Probability of combined events — Paper 2 · IGCSE Mathematics 0580

20 A gardener plants seeds from a packet of 25 seeds. For 14 of the seeds will give red flowers and 11 will give yellow flowers. Examiner's The gardener chooses two seeds at random. Use (a) Write the missing probabilities on the tree diagram below. First seed Second seed 13 24 Red 14 Red 25 … Yellow … Red 11 Yellow 25 … Yellow [2] (b) What is the probability that the gardener chooses two seeds which will give (i) two red flowers, Answer(b)(i) [2] (ii) two flowers of a different colour? Answer(b)(ii) [2]

6 marks

Mark scheme: 20 11 14 10 2 B1 any 2 correct and ISW (a) 24 24 24 91 (b) (i) o.e (= 0.303) 2* M1 14 x 13 only 300 25 24 77 (ii) o.e (= 0.513) 2* M1 for adding their R x Y and 150 Y x R probabilities

This question in 0580/21 Oct/Nov 2004

21 In this question, give all your answers as fractions. For Examiner's A box contains 3 red pencils, 2 blue pencils and 4 green pencils. Use Raj chooses 2 pencils at random, without replacement. Calculate the probability that (a) they are both red, Answer(a) [2] (b) they are both the same colour, Answer(b) [3] (c) exactly one of the two pencils is green. Answer(c) [3]

8 marks

Mark scheme: 1 3 2 21 (a) 2 M1 × 12 3 + 2 + 4 (their 9 ) − 1 5 4 × 3 2 (× 1) (b) 3 M2 their(a) + + 18 their 72 their 72 4 × 3 2 (× 1) or M1 or their 72 their 72 5 4 5 (c) 3 M2 2 × × 9 3 + 2 + 4 (their 9 ) − 1 4 5 or M1 × 3 + 2 + 4 (their 9 ) − 1

This question in 0580/21 May/June 2012

18 If it rains today the probability that it will rain tomorrow is 0.4 . If it does not rain today the probability that it will rain tomorrow is 0.2 . On Sunday it rained. (a) Complete the tree diagram for Monday and Tuesday. Monday Tuesday Rain 0.4 Rain 0.4 0.6 No rain Rain … … No rain … No rain [2] (b) Find the probability that it rains on at least one of the two days shown in the tree diagram. Answer(b) … [3] __________________________________________________________________________________________

5 marks

Mark scheme: 18 (a) 6.0 2.0 8.0 in correct places 2 B1 for 0.6 in correct place B1 for 0.2 and 0.8 in correct places (b) 0.52 oe nfww 3 M2FT for 1 – (their 0.6 × their 0.8) oe or M1FT for a correct product from their tree in (a)

This question in 0580/21 Oct/Nov 2014

20 The table shows the probability that a person has blue, brown or green eyes. Eye colour Blue Brown Green Probability 0.4 0.5 0.1 Use the table to work out the probability that two people, chosen at random, (a) have blue eyes, Answer(a) … [2] (b) have different coloured eyes. Answer(b) … [4] __________________________________________________________________________________________

6 marks

Mark scheme: 20 (a) 0.16 oe 2 M1 for 4.0 × 4.0 If zero scored SC1 for fully correct evaluated method involving a without replacement method (b) 0.58 oe 4 M3 for 1 − ( 4.0 2 + 5.0 2 + 1.02 ) oe or M2 for 4.0 2 + 5.0 2 + 1.0 2 ALT method M3 for 4.0 × ( 5.0 + )1.0 + 5.0 × ( 4.0 + )1.0 + 1.0 × ( 4.0 + 5.0) oe or M2 for addition of any three of: 4.0 × ,5.0 4.0 × ,1.0 5.0 × ,4.0 0.5 × 0.1, 1.0 × 4.0 and 1.0 × 5.0 or M1 for addition of any two of: 4.0 × ,5.0 4.0 × ,1.0 5.0 × ,4.0 5.0 × ,1.0 1.0 × 4.0 and 1.0 × 5.0 If zero scored SC2 for fully correct evaluated method involving a without replacement method [ ]

This question in 0580/21 Oct/Nov 2015

23 A box contains 6 red pencils and 8 blue pencils. A pencil is chosen at random and not replaced. A second pencil is then chosen at random. (a) Complete the tree diagram. First pencil Second pencil Red … Red 6 14 8 13 Blue Red … … Blue … Blue [2] (b) Calculate the probability that (i) both pencils are red, Answer(b)(i) … [2] (ii) at least one of the pencils is red. Answer(b)(ii) … [3]

