 6 6 3 1 (ii) oe (0.167) www 2 2 Allow 0.1666 – 0.1667, M1 for × × 2 oe 36 6 6 1 (c) (i) oe 1 4 1 (ii) oe 1 2 n 4  2  (d) 5 (but not from rounding) 2 M1 for repeating × oe e.g.   6  3  SC1 f l i −7  k 

This question in 0580/42 May/June 2010

9 (a) Emile lost 2 blue buttons from his shirt. For Examiner's A bag of spare buttons contains 6 white buttons and 2 blue buttons. Use Emile takes 3 buttons out of the bag at random without replacement. Calculate the probability that (i) all 3 buttons are white, Answer(a)(i) [3] (ii) exactly one of the 3 buttons is blue. Answer(a)(ii) [3] (b) There are 25 buttons in another bag. For Examiner's This bag contains x blue buttons. Use Two buttons are taken at random without replacement. 7 The probability that they are both blue is . 100 (i) Show that x2 O x O 42 = 0. Answer (b)(i) [4] (ii) Factorise x2 O x O 42. Answer(b)(ii) [2] (iii) Solve the equation x2 O x O 42 = 0. Answer(b)(iii) x = or x = [1] (iv) Write down the number of buttons in the bag which are not blue. Answer(b)(iv) [1]

14 marks

Mark scheme: 120 5 9 (a) (i) oe 0.357(1…) 3 Accept fraction, %, dec equivalents (3sf or better) 336 14 throughout but not ratio or words isw incorrect cancelling/conversion to other forms Pen –1 once for 2sf answers 6 5 4 M2 for × × 8 7 6 5 or M1 for seen 7 180 15 2 6 5 6 2 5 6 5 2 (ii) oe 0.536 or 0.5357… 3 M2 for × × + × × + × × 336 28 8 7 6 8 7 6 8 7 6 2 × 5 × 6 Accept 3 × 6 × 7 × 8 2 × 5 × 6 60 5 or M1 for oe seen ( oe ) 6 × 7 × 8 336 28 x x − 1 7 x x − 1 (b) (i) × = M2 M1 for or seen 25 24 100 25 24 x 2 −x 7 = 600 100 7 or x(x – 1) = × 25 × 24 M1 Or better, min requirement is x² – x = 7 × 6 100 x² – x – 42 = 0 E1 With no errors or omissions (ii) (x + 6) ( x – 7) B2 SC1 any other (x + a)(x + b) where a × b = –42 or a + b = –1 (iii) –6, 7 B1ft Correct or follow through dep on at least SC1 in (b)(ii) (iv) 18 B1ft Correct or ft 25 – their positive integer solution Dep on pos and neg answer to (b)(iii) Answer must be positive integer

This question in 0580/42 Oct/Nov 2011

9 For Examiner's Set A Use S U M S Set B M I N U S The diagram shows two sets of cards. (a) One card is chosen at random from Set A and replaced. (i) Write down the probability that the card chosen shows the letter M. Answer(a)(i) [1] (ii) If this is carried out 100 times, write down the expected number of times the card chosen shows the letter M. Answer(a)(ii) [1] (b) Two cards are chosen at random, without replacement, from Set A. Find the probability that both cards show the letter S. Answer(b) [2] (c) One card is chosen at random from Set A and one card is chosen at random from Set B. Find the probability that exactly one of the two cards shows the letter U. Answer(c) [3] (d) A card is chosen at random, without replacement, from Set B until the letter shown is either I or U. Find the probability that this does not happen until the 4th card is chosen. Answer(d) [2]

9 marks

Mark scheme: 9 Accept fraction, %, dec equivalents (3sf or better when not exact) throughout but not ratio or words isw incorrect cancelling/conversion to other forms 1 (a) (i) oe 1 4 1 (ii) 25 cao 1ft ft their × 100 to 3sf or better or rounding or 4 truncating to integer Not 25/100 2 2 1 (b) oe cao 2 M1 for × 0.167, 16.7% 12 4 3 7 1 4 3 1 (c) oe cao 3 M2 for × + × 20 4 5 4 5 1 4 3 1 or M1 for × or × 4 5 4 5 After 0, SC1 for 7 correct in list (condone UU in addition) 6 3 2 1  2  (d) oe cao 2 M1 for × × ×   60 5 4 3  2 

