Probability of combined events — Paper 4 · IGCSE Mathematics - International 0607

8 The Venn diagram shows the sets M, E and T. U M E 8 T 4 U = {students at a school} M = {students who study mathematics} E = {students who study English} T = {students who study technology} n M + E + T = 8 ^ h n M , E , T l = 4 ^ h n M + E = 12 , n M + T = 14 and n E + T = 20 ^ h ^ h ^ h n M = 25 , n E = 30 , n T = 35 and n U = 56 ^ h ^ h ^ h ^ h (a) Complete the Venn diagram. [3] (b) Find (i) n M + E l , T l ^ ^ hh, … [1] (ii) n M + T l ^ h. … [1] (c) One of these students is chosen at random. Find the probability that this student studies English and mathematics but not technology. … [2] (d) Two of the 56 students are chosen at random. Find the probability that they both study technology. … [2] (e) A student who studies mathematics is chosen at random. Find the probability that this student also studies technology but not English. … [2] (f) Two students who study English are chosen at random. Find the probability that they both study mathematics but not technology. … [3]

14 marks

Mark scheme: 8(a) Correct values inside circles 3 B2 for 4 or 5 regions correct B1 for 2 or 3 regions correct M E 7 4 6 6 [8] 12 [4] 9 T 8(b)(i) 17 1 FT their diagram 8(b)(ii) 11 1 FT their diagram 8(c) 4 2 FT their 4 oe their 4 p 56 M1 for (k > their 4) or (p < 56) k 56 8(d) 1190 2 35 34 oe M1 for × 3080 56 55 8(e) 6 2 FT their 6 oe their 6 p 25 M1 for (k > their 6) or (p < 25) k 25 8(f) 12 3 their 4 (their 4) − 1 oe M2 for × (their 4 < 30) 870 30 29 a a − 1 or M1 for × (their a < 30) 30 29

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11 A farmer sorts the grapefruit he grows into sizes, according to their diameter. The diameters, d cm, of 170 grapefruit are shown in the table. Size Small Medium Large Very Large Diameter (d cm) 9 1 d G 10 10 1 d G 12 12 1 d G 14 14 1 d G 17 Frequency 10 50 65 45 (a) Calculate an estimate of the mean diameter of the grapefruit. … cm [2] (b) On the grid, draw a histogram to represent this information. Complete the scale on the frequency density axis. Frequency density d 8 9 10 11 12 13 14 15 16 17 18 Diameter (cm) [4] (c) Two of the 170 grapefruit are chosen at random. Calculate the probability that (i) they are both Very Large, … [2] (ii) one is Small and the other is Medium. … [3]

11 marks

Mark scheme: 11(a) 12.9 or 12.86 to 12.87 2 M1 for evidence of at least three mid-interval values 9.5, 11, 13, 15.5 soi by 95, 550, 845, 697.5 or 2187.5 11(b) Correct Histogram 4 B1 for correct bar widths no gaps B3 for 4 correct heights and corresponding scale from 0 or B2 for 3 correct heights and corresponding scale from 0 or B1 for 2 correct heights and corresponding scale from 0 or B1 for 3 correct frequency densities soi 11(c)(i) 198 2 45 44 oe M1 for × 2873 170 169 11(c)(ii) 100 3 10 50 50 10 oe M2 for × + × oe 2873 170 169 170 169 10 50 or M1 for × 170 169

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9 In a survey, 40 students are asked if they like football, F, and if they like baseball, B. 22 like football, 19 like baseball and 6 do not like either football or baseball. (a) Complete the Venn diagram to show this information. U F B … … … 6 [2] (b) How many of these students (i) like both football and baseball, … [1] (ii) either like football or do not like baseball? … [1] (c) Find n (F + B l) . … [1] (d) Two of these students are chosen at random. Find the probability that they both like football. … [2] (e) (i) One of the 19 students who like baseball is chosen at random. Find the probability that this student also likes football. … [1] (ii) Two of the 19 students who like baseball are chosen at random. Find the probability that one likes football and one does not like football. … [3] (f) Another n students take part in the survey. They all like both baseball and football. A student is then chosen at random from the (40 + n) students. 5 The probability that a student likes both football and baseball is . 16 Find the value of n. n = … [3] (g) U F B On the Venn diagram, shade the region F l , B l. [1]

15 marks

Mark scheme: 9(a) 15, 7, 12 correctly placed 2 B1 for two correctly placed or M1 for 41 – (40 – 6) seen oe or correct equation 9(b)(i) 7 1 FT their Venn diagram 9(b)(ii) 28 1 FT their Venn diagram 9(c) 15 1 FT their Venn diagram 9(d) 462 2 22 21 oe M1 for × 1560 40 39 9(e)(i) 7 1 FT their Venn diagram 19 9(e)(ii) 168 3 their 7 their 12 their 12 their 7 oe M2 for × + × oe 342 19 18 19 18 or M1 for one of these products 9(f) 8 3 their 7 + n 5 M2 for = oe 40 + n 16 or M1 for at least two trials 9(g) 1

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8 A fair 6-sided die is numbered 0, 1, 1, 2, 3, 3. (a) The die is rolled and the number it shows is recorded. Find the probability that the number is (i) 3, … [1] (ii) not 3, … [1] (iii) an odd number. … [1] (b) The die is rolled twice. Find the probability that (i) both numbers are 0, … [2] (ii) one number is 2 and the other is 3. … [3] (c) The die is rolled three times and the three numbers shown are added. Find the probability that the total is not 0. … [2]

10 marks

Mark scheme: 8(a)(i) 2 1 oe 6 8(a)(ii) 4 1 oe 6 8(a)(iii) 4 1 oe 6 8(b)(i) 1 2 1 1 oe M1 for × 36 6 6 8(b)(ii) 4 3 1 2 2 1 oe M2 for × + × oe 36 6 6 6 6 1 or M1 for one product soi by oe 18 8(c) 215 2 1 1 1 oe M1 for 1 − × × oe 216 6 6 6

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4 The masses of 120 peaches are recorded in the table. Mass (m grams) Frequency 0 1 m G 120 12 120 1 m G 150 27 150 1 m G 180 33 180 1 m G 210 15 210 1 m G 250 28 250 1 m G 300 5 (a) Calculate an estimate of the mean mass of a peach. Give your answer correct to the nearest gram. … g [3] (b) Two peaches are chosen at random. Find the probability that they both have a mass of more than 210 g. Give your answer as a fraction in its simplest form. … [3] (c) (i) Complete the frequency density column in this table. Frequency Mass (m grams) Frequency density 0 1 m G 120 12 120 1 m G 150 27 150 1 m G 180 33 180 1 m G 210 15 210 1 m G 250 28 250 1 m G 300 5 0.1 [2] (ii) On the grid, draw an accurate histogram to show this information. Frequency density 0 m 0 50 100 150 200 250 300 Mass (grams) [4]

12 marks

Mark scheme: 4(a) 171 cao nfww 3 B2 for 171.25 or 171.3 or M2 for complete method with 1 numerical error or M1 for at least 3 mid-pts (60, 135, 165, 195, 230, 275) soi 4(b) 44 3 1056 cao B2 for oe accept 0.0739 or 0.07394 to 595 14280 0.07395 33 32 or M1 for × 120 119 4(c)(i) 0.1, 0.9, 1.1, 0.5, 0.7, [0.1] 2 B1 for 3 or 4 correct 4(c)(ii) Correct histogram 4 B1 for suitable scale B1 for correct column widths B1FT for 4 or more correct heights

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7 1 cent 1 cent 5 cents 10 cents 10 cents 10 cents The diagram shows coins of values 1 cent, 5 cents and 10 cents. Two of these coins are chosen at random. Find the probability that (a) each coin has a value of less than 10 cents, … [2] (b) the total value of the two coins is 11 cents, … [3] (c) the total value of the two coins is more than 2 cents. … [2]

