Introduction to probability — Paper 4 · IGCSE Mathematics - International 0607

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

This question in 0607/42 Oct/Nov 2017

2 Here are 12 numbers. 15 9 6 14 6 8 12 21 11 19 6 12 (a) For these numbers find (i) the range, … [1] (ii) the mode, … [1] (iii) the median, … [1] (iv) the mean, … [1] (v) the inter-quartile range. … [2] (b) Dee chooses a number at random from these numbers. Find the probability that it is a prime number. … [1]

7 marks

Mark scheme: 2(a)(i) 15 1 2(a)(ii) 6 1 2(a)(iii) 11.5 1 2(a)(iv) 11.6 or 11.58... 1 2(a)(v) 7.5 2 B1 for 7 or 14.5 seen 2(b) 2 1 oe 12

This question in 0607/42 Oct/Nov 2018

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

This question in 0607/41 May/June 2019

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

This question in 0607/43 May/June 2019

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

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

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

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

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

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

6 (a) Solve. 7x - 5 = 3x + 13 x = … [2] (b) Solve. 4 ( 2x - 3) = 3 ( 1 - 2 x) x = … [3] (c) Solve. 3x + 2 2 = 8 3x + 2 x = … or x = … [3] (d) Solve. 1 - 2 x 2 = 5x - 1 Give your answer correct to two decimal places. x = … or x = … [3] (e) log x = 1 + 4 log y Find x in terms of y. x = … [3] (f) There are 12 balls in a bag, n of them are blue. A ball is taken from the bag at random and replaced. The probability that the ball is blue is p. 6 more blue balls are added to the bag. A ball is taken from the bag at random. The probability that this ball is blue is 2p. Find the value of p. p = … [4]

18 marks

Mark scheme: 6(a) 4.5 oe 2 B1 for 7 x  3x = 13  5 oe 6(b) 15 3 B1 for 8 x  12 3 6 x oe oe 14 M1 for correctly collecting terms in an equation 6(c) 2 3 B2 for 3 x  2 4 oe ,  2 oe 3 or for 3 3 x  2  x  2   0  oe 4 4(3)( 4) or for oe 2(3) or M1 for  3 x  2  2  8  2 oe 6(d) 0.35 –2.85 3 B2 for –2.851 to –2.850 and 0.350 to 0.351 OR M2 for correct sketch indicating both roots 5 5 2  4(2)(  2) or for 2(2) or M1 for 2 x 2  5 x  2   0  or  2 x 2  5 x  2   0  6(e)  x  10 y 4 3 M1 for logy4 B1 for 1 = log10 6(f) 1 4 B3 for n = 3 oe 4 n n  6 or M2 for 212  18 or for 12p = 36p – 6 oe n n  6 or M1 for p  or 2 p  12 18 n n  6 or for and seen 12 18

This question in 0607/43 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

7 (a) Spinner A and spinner B are each fair 5-sided spinners. Spinner A is numbered 1, 2, 2, 3, 4. Spinner B is numbered 1, 2, 3, 4, 4. The two spinners are each spun once and the number on each spinner is recorded. Find the probability that (i) the number on spinner A is 6 … [1] (ii) the number on spinner B is not 4 … [1] (iii) the number on spinner A is the same as the number on spinner B … [3] (iv) the sum of the two numbers is 6. … [3] (b) (i) On the Venn diagram, shade A , B . U A B [1] (ii) Describe the shaded region using set notation. U A B … [1] (iii) The Venn diagram below shows the number of elements in each subset. U A B 3 8 4 1 6 9 7 5 C Find n ( A + B) + C l ` j. … [1]

11 marks

Mark scheme: 7(a)(i) 0 1 7(a)(ii) 3 1 oe 5 7(a)(iii) 6 3  1 1   2 1   1 1   1 2  oe M2 for                25  5 5   5 5   5 5   5 5  or correct sample space showing all 6 points or M1 for 2 correct products or correct sample space showing at least 3 points or list of all correct pairs 12 If 0 scored, SC1 for 25 7(a)(iv) 6 3  2 2   1 1   1 1  M2 for            25  5 5   5 5   5 5  or correct sample space showing all 6 points or M1 for 2 correct products or correct sample space showing at least 3 or for listing pairs that sum to 6 7(b)(i) 1 7(b)(ii) A  B / 1 7(b)(iii) 8 1

This question in 0607/41 May/June 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

1 150 students are each asked how many texts they sent the previous day. The results are shown in the table. Number of texts 0 1 2 3 4 5 6 Frequency 18 45 37 24 15 8 3 (a) Find (i) the mode … [1] (ii) the median … [1] (iii) the range … [1] (iv) the upper quartile. … [1] (b) One of the 150 students is selected at random. Find the probability that this student sent fewer than 3 texts. … [1]

5 marks

Mark scheme: Question Answer Marks Partial Marks 1(a)(i) 1 1 1(a)(ii) 2 1 1(a)(iii) 6 1 1(a)(iv) 3 1 1(b) 2 1 oe 3

This question in 0607/41 May/June 2025

2 The table shows the marks scored by 100 students in a test. Mark 1–10 11–20 21–40 41–60 61–80 81–90 91–100 Number of students 2 8 17 21 14 25 13 (a) One of these students is chosen at random. Find the probability that this student scored more than 90 marks. … [1] (b) Write down the group that contains the median. … [1] (c) Calculate an estimate for the mean. … [2]

4 marks

Mark scheme: 2(a) 0.13 oe 1 2(b) 61 - 80 1 2(c) 60.8 2 M1 for 5 midpoints soi If 0 scored SC1 for 60.3

This question in 0607/42 Oct/Nov 2025