This question in 0580/43 Oct/Nov 2011

8 In all parts of this question give your answer as a fraction in its lowest terms. For Examiner's Use 1 . (a) (i) The probability that it will rain today is 3 What is the probability that it will not rain today? Answer(a)(i) [1] 2 (ii) If it rains today, the probability that it will rain tomorrow is . 5 1 If it does not rain today, the probability that it will rain tomorrow is . 6 Complete the tree diagram. Today Tomorrow Rain Rain No rain Rain No rain No rain [2] (b) Find the probability that it will rain on at least one of these two days. Answer(b) [3] (c) Find the probability that it will rain on only one of these two days. Answer(c) [3]

9 marks

Mark scheme: 8 Throughout question, penalise non-reduced fraction only once; isw any conversion and allow decimals in working and on branches but not final answers if fractions not seen. 2 (a) (i) 1 3 1 2 2 3 1 5 1 2 3 5 (ii) , , , , , 2 B1 for and and or correctly placed 3 3 5 5 6 6 3 3 5 6 correctly placed For method marks in (b) and (c), ft tree with each probability 0< p <1 4 2 5 1 2 1 (b) cao www 3 3 M2 for 1 – × or + × 9 3 6 3 3 6 1 2 1 3 2 1 or × + × + × 3 5 3 5 3 6 1 2 5 M1 for + × 3 3 6 1 2 1 3 2 1 or two of × , × , × added 3 5 3 5 3 6 14 1 3 2 1 4 1 2 (c) cao www 3 3 M2 for × + × or their – × 45 3 5 3 6 9 3 5 1 3 2 1 M1 for one of × or × from a 3 5 3 6 maximum of two products added.

This question in 0580/42 May/June 2012

6 In a box there are 7 red cards and 3 blue cards. For Examiner′s A card is drawn at random from the box and is not replaced. Use A second card is then drawn at random from the box. (a) Complete this tree diagram. First card Second card … Red 7 Red 10 Blue … … Red … Blue Blue … [3] (b) Work out the probability that the two cards are of different colours. Give your answer as a fraction. Answer(b) … [3] _____________________________________________________________________________________

6 marks

Mark scheme: 6 (a) 3 correctly placed 10 1 Accept 0.3 6 3 and correctly placed 1 Accept 0.667 or better and 0.333 or better 9 9 7 2 and correctly placed 9 9 1 Accept 0.778 or better and 0.222 or better 42 21 14 7 7 3 3 7 (b) or or or 3 M2 for × + × soi by 0.467 or 90 45 30 15 10 9 10 9 better 7 3 3 7 or M1 for × or × soi by 0.233 or 10 9 10 9 better

This question in 0580/43 May/June 2013

4 T B 11 9 x 6 – x 4 P In the Venn diagram, = {children in a nursery} B = {children who received a book for their birthday} T = {children who received a toy for their birthday} P = {children who received a puzzle for their birthday} x children received a book and a toy and a puzzle. 6 children received a toy and a puzzle. (a) 4 children received a book and a toy. 5 children received a book and a puzzle. 7 children received a puzzle but not a book and not a toy. Complete the Venn diagram above. [3] (b) There are 40 children in the nursery. Using the Venn diagram, write down and solve an equation in x. Answer(b) [3] (c) Work out (i) the probability that a child, chosen at random, received a book but not a toy and not a puzzle, Answer(c)(i) … [1] (ii) the number of children who received a book and a puzzle but not a toy, Answer(c)(ii) … [1] (iii) n(B), Answer(c)(iii) … [1] (iv) n(B ∪ P), Answer(c)(iv) … [1] (v) n(B ∪ T ∪ P)'. Answer(c)(v) … [1] (d) T B P Shade the region B ∩ (T ∪ P)'. [1] __________________________________________________________________________________________

12 marks

Mark scheme: 4 (a) 4 – x correctly placed 1 SC3 for 1, 2 and 7 all correctly placed 5 – x correctly placed 1 instead of expressions in x 7 correctly placed 1 (b) 4 + 11 + (6 – x) + x + 9 + (4 – x) + M1 FT from their Venn diagram, condone (5 – x) + 7 = 40 oe omission of one subset 46 – 2x = 40 nfww A1 Must be in the form a + bx = c, ie each side simplified, or better x = 3 B1 (c) (i) 9 1 ISW cancelling or conversion after correct or 0.225 or 22.5% 40 answer seen (ii) 2 1FT FT from their Venn diagram and their x provided n(B ∩ P ∩ T’) ≠ 5 (iii) 15 1FT FT from their Venn diagram (iv) 25 1FT FT from their Venn diagram (v) 4 1 (d) Correct region shaded. 1 T B P IGCSE – May/June 2014 0580 41 Qu Answers Mark Part Marks