7 marks

Mark scheme: 7(a) 6 2 3 2 oe M1 for × oe 30 6 5 7(b) 12 3 2 3 oe M2 for × × 2 oe 30 6 5 2 3 or M1 for × oe 6 5 7(c) 28 2 2 1 oe M1 for 1 − × oe 30 6 5

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8 Rashid takes a language examination that has three tests. The probability that Rashid passes the Listening and Reading test is 0.9 . The probability that Rashid passes the Speaking test is 0.8 . The probability that Rashid passes the Writing test is 0.7 . (a) Complete the tree diagram to show the probabilities of passing (P) and failing (F) each part. Listening Speaking Writing and Reading P … P 0.8 … F P P … 0.9 … F … F P … P … … … F F P … … F … F [3] (b) To pass the whole examination Rashid has to pass all three tests. Calculate the probability that he passes the whole examination. … [2] (c) If Rashid only fails one test, he can take that test again. Calculate the probability that Rashid needs to take one test again. … [4]

9 marks

Mark scheme: 8(a) Fully correct tree diagram 3 B1 for each column correct of 0.9 and 0.1 correctly placed (L &R) 0.8 and 0.2 correctly placed (S) 0.7 and 0.3 correctly placed (W) 8(b) 0.504 2 M1 for 0.9 × 0.8 × 0.7 8(c) 0.398 4 M3 for 0.9 × 0.8 × 0.3 + 0.9 × 0.2 × 0.7 + 0.1 × 0.8 × 0.7 or M2 for 2 of above products or M1 for 1 of above products

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2 The table shows the marks for 75 students in a test. Mark 0 1 2 3 4 5, 6 or 7 8 Number of students 6 18 16 8 15 5 7 (a) Write down the mode. … [1] (b) Find the range. … [1] (c) Find the median. … [1] (d) Find the inter-quartile range. … [2] (e) Calculate an estimate of the mean. … [2] (f) Give a reason why your answer to part (e) is an estimate. … … [1] (g) Two of these students are chosen at random. Find the probability that the highest mark of these students is 2. … [3]

11 marks

Mark scheme: 2(a) 1 1 2(b) 8 1 2(c) 2 1 2(d) 3 2 B1 for either [LQ =] 1, or [UQ =] 4 2(e) 2.93 or 2.933… 2 M1 for ‘ ∑ fx ’ values 2(f) Assumed all scored 6 oe 1 2(g) 1008 3 M2 for 0.182 or 0.1816... or oe 16 15 16 24 24 16 5550 × + × + × oe 75 74 75 74 75 74 or M1 for one correct product 1560 52 SC1 for = = 0.28108... 5550 185

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8 The 150 members of a sports club were asked if they played cricket (C), hockey (H) or tennis (T). Some members play none of the three sports. The Venn diagram shows the numbers of members who play the three sports. U H C 12 35 24 8 13 15 27 T (a) Calculate the number of members who play none of the three sports. … [1] (b) Two of the 150 members are picked at random. Calculate the probability that (i) they both play hockey and tennis but not cricket, … [2] (ii) they are both members of the set (C , H ) + T l. … [3] (c) Three of the members who play tennis are chosen at random. Calculate the probability that none of them play cricket. … [3]

9 marks

Mark scheme: 8(a) 16 1 8(b)(i) 7 2 15 14 oe M1 for × oe with no extra 745 150 149 products 8(b)(ii) 497 3 71 70 oe M2 for × oe with no extra 2235 150 149 products or M1 for 35 + 12 + 24 soi by 71 8(c) 1640 3 42 41 40 oe M2 for × × oe with no extra 5673 63 62 61 products 15 + 27 42 or M1 for soi by 15 + 27 + 8 + 13 63

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5 Jian asks 60 people what their favourite type of television programme is. These are the results. Type of programme Number of people Factual 15 Sport 18 Drama 12 Game Show 10 Other 5 (a) Jian draws a pie chart to show these results. Calculate the sector angle for Drama. … [2] (b) Jian chooses one of the 60 people at random. Write down the probability that the person says Factual. … [1] (c) Jian chooses two of the 60 people at random. (i) Find the probability that one of them says Drama and the other says Game Show. … [3] (ii) Find the probability that at least one person says Sport. … [3]

9 marks

Mark scheme: 5(a) 72 2 12 M1 for × 360 60 5(b) 1 1 oe 4 5(c)(i) 4 3 12 10 10 12 oe M2 for × + × oe 59 60 59 60 59 12 10 10 12 2 or M1 for × or × soi 60 59 60 59 59 5(c)(ii) 303 3  42 41  oe M2 for 1 –  ×  oe 590  60 59  18 42 42 18 18 17 or × + × + × 60 59 60 59 60 59 18 42 42 18 18 17 or M1 for × or × or × 60 59 60 59 60 59 42 41 or × 60 59

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3 Jono walks to school when the weather is fine. When the weather is not fine, Jono takes the bus. If Jono walks to school, the probability that he is late is 0.2 . If Jono takes the bus, the probability that he is late is 0.05 . On any day, the probability that the weather is fine is 0.7 . (a) Complete the tree diagram. Late 0.2 Fine 0.7 Not … late Late … … Not fine Not … late [3] (b) (i) Find the probability that, on any day, Jono is late. … [3] (ii) Jono attends school on 200 days. Find the expected number of days that Jono is late. … [1]

7 marks

Mark scheme: 3(a) 0.3 3 B1 0.8 B1 0.05 and 0.95 B1 3(b)(i) 0.155 oe 3 M2 for 0.7 × 0.2 + their 0.3 × their 0.05 or M1 for 0.7 × 0.2 or their 0.3 × their 0.05 3(b)(ii) 31 1 FT 200 × their (i)

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9 240 students take part in a charity run. The table shows information about the times, t minutes, taken to complete the run. Time (t minutes) 20 1 t G 40 40 1 t G 50 50 1 t G 55 55 1 t G 75 Number of students 20 70 120 30 (a) Write down the time interval that contains the median. … 1 t G … [1] (b) Calculate an estimate of the mean. … min [2] (c) Complete the histogram to show the information in the table. 25 20 Frequency 15 density 10 5 0 t 20 30 40 50 60 70 80 Time (minutes) [4] (d) (i) One of the 240 students is chosen at random. Find the probability that this student took more than 55 minutes to complete the run. … [1] (ii) Two students are chosen at random from the 240 students. Calculate the probability that they both took more than 50 minutes. … [2] (iii) Two students are chosen at random from the 240 students. Complete the statement. 161 The probability that they both had times in the interval … 1 t G … is . 1912 [2]

12 marks

Mark scheme: 9(a) 50 < t ⩽ 55 1 Allow e.g. 50 to 55 9(b) 50 2 M1 for at least three of 30, 45, 52.5, 65 soi 9(c) Correct histogram 4 B1 for each correct column height. B1 for all widths correct If 0 scored, SC1 for 7, 24, 1.5 seen 9(d)(i) 30 1 oe 240 9(d)(ii) 22350 2 150 149 oe M1 for × 57360 240 239 9(d)(iii) 40 ... 50 2 k k − 1 M1 for × attempted 240 239

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3 There are 120 students at a school. There are 30 students in each class. The number of boys and the number of girls in each class is shown in the table. Class 1 Class 2 Class 3 Class 4 Boys 16 19 12 13 Girls 14 11 18 17 (a) A student is chosen at random from the 120 students. Calculate the probability that the student chosen is (i) a boy from Class 2, … [1] (ii) not from Class 3. … [1] (b) A boy is chosen at random. Calculate the probability that he is from Class 4. … [2] (c) Three students from Class 1 are chosen at random. Calculate the probability 3 girls are chosen. … [3]