7 marks

Mark scheme: 8 523 (a) and 1 14 13 6 7 1 and 13 13 30 (b) (i) oe 2 M1FT for 6 × 5 182 14 their13 126 (ii) oe 3 M2FT for 182 8 7 1 − × 14 13 6 5 6 8 8 6 or × + + × 14 13 14 ×13 14 13 6 8 6 or + × oe 14 14 13 or M1FT for sum of any two of 6 5 6 8 8 6 × or × or 14 13 14 13 14 ×13

This question in 0580/22 Oct/Nov 2015

18 Samira takes part in two charity runs. The probability that she finishes each run is 0.8 . First run Second run 0.8 finishes finishes 0.8 0.2 does not finish 0.2 0.8 finishes does not finish 0.2 does not finish Find the probability that Samira finishes at least one run. Answer … [3] __________________________________________________________________________________________

3 marks

Mark scheme: 18 0.96 oe 3 M2 for 1 − 2.0 × 2.0 or 0.8 + 0.2 × 0.8 or 0.8 × 0.8 + 0.8 × 0.2 + 0.2 × 0.8 or B1 for one of 2.0 × 2.0 , 8.0 × ,8.0 8.0 × ,2.0 2.0 × 8.0 seen 18 k

This question in 0580/23 Oct/Nov 2015

21 Dan either walks or cycles to school. The probability that he cycles to school is 1 . 3 (a) Write down the probability that Dan walks to school. … [1] (b) When Dan cycles to school the probability that he is late is 1 . 8 When Dan walks to school the probability that he is late is 3 . 8 Complete the tree diagram. 1 Late 8 Cycles 1 3 Not late … Late 3 8 … Walks Not late … [2] (c) Calculate the probability that (i) Dan cycles to school and is late, … [2] (ii) Dan is not late. … [3]

8 marks

Mark scheme: 2 21 (a) oe 1 3 2 7 5 7 5 (b) their , , oe 2 B1 for either or 3 8 8 8 8 1 1 1 (c) (i) oe 2 M1 for × seen 24 3 8 17 1 7 2 5 (ii) oe 3 M2FT for × + × 24 3 8 3 8 1 7 2 5 or M1FT for × or × 3 8 3 8

This question in 0580/22 Feb/March 2016

19 The probability of a cricket team winning or losing in their first two matches is shown in the tree diagram. First match Second match 3 win 4 1 win 3 1 lose 4 3 win 4 2 3 lose 1 lose 4 Find the probability that the cricket team wins at least one match. … [3]

3 marks

Mark scheme: 5 2 1 1 2 3 19 oe 3 M2 for 1 − × or + × 6 3 4 3 3 4 1 3 1 1 2 3 or × + × + × 3 4 3 4 3 4 2 1 1 1 2 3 or M1 for × or × + × 3 4 3 4 3 4 6π

This question in 0580/21 May/June 2016

8 Simon has two boxes of cards. In one box, each card has one shape drawn on it that is either a triangle or a square. In the other box, each card is coloured either red or blue. Simon picks a card from each box at random. The probability of picking a triangle card is t. The probability of picking a red card is r. Complete the table for the cards that Simon picks, writing each probability in terms of r and t. Event Probability Triangle and red Square and red (1 - t) r Triangle and blue Square and blue [3]

3 marks

Mark scheme: 8 3 B1 for each rt (1 – t) r (1 − r)t oe (1 – r)(1 – t) oe

This question in 0580/21 May/June 2017

26 The probability that Pedro scores a goal in any match is . 5 Calculate the probability that Pedro scores a goal in each of the next two matches. … [2]

2 marks

Mark scheme: 6 4 2 2 2 oe M1 for × oe or denominator 52 oe 25 5 5

This question in 0580/23 May/June 2017

20 (a) A box contains 3 blue pens, 4 red pens and 8 green pens only. A pen is chosen at random from the box. Find the probability that this pen is green. … [1] (b) Another box contains 7 black pens and 8 orange pens only. Two pens are chosen at random from this box without replacement. Calculate the probability that at least one orange pen is chosen. … [3]

4 marks

Mark scheme: 20(a) 8 1 oe 15 20(b) 168 3 M2 for oe 210 7 6 7 × 8 1 − × oe or 3( ) oe 15 14 15 × 14 or M1 for 7 6 7 8 8 7 × or × or × oe 15 14 15 14 15 14

This question in 0580/21 May/June 2018

24 Box A and box B each contain blue and green pens only. Raphael picks a pen at random from box A and Paulo picks a pen at random from box B. 2 The probability that Raphael picks a blue pen is . 3 8 The probability that both Raphael and Paulo pick a blue pen is . 15 (a) Find the probability that Paulo picks a blue pen. … [2] (b) Find the probability that both Raphael and Paulo pick a green pen. … [3]