This question in 0580/41 May/June 2014

6 (a) A square spinner is biased. The probabilities of obtaining the scores 1, 2, 3 and 4 when it is spun are given in the table. Score 1 2 3 4 Probability 0.1 0.2 0.4 0.3 (i) Work out the probability that on one spin the score is 2 or 3. Answer(a)(i) … [2] (ii) In 5000 spins, how many times would you expect to score 4 with this spinner? Answer(a)(ii) … [1] (iii) Work out the probability of scoring 1 on the fi rst spin and 4 on the second spin. Answer(a)(iii) … [2] (b) In a bag there are 7 red discs and 5 blue discs. From the bag a disc is chosen at random and not replaced. A second disc is then chosen at random. Work out the probability that at least one of the discs is red. Give your answer as a fraction. Answer(b) … [3] __________________________________________________________________________________________

8 marks

Mark scheme: 6 (a) (i) 0.6 oe 2 M1 for 0.2 + 0.4 (ii) 1500 1 (iii) 0.03 oe 2 M1 for 0.1 × 0.3 (b) 112 28 3 5 4 oe = 0.848[4…] M2 for 1 – 132 33 12 ×11 7 5 5 7 7 6 or × + × + × 12 11 12 11 12 11 7 5 7 or + × 12 12 11 or M1 for addition of any two of 7 5 5 7 7 6 × , × , × 12 11 12 11 12 11 or sum of 3 products with an error in the numerator of one product 5 4 or for × identified 12 11 IGCSE – May/June 2014 0580 41 Qu Answers Mark Part Marks 5  k 

This question in 0580/41 May/June 2014

1 9 If the weather is fi ne the probability that Carlos is late arriving at school is . 10 1 If the weather is not fi ne the probability that he is late arriving at school is . 3 3 The probability that the weather is fi ne on any day is . 4 (a) Complete the tree diagram to show this information. Weather Arriving at school 1 10 Late 3 Fine 4 Not late … … Late … Not fine Not late … [3] (b) In a school term of 60 days, fi nd the number of days the weather is expected to be fi ne. Answer(b) … [1] (c) Find the probability that the weather is fi ne and Carlos is late arriving at school. Answer(c) … [2] (d) Find the probability that Carlos is not late arriving at school. Answer(d) … [3] (e) Find the probability that the weather is not fi ne on at least one day in a school week of 5 days. Answer(e) … [2] __________________________________________________________________________________________

11 marks

Mark scheme: 1 9 1 2 1 9 1 2 9 (a) , , , 3 B1 for B1 for B1 for and 4 10 3 3 4 10 3 3 (b) 45 1 3 3 1 (c) oe 2 M1 for × oe 40 4 10 101 3 9 1 2 (d) oe 3 M2 for × + × only 120 4 10 4 3 1 1 or 1 −their (c) − × only 4 3 3 9 1 2 or M1 for × or × 4 10 4 3 1 1 or their (c) + × 4 3 5 781  3  (e) oe 2 M1 for 1 −   oe 1024  4  IGCSE – May/June 2014 0580 42 Qu Answers Mark Part Marks  1  1 1

This question in 0580/42 May/June 2014

6 In this question, give all your answers as fractions. N A T I O N The letters of the word NATION are printed on 6 cards. (a) A card is chosen at random. Write down the probability that (i) it has the letter T printed on it, Answer(a)(i) … [1] (ii) it does not have the letter N printed on it, Answer(a)(ii) … [1] (iii) the letter printed on it has no lines of symmetry. Answer(a)(iii) … [1] (b) Lara chooses a card at random, replaces it, then chooses a card again. Calculate the probability that only one of the cards she chooses has the letter N printed on it. Answer(b) … [3] (c) Jacob chooses a card at random and does not replace it. He continues until he chooses a card with the letter N printed on it. Find the probability that this happens when he chooses the 4th card. Answer(c) … [3] __________________________________________________________________________________________