7 marks

Mark scheme: 3(a)(i) 19 1 oe 120 3(a)(ii) 3 1 oe 4 3(b) 13 2 k oe M1 for 60 16 + 19 + 12 + 13 3(c) 13 2184 3 14 13 12 or oe M2 for × × 145 24 360 30 29 28 or M1 for 14 ×13 ×12 oe seen or 30 × 29 × 28 oe seen

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6 The diagram shows a six-sided die and a coin. The numbers on the faces of the die are 1, 1, 1, 2, 2, 3. When the die is rolled it is equally likely for any of the six faces to be on the top. When the coin is spun it is equally likely to land showing heads or tails. (a) Abi rolls the die. Write down the probability that it shows the number 3 on the top. … [1] (b) Beatrice rolls the die and spins the coin. (i) Find the probability that the die shows the number 2 on the top and the coin shows heads. … [2] (ii) Find the probability that the die shows the number 2 on the top or the coin shows heads or both. … [2] (c) Carl spins the coin 3 times. Find the probability that the coin shows heads at least once. … [2] (d) Drew rolls the die 3 times and records the numbers on the top. Find the probability that the die shows each of the numbers, 1, 2 and 3, once. … [3] (e) Eva spins the coin n times. 1 The probability that the coin shows tails each time is . 64 Find the value of n. n = … [1] (f) Frank rolls the die twice and records the two numbers. 1 The probability of these two numbers occurring is . 3 Find these two numbers. … and … [2]

13 marks

Mark scheme: 6(a) 1 1 oe 6 6(b)(i) 2 2 2 1 oe M1 for × oe 12 6 2 6(b)(ii) 8 2 2 1 2 1 oe M1 for + – × or indicating all 8 12 6 2 6 2 2 1 4 1 2 1 outcomes or × + × + × oe 6 2 6 2 6 2 6(c) 7 2 1 1 1 oe M1 for 1 − × × oe 8 2 2 2 6(d) 36 3 3 2 1 oe M2 for × × × 6 oe 216 6 6 6 or M1 for one product 6(e) 6 1 6(f) 1, 2 2 M1 for probability of 1 then 2 or 2 then 1 is 1 1 1 1 × or × 2 3 3 2 1 1 1 1 or for 2 × × or 2 × × seen 2 3 3 2 or for clear list

This question in 0607/41 Oct/Nov 2019

6 Spinner A is numbered 1, 2, 3, 4. Spinner B is numbered 1, 2, 3, 4, 5, 6. Each spinner is equally likely to land on any of its numbers. The two spinners are each spun once and the number that each spinner lands on is recorded. Find the probability that (a) the number on spinner A is greater than 4, … [1] (b) the number on spinner B is not a 3, … [1] (c) the number on spinner A is the same as the number on spinner B, … [2] (d) one number is odd and one number is even, … [3] (e) the sum of the numbers is 6. … [2]

9 marks

Mark scheme: 6(a) 0 cao 1 6(b) 5 1 oe 6 6(c) 4 2 1 1 oe M1 for × 24 4 6 k or B1 for soi k integer from 1 to 23 24 6(d) 12 3 2 3 2 3 oe M2 for × + × oe 24 4 6 4 6 or for 12 pairs listed or indicated 2 3 or M1 for × oe 4 6 or for 10 or 11 pairs listed or indicated 6(e) 4 2 1 1 oe M1 for × 24 4 6 or for (1, 5) (2, 4) (3, 3) (4, 2) listed or indicated

This question in 0607/42 Oct/Nov 2019

10 The mass of each of 80 apples is shown in the table. Mass (m grams) Frequency 0 1 m G 100 6 100 1 m G 120 22 120 1 m G 140 31 140 1 m G 160 13 160 1 m G 250 8 (a) Calculate an estimate of the mean mass of an apple. … g [2] (b) Find the interval which contains the upper quartile. … 1 m G … [1] (c) Two of these apples are chosen at random. Find the probability that they both have a mass of 120 g or less. Give your answer as a fraction in its simplest form. … [3] (d) (i) Complete the frequency density column in this table. Mass (m grams) Frequency Frequency density 0 1 m G 100 6 100 1 m G 120 22 120 1 m G 140 31 140 1 m G 160 13 160 1 m G 250 8 [2] (ii) On the grid, draw a histogram to show this information. 2.0 1.8 1.6 1.4 1.2 Frequency 1.0 density 0.8 0.6 0.4 0.2 0 m 0 50 100 150 200 250 Mass (grams) [3]

11 marks

Mark scheme: 10(a) 129.25 2 M1 for at least 3 of 50, 110, 130, 150, 205 seen 10(b) 140[ < m ⩽] 160 1 10(c) 189 3 756 B2 for 0.12[0] or 0.1196... or oe 1580 6320 28 27 or M1 for × 80 79 10(d)(i) 0.06, 1.1, 1.55, 0.65, 0.0889 or 2 B1 for 3 or 4 correct 0.08888 to 0.08889 10(d)(ii) Correct histogram 3 B1 for correct widths B2 FT for all heights correct or B1 FT for 3 or 4 correct heights

This question in 0607/42 Oct/Nov 2019

8 A dance club has 90 members. Here is some information about types of dancing members like. 50 like Ballroom (B) 37 like Latin (L) 47 like Modern (M) 18 like Ballroom and Latin 15 like Ballroom and Modern 22 like Latin and Modern 8 like Ballroom, Latin and Modern (a) Complete the Venn diagram. U B L 10 8 M [2] (b) Write down the number of members who do not like any of these three types of dancing. … [1] (c) Two of the 90 members are chosen at random. Find the probability that they both like Ballroom and Latin but not Modern. … [2] (d) Two of the members who like Ballroom are chosen. Find the probability that one of these members likes Latin but not Modern and the other likes Modern but not Latin. … [3]

8 marks

Mark scheme: 8(a) Correct Diagram 2 Condone 3 omitted from outside region B1 for 3 subsets correct 25 (10) 5 7 (8) 14 3 18 8(b) 3 1 8(c) 90 2 10 9 oe M1 for × 8010 90 89 8(d) 140 3 10 their 7 their 7 10 M2 for × + × 2450 50 49 50 49 oe or M1 for one of these products

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8 (a) When the weather is fine, the probability that Sara goes to the park is 0.9 . When the weather is not fine, the probability that Sara goes to the park is 0.2 . On any day, the probability that the weather is fine is 0.7 . (i) Complete the tree diagram. Weather Park Sara … goes Fine 0.7 … Sara does not go Sara … goes … Not fine … Sara does not go [3] (ii) Find the probability that, on any day, Sara goes to the park. … [3] (b) 30 students are asked if they like Mathematics (M) and if they like English (E). The Venn diagram shows the number of students in each subset. U M E 7 17 5 1 (i) Find n ( M , E l) . … [1] (ii) Two students are chosen at random. Find the probability that they both like Mathematics but not English. … [3]

10 marks

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1 A class of 40 students complete a science test. The table shows the marks of the 40 students. Mark 0 1 2 3 4 5 6 7 8 9 10 Number of students 1 1 2 5 5 5 6 3 9 2 1 (a) Write down the mode. … [1] (b) Work out the range. … [1] (c) Find the median. … [1] (d) Find the interquartile range. … [2] (e) Calculate the mean. … [2] (f) Two of the students are chosen at random. Find the probability that the difference in their marks is 8. … [3]

10 marks

Mark scheme: Question Answer Marks Partial Marks 1(a) 6 1 1(b) 10 1 1(c) 6 1 1(d) 4 2 B1 for LQ = 4 or UQ = 8 1(e) 5.55 2 M1 for attempt at  fx (3 or more terms correct) 1(f) 1 13 3 M2 for two of = or 0.0083 recurring or 120 1560  1 2   2 1   9 1   ×  +  ×  +  ×  0.008333...  40 39   40 39   40 39  or M1 for any one of above products