5 marks

Mark scheme: 24(a) 4 2 2 8 oe M1 for × p = or better 5 3 15 24(b) 1 3 4 1 oe 3FT (1 – their ) × correctly evaluated 15 5 3 4 2 M2 for (1 – their ) × (1 – ) oe 5 3 4 2 or M1 for 1 – their or 1 – 5 3

This question in 0580/22 May/June 2018

22 A group of 200 people were asked which city they would like to visit next. The table shows the results. City London Paris New York Tokyo Number of people 50 48 56 46 (a) A person from the group is chosen at random. Write down the probability that this person would like to visit either Paris or Tokyo next. … [2] (b) Two people are chosen at random from the group of 200. Find the probability that one person would like to visit London next and the other person would like to visit New York next. Give your answer as a percentage. … % [3]

5 marks

Mark scheme: 22(a) 94 2 46 48 oe M1 for + oe 200 200 200 22(b) 14.1 or 14.07… 3  50 56  M2 for 2  ×  oe  200 199  50 56 or M1 for × oe 200 199

This question in 0580/21 Oct/Nov 2018

11 1 2 3 4 5 The diagram shows five cards. Two of the cards are taken at random, without replacement. Find the probability that both cards show an even number. … [2]

2 marks

Mark scheme: 11 2 2 2 1 oe M1 for × oe 20 5 4

This question in 0580/21 May/June 2019

920 The probability that the school bus is late is . 10 15 If the school bus is late, the probability that Seb travels on the bus is . 16 3 If the school bus is on time, the probability that Seb travels on the bus is . 4 Find the probability that Seb travels on the bus. … [3]

3 marks

Mark scheme: 20 147 3 1 3 9 15 oe M2 for × + × 160 10 4 10 16 1 3 9 15 or M1 for × or × 10 4 10 16

This question in 0580/21 Oct/Nov 2019

18 Harris is taking a driving test. The probability that he passes the driving test at the first attempt is 0.6 . If he fails, the probability that he passes at any further attempt is 0.75 . Calculate the probability that Harris (a) passes the driving test at the second attempt, … [2] (b) takes no more than three attempts to pass the driving test. … [2]

4 marks

Mark scheme: 18(a) 0.3 oe 2 M1 for 0.4 × 0.75 18(b) 0.975 oe 2 M1 for 1 – 0.4 × 0.25 × 0.25 oe or 0.6 + 0.4 × 0.75 + 0.4 × 0.25 × 0.75 or 0.6 + their (a) + 0.4 × 0.25 × 0.75

This question in 0580/22 Oct/Nov 2019

17 A bag contains 3 blue buttons, 8 white buttons and 5 red buttons. Two buttons are picked at random from the bag, without replacement. Work out the probability that the two buttons are either both red or both white. … [3]

3 marks

Mark scheme: 17 19 3 8 7 5 4 oe M2 for × + × 60 16 15 16 15 8 7 5 4 or M1 for × or × 16 15 16 15 89 If 0 scored SC1 for oe 256

This question in 0580/21 May/June 2021

26 Malik goes to a shop every day to buy bread. On any day, the probability that Malik goes to the shop in the morning is 0.7 . If he goes in the morning, the probability that there is bread for Malik to buy is 0.95 . If he goes later, the probability that there is bread for Malik to buy is 0.6 . Calculate the probability that, on any day, there is bread for Malik to buy. … [3]

3 marks

Mark scheme: 26 0.845 oe 3 M2 for 0.7 × 0.95 + (1 – 0.7) × 0.6 oe or M1 for one of these products

This question in 0580/23 May/June 2021

16 Sachin picks a number at random from the first three multiples of 3. He then picks a number at random from the first three prime numbers. He adds the two numbers to find a score. (a) Complete the table. Multiples of 3 3 9 2 5 11 Prime 3 6 numbers [2] (b) Given that the score is even, find the probability that one of the numbers he picks is 9. … [2]

4 marks

Mark scheme: 16(a) Multiples of 3 2 B1 for at least 4 correct entries + 3 6 9 2 5 8 11 Prime numbers 3 6 9 12 5 8 11 14 16(b) 2 2 their 2 oe B2FT for 5 their 5 their 2 or B1FT for k is any integer in the k range 1 ⩽ k ⩽ 7 c or c is 0, 1 or 2 their 5

This question in 0580/22 Oct/Nov 2021

24 The probability of Jamie hitting a target is 1. 3 64 The probability that he hits the target for the first time on his nth attempt is . 2187 Find the value of n. n = … [2] Question 25 is printed on the next page.