9 marks

Mark scheme: 16 (a) (i) 1 6 4 (ii) oe 1 6 2 (iii) oe 1 6 16 2 4 4 2 (b) oe 3 M2 × + × only oe 36 6 6 6 6 2 4 4 2 2 or M1 for one of × or × soi by 6 6 6 6 9 48 4 3 2 2 (c) oe 3 M2 for × × × only oe 360 6 5 4 3 or M1 for denominators 6, 5, 4, 3 soi in product of four fractions

This question in 0580/43 May/June 2014

10 Kenwyn plays a board game. Two cubes (dice) each have faces numbered 1, 2, 3, 4, 5 and 6. In the game, a throw is rolling the two fair 6-sided dice and then adding the numbers on their top faces. This total is the number of spaces to move on the board. For example, if the numbers are 4 and 3, he moves 7 spaces. (a) Giving each of your answers as a fraction in its simplest form, fi nd the probability that he moves (i) two spaces with his next throw, Answer(a)(i) … [2] (ii) ten spaces with his next throw. Answer(a)(ii) … [3] (b) What is the most likely number of spaces that Kenwyn will move with his next throw? Explain your answer. Answer(b) … because … … [2] (c) 95 96 97 98 99 100 Go back WIN 3 spaces To win the game he must move exactly to the 100th space. Kenwyn is on the 97th space. If his next throw takes him to 99, he has to move back to 96. If his next throw takes him over 100, he stays on 97. Find the probability that he reaches 100 in either of his next two throws. Answer(c) … [5] __________________________________________________________________________________________

12 marks

Mark scheme: 110 (a) (i) final answer 2 M1 for 1 × 1 6 6 36 1  1 1  (ii) final answer 3 M2 for 3 ×  oe 12  6 6  or M1 for identifying 3 correct pairs (4, 6), (6, 4) and (5, 5) (b) 7 1 Refers to most combinations oe 1 Dependent on previous mark 141  47  2  3  2   1 3  (c) oe 5 M4 for + −1 × +  ×  oe   1296  432  36   36  36   36 36  or M3 for 2 correct probabilities shown added from those above 3  2 or M1 for − 1  × seen oe  36  36 1 3 And M1 for × seen oe 36 36 1 1 1 1 or × × × oe alone or added to a 6 6 6 6 n probability not of the form 36

This question in 0580/42 Oct/Nov 2014

4 Yeung and Ariven compete in a triathlon race. 3 The probability that Yeung fi nishes this race is . 5 2 The probability that Ariven fi nishes this race is . 3 (a) (i) Which of them is more likely to fi nish this race? Give a reason for your answer. Answer(a)(i) … because … … [1] (ii) Find the probability that they both fi nish this race. Answer(a)(ii) … [2] (iii) Find the probability that only one of them fi nishes this race. Answer(a)(iii) … [3] (b) After the fi rst race, Yeung competes in two further triathlon races. (i) Complete the tree diagram. First race Second race Third race 7 10 Finishes 6 Finishes 7 Does not … finish Finishes 7 3 10 Finishes 5 … Does not finish Does not … finish 7 10 Finishes 6 Finishes 7 … Does not … finish Does not finish 7 10 Finishes … Does not finish Does not … finish [3] (ii) Calculate the probability that Yeung fi nishes all three of his races. Answer(b)(ii) … [2] (iii) Calculate the probability that Yeung fi nishes at least one of his races. Answer(b)(iii) … [3] __________________________________________________________________________________________

14 marks

Mark scheme: 4 (a) (i) Ariven with comparable form for 1 Accept probabilities changed to decimals or both shown or difference between percentages (to 2sf or better) the two fractions shown 6 3 2 (ii) oe 2 M1 for × 15 5 3 7 3 1 2 2 2 1 (iii) oe 3 M2 for × + × oe 1 − their (a)(ii) − × 15 5 3 5 3 5 3 or 3 1 2 2 M1 for × or × seen 5 3 5 3 (b) (i) Completes tree diagram correctly 3 B2 for 5 values correct or B1 for 1 value correct 126  9  3 6 7 (ii) oe 2 M1 for × × 350  25  5 7 10 344 2 1 3 (iii) oe 3 M2 for 1 − their × their × their oe 5 7 10 350 3 2 6 2 1 7 or + × + × × 5 5 7 5 7 10 2 1 3 M1 for their × their × their oe 5 7 10 or identifies the 7 routes or attempt to add 7 probabilities with at least 5 correct 9 27 3 9 6 18 1 + + + + + + oe 25 175 50 350 25 175 25