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5 Fifty students, 25 boys and 25 girls, were asked which sport they prefer. The results are shown in the table. Athletics Football Swimming Tennis Boy 4 9 2 10 Girl 3 3 12 7 (a) A student is selected at random. Calculate the probability that the student chosen is (i) a girl who prefers swimming, … [1] (ii) a boy who does not prefer football, … [1] (iii) a student who prefers athletics. … [1] (b) Two of the girls are chosen at random. Calculate the probability they both prefer tennis. … [2] (c) Two of the students who prefer athletics are chosen at random. Calculate the probability that one is a boy and one is a girl. … [3] (d) Three of the 50 students are chosen at random. Calculate the probability that one is a boy and two are girls and they all prefer swimming. … [4]

12 marks

Mark scheme: 5(a)(i) 12 1 oe 50 5(a)(ii) 16 1 oe 50 5(a)(iii) 7 1 oe 50 5(b) 42 2 7 oe M1 for soi 600 25 5(c) 24 3  4 3   3 4  oe M2 for  ×  +  ×  42  7 6   7 6  4 3 3 4 or M1 for × or × 7 6 7 6 5(d) 792 33 4 M3 for or oe 117600 4900  2 12 11   12 2 11   × ×  +  × ×   50 49 48   50 49 48   11 12 2  +  × ×  oe  50 49 48  or M2 for any two products or M1 for any one of above products

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8 The number of people living in each house in a street of 100 houses is recorded. The results are shown in the table. Number of people Frequency 1 5 2 16 3 28 4 32 5 17 6 2 (a) Find (i) the range, … [1] (ii) the median, … [1] (iii) the mean. … [2] (b) Two of the houses are selected at random. Find the probability that (i) both had exactly one person living in them, … [2] (ii) one had exactly 2 people living in it and the other had exactly 3 people living in it, … [3] (iii) at least one house had fewer than 5 people living in it. … [2]

11 marks

Mark scheme: 8(a)(i) 5 1 8(a)(ii) 4 1 fx8(a)(iii) 3.46 2  M1 for 100 8(b)(i) 20 2 5 4 oe M1 for × oe 9900 100 99 8(b)(ii) 896 3 16 28 28 16 M2 for × + × oe 9900 100 99 100 99 or M1 for one of the above products 8(b)(iii) 9558 2 19 18 M1 for 1 – × oe 9900 100 99

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11 A bag contains 4 red balls, 5 black balls and 3 white balls only. (a) In an experiment, one ball is chosen at random. (i) Find the probability that the ball chosen is not black. … [1] (ii) This experiment is carried out 1440 times. Find the expected number of times the ball chosen is not black. … [1] (b) In a different experiment, one ball is chosen at random, the colour is noted, and the ball is replaced in the bag. Another ball is then chosen at random and the colour is noted. Find the probability that the balls chosen are (i) both white, … [2] (ii) both the same colour, … [3] (iii) different colours. … [1] (c) In another experiment, three balls are chosen at random without replacement. (i) Find the probability that the first ball is not black, the second ball is black and the third ball is white. … [3] (ii) Find the probability that exactly two of the balls are red. … [4] Question 12 is printed on the next page.

15 marks

Mark scheme: 11(a)(i) 7 1 oe 12 11(a)(ii) 840 1 FT their (i) 11(b)(i) 1 2 3 3 oe M1 for × 16 12 12 11(b)(ii) 25 3 3 3 4 4 5 5 oe M2 for × + × + × 72 12 12 12 12 12 12 or M1 for any one of these products seen 11(b)(iii) 47 1 FT 1 – their (ii) oe 4 8 5 7 3 9 72 or × + × + × 12 12 12 12 12 12 11(c)(i) 3 3 4 5 3 3 5 2 oe M2 for × × or × × 44 12 11 10 12 11 10 or M1 for any product of three proper fractions with denominators 12, 11 and 10 11(c)(ii) 12 4 4 3 8 oe M3 for × × × 3 oe 55 12 11 10 4 3 8 or M2 for × × oe 12 11 10 or M1 for product of three fractions with numerators 4, 3, 8 oe

This question in 0607/41 Oct/Nov 2020

9 (a) The Venn diagram shows information about 115 people who play musical instruments. F = {people who play the flute} D = {people who play the drums} U F D 3x x + 2 4x + 1 8 (i) Calculate the number of people who play both the flute and the drums. … [3] (ii) On the Venn diagram, shade F l + D . [1] (iii) Briony plays both the flute and the drums. Use set notation to complete the statement. Briony … ( F + D ) [1] (b) Briony has 6 red socks, 4 green socks and 8 white socks. (i) She picks a sock at random. Find the probability that the sock is green. … [1] (ii) Briony replaces the sock. She now picks two socks at random, without replacement. Calculate the probability that the two socks are different colours. … [4]

10 marks

Mark scheme: 9(a)(i) 15 nfww 3 M2 for 8x = 104 or better or M1 for 3x + x + 2 + 4x + 1 + 8 [=115] oe If 0 scored, SC1 for 16 as final answer 9(a)(ii) correct shading 1 9(a)(iii) ∈ 1 9(b)(i) 2 1 oe 9 9(b)(ii) 104 4 M3 for oe 6 4 6 8 4 6 4 8 8 6 8 4 153 × + × + × + × + × + × 18 17 18 17 18 17 18 17 18 17 18 17 oe or M2 for 4 or 5 correct products added or M1 for 2 or 3 correct products added OR 6 12 4 14 8 10 M3 for × + × + × 18 17 18 17 18 17 or M2 for 2 correct products added or M1 for 1 correct product OR  6 5 4 3 8 7  M3 for 1 –  × + × + ×   18 17 18 17 18 17  or M2 for 1 – (two correct products added) or M1 for 1 – one correct product 52 If 0 scored SC1 for final answer oe 81

This question in 0607/42 Feb/March 2021

8 Spinner A is numbered 2, 3, 4, 5, 6, 7. Spinner B is numbered 2, 3, 4, 5. Each spinner is equally likely to stop on any of its numbers. The two spinners are each spun once and the number that each spinner stops on is recorded. Find the probability that (a) spinner A stops on a number less than 4, … [1] (b) spinner B stops on 6, … [1] (c) spinner A and spinner B both stop on the same number, … [2] (d) one number is prime and one number is not prime, … [3] (e) the sum of the numbers is a multiple of 3. … [2]

9 marks

Mark scheme: 8(a) 1 1 oe 3 8(b) 0 1 8(c) 1 2 1 1 1 1 1 1 1 1 oe M1 for × + × + × + × oe 6 6 4 6 4 6 4 6 4 k or for where k < their (6 × 4) 6 × 4 or for table of outcomes with correct 4 identified 8(d) 5 3 4 1 2 3 oe M2 for × + × 12 6 4 6 4 or M1 for either of these products seen OR M2 for table of outcomes with correct 10 identified or M1 for table of outcomes with 8 or 9 correct identified 8(e) 1 2 M1 for at least 6 of (2, 4) (3, 3) (4, 2) oe (4, 5) (5, 4) (6, 3) (7, 2) (7, 5) 3 identified

This question in 0607/41 May/June 2021

10 Hua travels to school by bus or she cycles or she walks. If it rains, the probability that she travels by bus is 0.7 and the probability that she cycles is 0.25 . If it does not rain, the probability that she cycles is 0.55 and the probability that she walks is 0.25 . On any day, the probability that it rains is 0.6 . (a) Complete the tree diagram to show the probabilities of the three methods of travel. Bus 0.7 Rain Cycle … 0.6 … Walk Bus … … 0.55 Not Rain Cycle … Walk [2] (b) Calculate the probability that, on any day, (i) Hua walks to school, … [3] (ii) Hua does not cycle. … [3] (c) Last week it rained every day of the 5 school days. Calculate the probability that Hua travelled by bus on exactly 4 of the 5 days. … [3]