2 marks

Mark scheme: 24 7 2 B1 for answer 6  2 k 1 or M1 for   shown with k > 1  3 3 an + b  2  1 64 or = oe     3  3 2187 or for 3n = 2187 soi or 2n – 1 = 64 or 3n – 1 = 729 or better

This question in 0580/22 Feb/March 2023

15 A bag contains 5 green buttons, 2 blue buttons and 6 white buttons. Maya takes two buttons at random from the bag, without replacement. Calculate the probability that one button is green and the other button is not green. … [3]

3 marks

Mark scheme: 15 20 3 5 8 oe M2 for   2  oe 39 13 12 5 8 5 8 or M1 for or or or 13 12 12 13 80 If 0 scored SC1 for answer oe 169

This question in 0580/21 May/June 2023

23 Bag A and bag B each contain red sweets and yellow sweets. Anna picks a sweet at random from bag A. Ben picks a sweet at random from bag B. The probability that Anna picks a red sweet is 2. 5 1 The probability Anna and Ben both pick a yellow sweet is . 10 Find the probability that Anna and Ben both pick a red sweet. … [3]

3 marks

Mark scheme: 23 1 3  2  1 oe M1 for  1   × p = oe 3  5  10 2 M1 for  1 their p  where 0 < their p < 1 5

This question in 0580/22 May/June 2023

25 A bag contains 2 green buttons, 5 red buttons and 6 blue buttons. Two buttons are taken at random from the bag without replacement. Calculate the probability that the two buttons are different colours. … [4]

4 marks

Mark scheme: 25 2 4 2 11 5 8 6 7 oe nfww M3 for  +  +  oe 3 13 12 13 12 13 12  2 1 5 4 6 5  or 1 −   +  +   oe  13 12 13 12 13 12  or M2 for sum of three or more correct product pairs and no incorrect pairs 2 1 5 4 6 5 or for  +  +  and no 13 12 13 12 13 12 other pairs j k or M1 for  13 12 104 If 0 scored SC1 for answer oe 169

This question in 0580/22 Feb/March 2024

22 Bag A and bag B each contain red counters and blue counters only. Stephan picks a counter at random from bag A and Jen picks a counter at random from bag B. The probability that Stephan picks a red counter is 0.4 . The probability that Stephan and Jen both pick a red counter is 0.25 . Find the probability that Stephan and Jen both pick a blue counter. … [4]

4 marks

Mark scheme: 22 0.225 oe 4  0.25 oe M3 for  1 –  1 – 0.4   0.4  OR 0.25 M2 for 0.4 or M1 for 0.4 × p = 0.25 oe M1 for (1 –theirP(Jen red))×(1 – 0.4) oe

This question in 0580/21 May/June 2024

16 The stem-and-leaf diagram shows the mass of each of 13 packets. 3 1 2 8 4 0 1 2 3 3 8 5 1 2 3 4 Key: 3 q1 represents 31 g (a) Work out the interquartile range. … g [3] (b) Two of these packets are chosen at random. Find the probability that the one packet has a mass of more than 50 g and the other packet has a mass of less than 50 g. … [3]

6 marks

Mark scheme: 16(a) 12.5 3 M2 for 51.5 – 39 oe OR B1 for [UQ =] 51.5 B1 for [LQ =] 39 OR M1 for k – c where 50.25 ⩽ k ⩽ 52 and 38 ⩽ c ⩽ 40 16(b) 6 3  4 9  oe M2 for  2      oe 13  13 12  or B1 for 4 9 9 4 and or and 13 12 13 12 k c or M1 for  13 12 where 0 < k < 13 and 0 < c < 12  4 9  If 0 scored, SC1 for 2     13 13 

This question in 0580/22 Feb/March 2025

20 Bag A Bag B Bag A contains 5 white balls and 3 black balls. Bag B contains 3 white balls and 1 black ball. (a) Two balls are picked at random from bag B without replacement. Find the probability that both balls are black. … [1] (b) The balls are replaced into bag B. Kyle picks a ball at random from each bag. (i) Complete the tree diagram. Bag A Bag B white … white 5 8 black … white … … black black … [2] (ii) Find the probability that the two balls are the same colour. … [3] (c) The balls are replaced into their bags. Jo picks a ball at random from bag A and places it into bag B. She then picks a ball at random from bag B. Find the probability that she picks a black ball from bag B. … [3]

9 marks

Mark scheme: 20(a) 0 1 20(b)(i) 2 3 B1 for 8 3 1 or for and correctly placed on the 4 4 same pair of branches 20(b)(ii) 18 3 5 3 3 1 oe M2 for  their + their  their oe 32 8 4 8 4 or M1 for one correct branch e.g. 5 3 3 1  their or their  their oe 8 4 8 4 20(c) 11 3 5 1 3 2 oe M2 for  +  oe 40 8 5 8 5 or M1 for one product of the form 5 k 3 k  or  oe (k = 1, 2, 3 or 4) 8 5 8 5

This question in 0580/21 Oct/Nov 2025