This question in 0580/43 Oct/Nov 2014

6 In this question write any probability as a fraction. Navpreet has 15 cards with a shape drawn on each card. 5 cards have a square, 6 cards have a triangle and 4 cards have a circle drawn on them. (a) Navpreet selects a card at random. Write down the probability that the card has a circle drawn on it. Answer(a) … [1] (b) Navpreet selects a card at random and replaces it. She does this 300 times. Calculate the number of times she expects to select a card with a circle drawn on it. Answer(b) … [1] (c) Navpreet selects a card at random, replaces it and then selects another card. Calculate the probability that (i) one card has a square drawn on it and the other has a circle drawn on it, Answer(c)(i) … [3] (ii) neither card has a circle drawn on it. Answer(c)(ii) … [3] (d) Navpreet selects two cards at random, without replacement. Calculate the probability that (i) only one card has a triangle drawn on it, Answer(d)(i) … [3] (ii) the two cards have different shapes drawn on them. Answer(d)(ii) … [4] __________________________________________________________________________________________

15 marks

Mark scheme: 46 (a) 1 15 (b) 80 1FT FT 300 × their (a) 40  8  5 4 (c) (i) oe 3 M2 for × × 2 oe 225  45  15 15 5 4 or M1 for × 15 15 121 11 (ii) 3 M2 for 11× oe 225 15 15 11 4 or M1 for or 1− seen 15 15 108  18  6 9 9 6 (d) (i) oe 3 M2 for × + × oe 210  35  15 14 15 14 6 9 9 6 or M1 for × oe or × oe 15 14 15 14 6 5 6 4 or × oe or × oe 15 14 15 14 148  74  5 10 6 9 4 11 (ii) oe 4 M3 for × + × + × oe 210  105  15 14 15 14 15 4 5 4 6 5 4 3 or 1 − × − × − × 15 14 15 14 15 14 or M2 for equivalent of 2 of above products added together oe or M1 for one correct relevant product oe

This question in 0580/42 Feb/March 2015

4 30 students were asked if they had a bicycle (B), a mobile phone (M) and a computer (C). The results are shown in the Venn diagram. B M 2 4 x 7 1 6 3 2 C (a) Work out the value of x. Answer(a) x = … [1] (b) Use set notation to describe the shaded region in the Venn diagram. Answer(b) … [1] (c) Find n(C (M B)). Answer(c) … [1] (d) A student is chosen at random. (i) Write down the probability that the student is a member of the set M  . Answer(d)(i) … [1] (ii) Write down the probability that the student has a bicycle. Answer(d)(ii) … [1] (e) Two students are chosen at random from the students who have computers. Find the probability that each of these students has a mobile phone but no bicycle. Answer(e) … [3]

8 marks

Mark scheme: 4 (a) 5 1 (b) C ∩ M oe 1 Allow e.g. (B ∩ C ∩ M) ∪ (C ∩ M) (c) 3 1 8 (d) (i) oe 1 0.267 or better 30 14 (ii) oe 1 0.467 or better 30 30 6 5 (e) oe 3 M2 for × 272 17 16 6 or M1 for seen 17 0.110[2…] or better 55

This question in 0580/41 May/June 2015

5 A A A A B B C (a) One of these 7 cards is chosen at random. Write down the probability that the card (i) shows the letter A, Answer(a)(i) … [1] (ii) shows the letter A or B, Answer(a)(ii) … [1] (iii) does not show the letter B. Answer(a)(iii) … [1] (b) Two of the cards are chosen at random, without replacement. Find the probability that (i) both show the letter A, Answer(b)(i) … [2] (ii) the two letters are different. Answer(b)(ii) … [3] (c) Three of the cards are chosen at random, without replacement. Find the probability that the cards do not show the letter C. Answer(c) … [2] __________________________________________________________________________________________