11 marks

Mark scheme: 10(a) (0.7) B1 0.25 oe 0.05 oe 0.2 oe B1 0.4 oe (0.55) 0.25 oe 10(b)(i) 0.13 oe 3 M2 for 0.6 × their 0.05 + their 0.4 × their 0.25 oe or M1 for one of above products 10(b)(ii) 0.63 oe 3 M2 for their (b)(i) + 0.6 × 0.7 + their 0.4 × their 0.2 or M1 for 0.6 × 0.7 + their 0.4 × their 0.2 OR M2 for 0.6 × their 0.75 + their 0.4 × their 0.45 oe or M1 for one of above products OR M2 for 1 – 0.6 × their 0.25 – their 0.4 × 0.55 or M1 for 0.6 × their 0.25 and their 0.4 × 0.55 10(c) 0.36[0] or 0.3601 to 0.3602 oe 3 M2 for 5 × 0.74 × 0.3 oe or M1 for 0.74 × 0.3 oe

This question in 0607/42 May/June 2021

10 (a) Kris can go to school by bus or by taxi. On any day the probability that Kris goes by bus is 0.9 . When Kris goes by bus, the probability that she is late for school is 0.06 . When she goes by taxi, the probability that she is late for school is 0.01 . (i) Find the probability that, on any day, Kris is late for school. … [3] (ii) Find the probability that, on any day, Kris is not late for school. … [1] (iii) In one year, Kris attends school on 200 days. Find the number of days Kris is expected not to be late. … [1] (b) Alex also goes to school by bus or by taxi. The probability that Alex goes by bus is 0.8 . The probability that Alex goes by bus and is late is 0.12 . Find the probability that Alex is late when he goes by bus. … [2]

7 marks

Mark scheme: 10(a)(i) 0.055 oe 3 M2 for 0.9 × 0.06 + 0.1 × 0.01 oe or M1 for 0.9 × 0.06 oe or 0.1 × 0.01 oe 10(a)(ii) 0.945 oe 1 FT 1 – their (a)(i) 10(a)(iii) 189 1 FT 200 × their (a)(ii) 10(b) 0.15 oe 2 M1 for 0.8 × p = 0.12 oe

This question in 0607/43 May/June 2021

13 Two bags each contain only blue balls and red balls. Bag 1 contains 7 blue balls and 3 red balls. Bag 2 contains 3 blue balls and 7 red balls. Maria chooses a ball at random from Bag 1 and puts it into Bag 2. (a) Find the probability that the ball chosen is blue. … [1] (b) Maria now chooses a ball at random from Bag 2 and puts it into Bag 1. (i) Find the probability that both balls chosen are red. … [2] (ii) Find the probability that one of the balls chosen is red and the other is blue. … [3] (iii) Find the probability that there are now exactly 7 blue balls in Bag 1. … [3]

9 marks

Mark scheme: 13(a) 7 1 oe 10 13(b)(i) 12 2 3 8 oe M1 for × 55 10 11 13(b)(ii) 29 3 3 3 7 7 oe M2 for × + × 55 10 11 10 11 or M1 for either of these products seen 13(b)(iii) 26 3 M2 for 1 – their (b)(ii) oe 55 OR 7 4 3 8 M2 for × + × 10 11 10 11 or M1 for either of these products seen

This question in 0607/43 Oct/Nov 2021

10 When Zena wears a sweatshirt, the probability that she goes for a walk is 10. 9 When Zena does not wear a sweatshirt, the probability that she goes for a walk is . 10 On any day, the probability that she wears a sweatshirt is 1. 5 (a) Complete the tree diagram. Wears a sweatshirt Goes for a walk Yes … Yes 1 5 … No Yes … … No … No [3] (b) (i) Find the probability that on one day Zena does not wear a sweatshirt and she goes for a walk. … [2] (ii) Find the probability that on one day Zena goes for a walk. … [2] (c) In the tree diagram below, the value of J is the answer to part (b)(i) and the value of K is the answer to part (b)(ii). Goes for a walk Wears a sweatshirt No J … Yes K … Yes No … … No … Yes (i) Find the probability that Zena does not wear a sweatshirt when she goes for a walk. … [2] (ii) Complete the tree diagram above. [3]

12 marks

Mark scheme: 10(a) 4 3 oe B1 for each pair of branches 5 7 3 , oe 10 10 9 1 , oe 10 10 10(b)(i) 18 2 4 9 oe M1 for their ×their 25 5 10 10(b)(ii) 43 2 1 7 oe M1 for their(b)(i) + ×their oe 50 5 10 10(c)(i) 36 2 M1 for their(b)(ii) × p = their(b)(i) or better oe 43 10(c)(ii) their(b)(ii), 1 – their (b)(ii) oe 3 B1 for each pair of branches their (c)(i), 1 – their(c)(i) oe 4 3 , oe 7 7

This question in 0607/42 Feb/March 2022

5 A sequence of patterns is made using grey tiles and white tiles. Pattern 1 Pattern 2 Pattern 3 (a) Complete the table. Pattern number 1 2 3 4 n Number of grey tiles 6 10 Number of white tiles 0 2 [6] (b) Find and simplify an expression for the total number of tiles in Pattern n. … [1] (c) Pattern k has a total of 600 tiles. Find the number of grey tiles in Pattern k. … [4] (d) The tiles in a pattern are put in a bag. 5 The probability of taking a grey tile from the bag at random is . 12 A tile is taken from the bag at random and replaced. This is repeated 3 times. Find the probability that all 3 tiles are white. … [2] (e) All the grey tiles from Pattern 4 are put in a bag. Two tiles are taken from the bag at random without replacement. Find the probability that one tile came from a corner of the pattern and the other did not. … [3]

16 marks

Mark scheme: 5(a) 6 B2 for all four numbers correct 14 18 4n + 2 oe or B1 for at least two correct B2 for 4 n  2 oe 6 12 n2 – n oe or M1 for 4 n  k B2 for n 2  n oe or M1 for any quadratic or for second differences of 2 seen 5(b) n2 + 3n + 2 or (n + 1)(n + 2) 1 FT their grey + white if both in terms of n 5(c) 94 4 M1 for their k 2  3k  2 = 600 oe M1 for correct method for solving their quadratic M1 for substituting their positive integer (from a quadratic) k into their 4n+2 5(d) 343 2  5  3 oe M1 for 1  oe 1728    12  5(e) 56 3 FT their 18 from (a) for M marks only oe 153 4 14 14 4 M2 for    oe 18 17 18 17 M1 for one product

This question in 0607/41 May/June 2022

12 A bag contains 7 red balls, 4 blue balls and 1 green ball. In an experiment, three balls are chosen at random without replacement. (a) Find the probability that the three balls chosen are (i) all green, … [1] (ii) all red, … [2] (iii) two red and one blue. … [3] (b) This experiment is to be carried out 2640 times. Use your answer from part (a)(ii) to find the expected frequency of 3 red balls being chosen. … [1]