10 marks

Mark scheme: 4 5 (a) (i) oe 1 7 6 (ii) oe 1 7 5 (iii) oe 1 7 12 4 3 (b) (i) oe nfww 2 M1 for × 42 7 6 28 4 3 2 5 1 (ii) oe nfww 3 M2 for × + × + or 42 7 6 7 6 7 4 3 2 1 1 − × − × oe 7 6 7 6 or M1 for the sum of two terms of 4 3 2 5 1 × , × , 7 6 7 6 7 120 6 5 4 (c) oe nfww 2 M1 for × × 210 7 6 5  4 3 2   4 3 2   4 2 1  or  × ×  + 3 × ×  + 3 × ×   7 6 5   7 6 5   7 6 5  oe Qu Answers Mark Part Marks

This question in 0580/43 May/June 2015

6 The table shows information about the masses, m grams, of 160 apples. Mass (m grams) 30 < m  80 80 < m  100 100 < m  120 120 < m  200 Frequency 50 30 40 40 (a) Calculate an estimate of the mean. Answer(a) … g [4] (b) On the grid, complete the histogram to show the information in the frequency table. 2.5 2 1.5 Frequency density 1 0.5 m 0 40 80 120 160 200 Mass (grams) [3] (c) An apple is chosen at random from the 160 apples. Find the probability that its mass is more than 120 g. Answer(c) … [1] (d) Two apples are chosen at random from the 160 apples, without replacement. Find the probability that (i) they both have a mass of more than 120 g, Answer(d)(i) … [2] (ii) one has a mass of more than 120 g and one has a mass of 80 g or less. Answer(d)(ii) … [3] __________________________________________________________________________________________

13 marks

Mark scheme: 6 (a) 101.5625 or 102 or 101.5 to 101.6 4 M1 for 55, 90, 110, 160 soi nfww M1 for Σfm with frequencies and each m in or on a boundary of a correct interval 2750, 2700, 4400, 6400 M1 dep on 2nd M for ÷ 160 (b) Correct histogram drawn with 3 B1 for each correct block correct widths and heights If zero scored, SC1 for correct heights or 1, 1.5 and 2 (no gaps) frequency densities 40 (c) oe 1 160 1560 40 39 (d) (i) oe 2 M1 for × 25440 160 159 4000 40 50 50 40 (ii) oe 3 M2 for × + × oe 25 440 160 159 160 159 or M1 for one of these products soi

This question in 0580/43 Oct/Nov 2015

5 Kiah plays a game. The game involves throwing a coin onto a circular board. Points are scored for where the coin lands on the board. 5 10 20 If the coin lands on part of a line or misses the board then 0 points are scored. The table shows the probabilities of Kiah scoring points on the board with one throw. Points scored 20 10 5 0 Probability x 0.2 0.3 0.45 (a) Find the value of x. x = … [2] (b) Kiah throws a coin fifty times. Work out the expected number of times she scores 5 points. … [1] (c) Kiah throws a coin two times. Calculate the probability that (i) she scores either 5 or 0 with her first throw, … [2] (ii) she scores 0 with her first throw and 5 with her second throw, … [2] (iii) she scores a total of 15 points with her two throws. … [3] (d) Kiah throws a coin three times. Calculate the probability that she scores a total of 10 points with her three throws. … [5]

15 marks

Mark scheme: 5 (a) 0.05 oe 2 M1 for 1 – (0.2 + 0.3 + 0.45) oe (b) 15 1 (c) (i) 0.75 oe 2 M1 for 0.45 + 0.3 oe (ii) 0.135 oe 2 M1 for 0.45 × 0.3 oe (iii) 0.12 oe 3 M2 for 2(0.3 × 0.2) oe or M1 for 0.3 × 0.2 or 0.06 oe nfww (d) 0.243 oe 5 M4 for 3(0.45 × 0.45 × 0.2) + 3(0.3 × 0.3 × 0.45) oe or M3 for 3(0.45 × 0.45 × 0.2) or 3(0.3 × 0.3 × 0.45) oe or M2 for 0.45 × 0.45 × 0.2 and 0.3 × 0.3 × 0.45 or M1 for 0.45 × 0.45 × 0.2 or 0.3 × 0.3 × 0.45 oe or for identifying the correct 6 outcomes e.g. 10 0 0, 0 0 10, 0 10 0, 5 5 0, 5 0 5, 0 5 5