7 marks

Mark scheme: 12(a)(i) 0 1 12(a)(ii) 7 2 7 6 5 oe M1 for   44 12 11 10 12(a)(iii) 21 3 7 6 4 oe M2 for    3 oe 55 12 11 10 7 6 4 or M1 for   oe 12 11 10 12(b) 420 1 FT (their part (a)(ii)) × 2640 Additional Guidance 3 5 (a) (i) allow  0.6 then 0.6 × 1240 = 744 5 5 x 14880 3 1 1 allow  1240 then 5x = 14880 then x = = 2976 then  then  2976  744 12 5 12 4 4 5 x 3 1 1 but do not allow  1240 then 5x = 14880 then x = 2976 then  then  2976  744 12 12 4 4 (division by 5 not seen) 5 (b),(c) all parts use of e.g. 1 – 20% , do not allow M marks but correct answers can score. 7 (e) e.g. Finding angle D first B2 for [C =] 180 – 28 – (27.4 to 27.5) (124.5 to 124.6) 420sin 28 or M1 for sinD = 27.4 to 27.5 (i.e. explicit expression for sinD) (note: cosine rule their CD may be used for angle D.) and M1 for 360 – 90 – (180 – their acute C) or 360 – 90 – their obtuse C)

This question in 0607/42 May/June 2022

7 (a) Shade the region indicated below each of these Venn diagrams. U U A B P Q ( A , B ) l ( P + Q l ) , ( P l + Q ) [2] (b) Bag A Bag B Bag A contains 4 white balls and 3 black balls. Bag B contains 4 white balls and 5 black balls. A ball is taken at random from bag A. If the ball is white, it is replaced in Bag A. If the ball is black, it is put in bag B. A ball is then taken at random from bag B. Find the probability that (i) the ball taken from bag A is white, … [1] (ii) both balls are black, … [2] (iii) the balls are different colours. … [3]

8 marks

Mark scheme: 7(a) Correct shading 2 B1 for each 7(b)(i) 4 1 oe 7 7(b)(ii) 9 2 3 6 oe M1 for  oe 35 7 10 7(b)(iii) 22 3 3 4 4 5 oe M2 for  +  oe 45 7 10 7 9 or M1 for one of above products

This question in 0607/43 May/June 2022

10 A bag contains 5 red balls, 3 blue balls and 2 green balls. (a) Rosa chooses a ball at random from the bag, notes its colour and replaces it. She then chooses a ball at random from the bag a second time, notes its colour and replaces it. Find the probability that the two balls chosen are (i) both green, … [2] (ii) the same colour. … [2] (b) Savio chooses a ball at random from the bag and does not replace it. He then chooses another ball from the bag. Find the probability that the two balls chosen are different colours. … [3]

7 marks

Mark scheme: 10(a)(i) 1 2 2 2 oe M1 for  25 10 10 10(a)(ii) 19 2 5 5 3 3  2 2  oe M1 for two of  ,  , their    50 10 10 10 10  10 10  added oe 10(b) 31 3 5 5 3 7 2 8 oe M2 for  +  +  oe 45 10 9 10 9 10 9 5 5 3 7 2 8 or M1 for  or  or  10 9 10 9 10 9 OR  5 4 3 2 2 1  M2 for 1 −   +  +   oe  10 9 10 9 10 9  or M1 for two of 5 4 3 2 2 1  ,  ,  10 9 10 9 10 9

This question in 0607/41 Oct/Nov 2022

9 The table gives some information about a group of 200 people. Eye colour Total Brown Blue Green Right-handed 51 144 Left-handed 24 18 56 Total 69 20 200 (a) Complete the table. [2] (b) Find the probability that one of these people chosen at random has blue eyes. … [1] (c) Two of these people are chosen at random. Find the probability that they are both left-handed. … [2] (d) Two of the left-handed people are chosen at random. Find the probability that they both have brown eyes. … [2] (e) Two of the people with blue eyes are chosen at random. Find the probability that one is right-handed and the other is left-handed. … [3]

10 marks

Mark scheme: 9(a) 87 6 2 B1 for 2 or 3 correct 14 111 9(b) 69 1 oe 200 9(c) 77 2 56 55 oe M1 for  oe 995 200 199 9(d) 69 2 n n − 1 oe M1 for  oe 385 56 55 9(e) 9 3 51 18 18 51 oe M2 for  +  oe 23 69 68 69 68 or M1 for one of above products If 0 scored, SC1 for 204, 0.386 or 529 0.3856…

This question in 0607/42 Oct/Nov 2022

7 (a) The time, t hours, spent watching television in one week by each of 100 students is shown in the table. Time, t hours 0 1 t G 10 10 1 t G 20 20 1 t G 25 25 1 t G 30 30 1 t G 60 Frequency 3 11 42 40 4 (i) A pie chart is drawn to show the results. Calculate the sector angle for the number of students who spend more than 30 hours watching television. … [2] (ii) Calculate an estimate of the mean. … h [2] (b) A shopkeeper records the midday temperature, t °C, and the number of ice creams, n, sold each day in one week. The table shows the results. Midday 20 24 20 17 18 20 25 temperature, t °C Number of ice 103 106 95 91 93 98 114 creams, n (i) Write down the type of correlation shown in the table. … [1] (ii) Find the equation of the regression line, giving n in terms of t. n = … [2] (iii) Use your answer to part(b)(ii) to find the number of ice creams expected to be sold when the midday temperature is 22 °C. … [1] (iv) During this week, the shopkeeper sells 700 ice creams. She estimates that she will sell a total of 9800 ice creams during the next 14 weeks. Give a reason why this may not be a good estimate. … [1] (c) When the weather is fine, the probability that Lance goes cycling is 7. 9 When the weather is not fine, the probability that Lance goes cycling is 1. 5 The probability that the weather is fine is 3. 4 (i) Complete the tree diagram. Weather Cycles Yes 7 9 Fine 3 4 No … Yes … … Not fine … No [2] (ii) Find the probability that Lance goes cycling. … [3]

14 marks

Mark scheme: 7(a)(i) 14.4 2 4 M1 for  360 100 7(a)(ii) 24.05 or 24.1 2 M1 for at least 3 mid-values soi 7(b)(i) positive 1 7(b)(ii) n = 2.61t + 46.3 2 B1 for 2.61t + k or kt + 46.3 or 2.6t + 46 7(b)(iii) 103 or 104 1 FT their (c)(ii) but must be integer answer 7(b)(iv) small sample oe or reference to weather 1 7(c)(i) 1 2 1 4 2 B1 for 2 correct , , , 4 9 5 5 7(c)(ii) 19 3 3 7 1 1 oe M2 FT for  + their  their 30 4 9 4 5 or M1 FT for one of the products only FT probabilities < 1

This question in 0607/43 Oct/Nov 2022

12 A bag contains x red balls, y blue balls and z green balls. (a) Paula chooses a ball at random from the bag, notes its colour and replaces it in the bag. She then chooses a ball from the bag a second time and notes its colour. Giving your answers as unsimplified algebraic fractions, find the probability, in terms of x, y, and z, that the two balls chosen are (i) both red … [2] (ii) one blue and one green. … [2] (b) All of the green balls are removed from the bag. Novak now chooses a ball at random from the bag, notes its colour and replaces it in the bag. He then chooses a ball from the bag a second time and notes its colour. 49 The probability that the two balls chosen are both red is . 400 x Find, as a fraction, the value of . y x = … [3] y

7 marks

Mark scheme: 12(a)(i) x 2 2 x oe final answer M1 for ( x + y + z ) 2 ( x + y + z ) 12(a)(ii) 2 yz 2 y z oe final answer M1 for  ( x + y + z ) 2 x + y + z x + y + z 12(b) 7 3 x 7 M2 for = oe soi 13 x + y 20 x 2 49 or M1 for = ( x + y ) 2 400