This question in 0580/42 May/June 2016

4 Coins are put into a machine to pay for parking cars. The probability that the machine rejects a coin is 0.05 . (a) Adhira puts 2 coins into the machine. (i) Calculate the probability that the machine rejects both coins. … [2] (ii) Calculate the probability that the machine accepts at least one coin. … [1] (b) Raj puts 4 coins into the machine. Calculate the probability that the machine rejects exactly one coin. … [3] (c) The table shows the amount of money, $a, received for parking each day for 200 days. Amount ($a) 200 1 a G 250 250 1 a G 300 300 1 a G 350 350 1 a G 400 400 1 a G 450 450 1 a G 500 Frequency 13 19 27 56 62 23 Calculate an estimate of the mean amount of money received each day. $ … [4] (d) The histogram shows the length of time that 200 cars were parked. 1.2 1.1 1 0.9 0.8 0.7 Frequency density 0.6 0.5 0.4 0.3 0.2 0.1 0 50 100 150 200 250 300 350 400 Time in minutes (i) Calculate the number of cars that were parked for 100 minutes or less. … [1] (ii) Calculate the percentage of cars that were parked for more than 250 minutes. … % [2]

13 marks

Mark scheme: 14 (a) (i) 2 M1 for 0.052 oe 0.0025 or oe 400 399 (ii) 1FT FT for 1 – (their (a)(i)) oe 0.9975 or oe 400 6859 (b) 3 M2 for 4(0.05 × 0.953) oe 0.171 or 0.1714 to 0.1715 or 40 000 M1 for 0.05 × 0.953 oe seen or for the 4 combinations correctly identified

This question in 0580/43 May/June 2016

7 A train stops at station A and then at station B. If the train is late at station A, the probability that it is late at station B is 0.9 . If the train is not late at station A, the probability that it is late at station B is 0.2 . The probability that the train is late at station A is 0.3 . (a) Complete the tree diagram. Station A Station B late 0.9 late 0.3 not late … late … … not late not late … [2] (b) (i) Find the probability that the train is late at one or both of the stations. … [3] (ii) This train makes 250 journeys. Find the number of journeys that the train is expected to be late at one or both of the stations. … [1] (c) The train continues to station C. The probability that it is late at all 3 stations is 0.27 . Describe briefly what this probability shows. … … [1]

7 marks

Mark scheme: 7 (a) 0.7, 0.1 oe correctly placed 1 0.2, 0.8 oe correctly placed 1 (b) (i) 0.44 nfww oe 3 M2 for 1 − their 0.7 × their 0.8 or for 0.3 + their 0.7 × their 0.2 oe or M1 for their 0.7 × their 0.8 or for two of 0.3 × 0.9, 0.3 × their 0.1, their 0.7 × their 0.2 (ii) 110 1FT FT 250 × their (b)(i) (c) If late at first two stations then certain 1 Indication of certain event (allow 1 or 100% to be late at station C oe probability or sure) at third station if late at first two stations 323 323 323 323

This question in 0580/41 Oct/Nov 2016

75 The probability that a plant will produce flowers is . 8 The flowers are either red or yellow. 3 If the plant produces flowers, the probability that the flowers are red is . 4 (a) (i) Complete the tree diagram by writing a probability beside each branch. Red … Flowers … … Yellow … No flowers [2] (ii) Calculate the probability that a plant, chosen at random, will produce red flowers. … [2] (iii) Two plants are chosen at random. Calculate the probability that both will produce red flowers. … [2] (b) Alphonse buys 200 of these plants. Calculate the number of plants that are expected to produce flowers. … [2] (c) Gabriel has 1575 plants with red flowers. Estimate the total number of plants that Gabriel has. … [2]

10 marks

Mark scheme: 5 (a) (i) 3 1 2 B1 for any 2 correct , 4 4 7 1 , 8 8 21 7 3 (ii) oe 2 M1 for × oe 32 8 4 441  7 3  2 (iii) oe 2FT M1 for  ×  or their ((a)(ii))2 oe 1024  8 4  (b) 175 2 7 M1 for 200 × 8 (c) 2400 2 M1 for 1575 ÷ their(a)(ii)

This question in 0580/42 Oct/Nov 2016

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

Mark scheme: 1 1  1 1 1  4 (a) oe 3 M2 for  1 − − −  oe 8 2  6 4 3  1 1 1 or M1 for + + seen oe or idea that 6 4 3 all sum to 1 7 1 1 (b) oe 2 M1 for + oe 12 3 4 1 1 1 (c) (i) oe 2 M1 for × oe 16 4 4 2 1 1 (ii) oe 3 M2 for 2 × × oe 24 6 4 1 1 or M1 for × oe 6 4 (d) 12 1

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

12 A box contains 20 packets of potato chips. 6 packets contain barbecue flavoured chips. 10 packets contain salt flavoured chips. 4 packets contain chicken flavoured chips. (a) Maria takes two packets at random without replacement. 9 (i) Show that the probability that she takes two packets of salt flavoured chips is . 38 [2] (ii) Find the probability that she takes two packets of different flavoured chips. … [4] (b) Maria takes three packets at random, without replacement, from the 20 packets. Find the probability that she takes at least two packets of chicken flavoured chips. … [3]

9 marks

Mark scheme: 12(a)(i) 10 9 M2 9 × oe B1 for oe seen 20 19 19 12(a)(ii) 62 4 6 14 10 10 4 16 oe M3 for × + × + × 95 20 19 20 19 20 19 oe 6 5 10 9 4 3 or 1 – × – × – × oe 20 19 20 19 20 19 or M2 for the sum of two products of different flavours isw or M1 for one correct product of different flavours isw 12(b) 5 3 M2 for oe 57  4 3 16  4 3 2 N × × × + × × oe    20 19 18  20 19 18  4 3 16  or for 3 × × oe    20 19 18  or  4 16 15  16 15 14 1 – {N × × × + × × }    20 19 18  20 19 18 oe 4 3 k or M1 for × × oe seen 20 19 18

This question in 0580/42 Oct/Nov 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

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

3 The probability that Andrei cycles to school is r. (a) Write down, in terms of r, the probability that Andrei does not cycle to school. … [1] (b) The probability that Benoit does not cycle to school is 1.3 - r. The probability that both Andrei and Benoit do not cycle to school is 0.4 . (i) Complete the equation in terms of r. ( … ) # ( … ) = 0.4 [1] (ii) Show that this equation simplifies to 10r 2 - 23 r + 9 = 0 . [3] (iii) Solve by factorisation 10r 2 - 23r + 9 = 0 . r = … or r = … [3] (iv) Find the probability that Benoit does not cycle to school. … [1]

9 marks

Mark scheme: 3(a) 1 – r 1 3(b)(i) (1 – r) (1.3 – r) [= 0.4] 1 FT their(a) dep on (a) being an expression in r 3(b)(ii) 1.3 – 1.3r – r + r 2 or better nfww M1 FT their (b)(i) 0.9 − 2.3r + r 2 [ = 0] M1 Strict FT their expansion to a quadratic then equating to 0.4 and then collecting to 3 terms on ‘one side’ OR OR Strict FT their expansion to a quadratic = 0.4 all 13 – 13r – 10r + 10r2 = 4 oe multiplied by 10 2 A1 no errors or omissions seen 10 r − 23 r + 9 = 0 3(b)(iii) (5r − 9 )(2 r − )1 [= 0] B2 or B2 for e.g. 5r(2r – 1) – 9(2r – 1) and then 5r – 9 = 0 and 2r – 1 = 0 or B1 for 5r(2r – 1) – 9(2r – 1) [ = 0] or 2r(5r – 9) – 1(5r – 9) [ = 0] or (5r + a)(2r + b) [ = 0] where a, b are integers and ab = +9 or 2a + 5b = – 23 If 0 scored, SC1 for 5r – 9 and 2r – 1 seen but not in factorised form 9 1 B1 [r =] oe [r =] oe 5 2 3(b)(iv) 4 1 0.8 or oe 5

This question in 0580/42 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

20 Stephan has two bags, A and B, each containing red sweets and yellow sweets only. He picks two sweets at random, one from bag A and the other from bag B. The probability that Stephan picks a red sweet from bag A is 3. 5 2 The probability that Stephan picks a red sweet from bag A and a red sweet from bag B is . 15 (a) Find the probability that Stephan picks a red sweet from bag B. … [2] (b) Find the probability that Stephan picks a yellow sweet from bag A and a yellow sweet from bag B. … [3]

5 marks

Mark scheme: 20(a) 2 2 3 2 oe M1 for × p = oe or better 9 5 15 20(b) 14 3 2 2 oe FT (1 – their ) × correctly evaluated 45 9 5 2 3 M2 for (1 – their ) × (1 – ) oe 9 5 2 3 or M1 for 1 – their or 1 – oe 9 5

This question in 0580/43 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