This question in 0607/42 Feb/March 2023

9 P E R C E N T I L E Asa and Bernice have these 10 letter cards. A, E, I, O and U are vowels. All other letters are consonants. (a) Asa picks a card at random. Write down the probability that Asa’s card shows the letter T. … [1] (b) Asa replaces his card. Bernice picks two cards at random without replacement. Calculate the probability that both of Bernice’s cards are vowels. … [2] (c) Bernice replaces her cards. Asa picks 3 cards at random without replacement. Calculate the probability that Asa’s cards can be arranged to spell the word PEN. … [3] (d) Asa replaces his cards. Bernice picks cards at random with replacement until she first gets a consonant. 48 The probability that she first gets a consonant on her nth pick is . 3125 Find the value of n. … [3]

9 marks

Mark scheme: 9(a) 1 1 oe 10 9(b) 2 2 4 3 oe M1 for  15 10 9 9(c) 1 3 1 3 1 oe M2 for k ×   oe, k = 3, 4, 5 or 6 40 10 9 8 1 3 1 or M1 for   oe 10 9 8 If 0 scored SC1 for indicating 6 possibilities 9(d) 5 3 n1  4   6  48 M2 for       oe  10   10  3125  4  k  6  or M1 for      , k ⩾ 2 oe  10   10 

This question in 0607/42 May/June 2023

9 There are 80 students in a school year, 44 boys and 36 girls. Each student chooses their favourite sport. The number of boys and the number of girls choosing each sport is shown in the table. Athletics Football Hockey Swimming Boys 12 16 8 8 Girls 5 3 17 11 (a) A student is chosen at random from the 80 students. Find the probability that the student chosen is (i) a girl whose favourite sport is athletics … [1] (ii) a boy whose favourite sport is not football. … [1] (b) One of the girls is chosen at random. Find the probability that her favourite sport is hockey. … [2] (c) Three of the boys are chosen at random. (i) Find the probability that one of the boys chooses athletics, one of them chooses football and the other chooses swimming. … [1] (ii) Calculate the probability that the three boys each have a different favourite sport. … [3]

8 marks

Mark scheme: 9(a)(i) 5 1 oe 80 9(a)(ii) 28 1 oe 80 9(b) 17 2 k 17 oe M1 for where k < 36 or where 36 36 m 17  m  80 9(c)(i)  12 16 8  384 1    6  oe    44 43 42  3311 9(c)(ii) 3648 1216 3 M2 for or oe 9933 3311  12 16 8 12 16 8         44 43 42 44 43 42   k 6  or 0.367 or 0.3672 to 0.3673  12 8 8 16 8 8           44 43 42 44 43 42  or M1 for any two of these products seen

This question in 0607/43 May/June 2023

9 On any day the probability that Samira cycles to school is 6. When Samira cycles to school the probability that she arrives on time is 4. 5 When Samira does not cycle to school the probability that she arrives on time is 2. 5 (a) Find the number of days Samira is expected to cycle to school in a school term of 54 days. … [1] (b) Complete the tree diagram. Cycles to school Arrives on time Yes 4 5 Yes 5 6 … No Yes … … No … No [2] (c) Calculate the probability that on any day Samira arrives at school on time. … [3] (d) In a school week of 5 days, find the probability that Samira cycles to school on exactly 1 day. … [3]

9 marks

Mark scheme: 9(a) 45 1 9(b) 1 1 2 B1 and correctly placed 6 5 2 3 and correctly placed B1 5 5 9(c) 11 3 5 4 1 2 oe M2 for  + their  their 5 6 5 6 5 or 5 4 1 2 M1 for  or their  their 6 5 6 5 9(d) 25 3 4 oe  5  1 oe M2 for   their  [ 5] 7776  6  6  or  5  1  k M1 for   their  , k > 1  6  6 

This question in 0607/41 Oct/Nov 2023

9 (a) A bag contains 3 black discs and 5 white discs. Jani takes a disc from the bag at random. When the disc is black, he does not replace it. When the disc is white, he replaces it in the bag. Jani then takes a second disc at random. (i) Complete the tree diagram for the first and second discs. First disc Second disc Third disc Black … Black … … White Black … … White … White [3] (ii) Jani takes a third disc from the bag at random. Find the probability that he takes 2 black discs and 1 white disc. … [4] (b) Another bag contains 10 discs. x are red and the rest are green. (i) Write down an expression for the number of green discs. … [1] (ii) y blue discs are added to the bag. A disc is taken from the bag at random. (a) The probability of taking a red disc from the bag is 1. 3 Show that 3x = 10 + y . [1] (b) The probability of taking a green disc is 2. 9 Write another equation in x and y and find the number of red discs and the number of blue discs. number of red discs, x = … number of blue discs, y = … [5]

14 marks

Mark scheme: 9(a)(i) 3 5 2 5 3 5 3 B1 for each correct pair in correct position , , , , , 8 8 7 7 8 8 9(a)(ii) 365 4 FT their tree diagram probabilities for method or 0.233 marks 1568 3 2 5 3 5 2 5 3 2 M3 for   [ + ]   [ + ]   soi 8 7 6 8 7 7 8 8 7 without extras or M2 for two correct products soi or M1 for one correct product soi or for clear indication on tree diagram of all three combinations. or list of the options 9(b)(i) 10−x 1 9(b)(ii)(a) x 1 1 = oe seen 10 + y 3 leading to 3x = 10 + y 9(b)(ii)(b) [x = or red =] 6 5 B2 for 9 x + 2 y = 70 oe [y = or blue =] 8 10 − x 2 or B1 for = 10 + y 9 M1 for correct method to eliminate one variable A1 for [x = or red =] 6 or [y = or blue =] 8

This question in 0607/42 Oct/Nov 2023

6 (a) U = {integers from 1 to 15} P = {factors of 12} Q = {multiples of 3} (i) Complete the Venn diagram. U P Q [2] (ii) Write down the elements of P + Q . … [1] (iii) Find n ( P l + Q ) , ( P + Q l ) _ i. … [1] (b) Bag A Bag B Bag A contains 4 black balls and 3 white balls. Bag B contains 2 black balls and 4 white balls. (i) Amy picks a ball at random from bag A. She notes the colour of the ball and replaces it in bag A. Find the probability that Amy’s ball is black. … [1] (ii) Basma picks two balls at random from bag B. She notes the colour of each ball and replaces them in bag B. Find the probability that both balls are white. … [2] (iii) Basma chooses one bag at random. She picks one ball at random from this bag. Find the probability that the ball is white. … [3]

10 marks

Mark scheme: 6(a)(i) 2 B1 for 1 or 2 elements misplaced or omitted. 1 2 3 9 6 4 12 15 5 7 8 10 11 13 14 6(a)(ii) 3, 6, 12 1 FT their Venn diagram 6(a)(iii) 5 1 FT their Venn diagram 6(b)(i) 4 7 6(b)(ii) 2 2 4 3 oe M1 for  oe 5 6 5 6(b)(iii) 23 3 1 3 1 4 oe M2 for  +  42 2 7 2 6 or M1 for one of above products

This question in 0607/42 Feb/March 2024

10 A bag contains 5 red balls, 4 blue balls and 3 green balls. (a) (i) Tina picks one ball at random, notes the colour and replaces it in the bag. Find the probability that Tina picks a red ball. … [1] (ii) Tina repeats this 60 times. Find the number of times the ball she picks is expected to be red. … [1] (b) Eli picks two balls at random without replacement. Find the probability that (i) both balls are blue … [2] (ii) one ball is red and one ball is blue. … [3] (c) The balls are replaced in the bag. Ida picks one ball at random, notes the colour and replaces it in the bag. She then picks another ball at random. Find the probability that the two balls are the same colour. … [3]

10 marks

Mark scheme: 10(a)(i) 5 1 12 10(a)(ii) 25 1 FT 60 × their(a)(i) but must be an integer 10(b)(i) 1 2 4 3 oe M1 for  11 12 11 10(b)(ii) 10 3 5 4 oe M2 for   2 oe 33 12 11 5 4 or M1 for  oe 12 11 5 If 0 scored, SC1 for oe 18 10(c) 25 3 5 5 4 4 3 3 oe M2 for      oe 72 12 12 12 12 12 12 or M1 for two of these products 19 If 0 scored, SC1 for oe 66

This question in 0607/42 May/June 2024

3 (a) Noora throws a fair 6-sided die numbered from 1 to 6. Write down the probability that the die shows (i) a number less than 5 … [1] (ii) an even number. … [1] (b) Dilshan has two fair 6-sided dice each numbered from 1 to 6. He throws both dice. Find the probability that (i) both dice show a 6 … [2] (ii) at least one die does not show a 6. … [1] (c) The probability that it rains on Wednesday is 0.48 . If it rains, the probability that Hannah cycles to work is 0.28 . If it does not rain, the probability that Hannah cycles to work is 0.84 . (i) Complete this tree diagram. Cycles 0.28 Rains 0.48 Does not cycle … Cycles 0.84 … Does not rain Does not cycle … [2] (ii) Find the probability that, on Wednesday, it does not rain and Hannah cycles. … [2]

9 marks

Mark scheme: 3(a)(i) 2 1 oe 3 3(a)(ii) 1 1 oe 2 3(b)(i) 1 2 1 1 oe M1 for  36 6 6 3(b)(ii) 35 1 FT 1 – their (b)(i) oe 36 3(c)(i) 0.52 oe 2 B1 for one correctly placed 0.72 oe 0.16 oe Correctly placed 3(c)(ii) 0.4368 oe 2 M1 for their 0.52 × 0.84 oe

This question in 0607/43 May/June 2024

7 (a) There are 49 students in a year group. Each student studies at least one of the sciences, biology (B), chemistry (C) and physics (P). x students study all 3 sciences. y students study chemistry only. 12 students study physics only. 6 students study biology and chemistry but not physics. 11 students study biology and physics but not chemistry. 2 students study physics and chemistry but not biology. 25 students study only one science. (i) Show this information on the Venn diagram. U B x C P [2] (ii) Find the number of students who study all 3 sciences. … [2] (iii) The number of students that study biology is two times the number of students that study chemistry. Find the number of students who study (a) chemistry only … [2] (b) biology only. … [1] (b) A bag contains 7 red balls and 3 blue balls. In an experiment, three balls are chosen at random without replacement. Find the probability that at least two of the balls chosen are red. … [4]

11 marks

Mark scheme: 7(a)(i) 2 B1 for 4 correct regions 7(a)(ii) 5 2 M1 for 49 – 25 – 11 – 6 – 2 oe 7(a)(iii)(a) 3 2 M1 for either 30 + x − y or 2(8 + x + y ) oe 7(a)(iii)(b) 10 1 FT 13 – their 3, must be an integer 7(b) 49 4  7 6 5   7 6 3  oe M3 for     + 3      60  10 9 8   10 9 8   7 6 5   7 6 3  or M2 for     and      10 9 8   10 9 8   7 6 3  or 3       10 9 8   7 6 5   7 6 3  or M1 for     or      10 9 8   10 9 8 

This question in 0607/41 Oct/Nov 2024

11 32 students in a class are asked which of three activities they like. W = {students who like walking} S = {students who like swimming} C = {students who like cycling} The Venn diagram shows the number of students in each subset. U W S 8 2 3 9 4 0 5 1 C (a) One of these students is chosen at random. Complete the sentence. This student is most likely to belong in {students who like … }. [1] (b) Write down the number of students who like all three activities. … [1] (c) Find n (( S l , C) + W l ). … [1] (d) A student is chosen at random from the class. Find the probability that this student likes both walking and swimming. … [1] (e) Two of the students who like swimming are chosen at random. Find the probability that one of these students likes walking but not cycling and the other student only likes swimming. … [3] (f) Three of the 32 students are chosen at random. Find the probability that one student likes exactly one of the activities and the other two students like exactly two of the activities. … [3]

10 marks

Mark scheme: 11(a) walking 1 11(b) 9 1 11(c) 6 1 11(d) 11 1 oe 32 11(e) 6 3 2 3 3 2 oe M2 for  or  oe 91 14 13 14 13 k j or M1 for  14 13 2 3 or B1 for or seen 14 14 11(f) 3 3 16 6 5 oe M2 for [k ×] × × oe where k = 1, 2, 3 62 32 31 30 m n n − 1 or M1 for   oe 32 31 30

This question in 0607/43 Oct/Nov 2024

16 Rupesh plays a game 3 times. He can either win each game or lose each game. The probability that Rupesh wins each game is 0.2 . Find the probability that Rupesh loses the 3 games. … [2]

2 marks

Mark scheme: 16 0.512 2 n M1 for (1 − 0.2 ) oe n > 1

This question in 0607/42 Feb/March 2025

14 Ahmed records the mass of each of 50 pumpkins. The results are shown in the table. Mass (m kg) 0 1 m G 2 2 1 m G 5 5 1 m G 10 10 1 m G 15 15 1 m G 25 Frequency 13 17 10 7 3 (a) Calculate an estimate of the mean. … kg [2] (b) Ahmed picks two of the 50 pumpkins at random. Find the probability that one of these pumpkins has a mass greater than 10 kg and the other pumpkin has a mass of not more than 2 kg. … [3]

5 marks

Mark scheme: 14(a) 5.9 2 M1 for at least 3 correct midpoints soi 14(b) 26 3 10 13 oe M2 for [2  oe 245 ]50 49 10 13 13 10 or B1 for and oe or and oe 50 49 50 49 13 If 0 scored, SC1 for answer oe 125

This question in 0607/41 May/June 2025

10 The table shows the marks of each of 10 students in a physics exam and in a chemistry exam. Physics mark (x) 9 21 33 41 55 68 75 83 89 96 Chemistry mark (y) 31 46 42 50 50 61 69 68 72 90 (a) Find the mean physics mark. … [1] (b) (i) Find the equation of the regression line for y in terms of x. y = … [2] (ii) Use your answer to part (b)(i) to estimate the chemistry mark when the physics mark is 46. … [1] (c) Two of the students who scored more than 35 in the physics exam are chosen at random. Find the probability that they both scored more than 65 in the chemistry exam. … [3]

7 marks

Mark scheme: 10(a) 57 1 10(b)(i) y = 0.548 x + 26.6 2 B1 for y = kx + 26.6 or y = 0.548x + k or y = 0.55x + 27 10(b)(ii) 52 or 51.8 to 51.9 1 FT their (b)(i) 10(c) 2 3 4 3 oe M2 for  7 7 6 4 or M1 for seen 7 x x − 1 or  where x < y and y < 10 y y − 1

This question in 0607/42 May/June 2025

9 A bag contains 5 black balls and 7 white balls. One ball is chosen at random and not replaced. A second ball is then chosen at random. Find the probability that both balls are black. … [2]

2 marks

Mark scheme: 9 5 2  5 4  oe M1 for    33  12 11 

This question in 0607/42 Oct/Nov 2025

16 A spinner has 24 equal sections. Each section is coloured red or blue. There are x red sections and all other sections are blue. The spinner is equally likely to land on any of its sections. The spinner is spun twice and the colour that the spinner lands on each time is recorded. The probability that the spinner lands on blue exactly once is 4. 9 Find the possible values of x. x = … or x = … [5] Questions 17 is printed on the next page.

5 marks

Mark scheme: 16 8, 16 final answer 5 x 24 − x 4 M2 for  2   = oe 24 24 9 24 −x or B1 for seen 24 M2 for x 2 − 24 x + 128 = 0 oe 4  24  24 or M1 for x (24 − x ) = oe  2   9 If 0 scored SC1 for a single final answer of either 8 or 16

This question in 0607/42 Oct/Nov 2025