IGCSE
Cambridge (CIE)
Maths
30 questions
HARD U2-3 math Worksheet prob
Questions
Scan here for more resources
or visit savemyexams.com
Total Marks
/141
© 2026 Save My Exams, Ltd.
Get more and ace your exams at savemyexams.com
1
1 (a) The times taken for some students to complete a test are shown in the following table.
Time, t minutes
Frequency
0 ≤ t < 20
20 ≤ t < 35
35 ≤ t < 45
45 ≤ t < 50
7
18
36
x
A histogram is drawn to show the information in the frequency table.
The height of the bar representing the interval 20 ≤ t < 35 is 8.4 cm.
Calculate the height of the bar representing the interval 0 ≤ t < 20 .
............................................ cm
(2 marks)
(b) The height of the bar representing the interval 45 ≤ t < 50 is 1.4 cm.
Find the value of x .
(2 marks)
2 The manager at a doctor's surgery is monitoring the time that patients spend on the phone waiting to speak to
the receptionist. She takes a sample of 120 patients and draws a cumulative frequency diagram to show their
© 2026 Save My Exams, Ltd.
Get more and ace your exams at savemyexams.com
2
waiting times in seconds.
(i) Find an estimate of the median.
....................................... seconds [1]
(ii) Find an estimate of the interquartile range.
........................................seconds [2]
(iii) Estimate the percentage of patients that waited longer than a minute.
[3]
(6 marks)
3 Students in a class are asked if they like apples and if they like bananas.
The probability a student likes apples but not bananas is 0.2
© 2026 Save My Exams, Ltd.
Get more and ace your exams at savemyexams.com
3
The probability a student likes neither apples nor bananas is 0.05
The probability a student likes bananas but not apples is four times the probability a student likes both apples
and bananas.
Find the probability that a student likes both apples and bananas.
(3 marks)
4 (a) The histogram shows the distribution of the amounts spent by students on revision materials in a year.
How many students spent $40 or less?
© 2026 Save My Exams, Ltd.
Get more and ace your exams at savemyexams.com
4
(2 marks)
(b) Estimate the number of students who spent at least $55.
(2 marks)
5
The Venn diagram above shows information about the number of sports club members who took part in
various activities on an outing: Archery (A), Badminton (B) and Canoeing (C).
Two members are chosen at random from those who took part in Canoeing.
Calculate the probability that they both also took part in Archery.
(3 marks)
6
The diagram shows six discs. Each disc has a colour and a number.
Two of the six discs are picked at random without replacement. Find the probability that
© 2026 Save My Exams, Ltd.
Get more and ace your exams at savemyexams.com
5
i) both discs have the number 3,
[2]
ii) both discs have the same colour.
[3]
(5 marks)
7 (a) The speeds of cars, s metres per second, are recorded as they pass a speed camera. The frequencies are
shown in the table.
Speed (s m/s)
Frequency
0 <s ≤8
8 < s ≤ 10
10 < s ≤ 16
12
36
x
On the grid, complete the histogram to show the information in the table.
(3 marks)
© 2026 Save My Exams, Ltd.
Get more and ace your exams at savemyexams.com
6
(b) Find the value of x .
(1 mark)
8 A bag contains 5 blue marbles and 2 green marbles.
Bryn picks one marble at random without replacement.
If this marble is not green, he picks another marble at random without replacement.
He continues until he picks a green marble.
Find the probability that he picks a green marble on his first, second or third attempt.
(4 marks)
9 (a) 40 children were asked if they have a computer or a phone or both.
The Venn diagram shows the results.
A child is chosen at random from the children who have a computer.
Write down the probability that this child also has a phone.
(1 mark)
© 2026 Save My Exams, Ltd.
Get more and ace your exams at savemyexams.com
7
(b) Complete the Venn diagram.
(2 marks)
10 Esme has a bag with 5 green counters and 4 red counters.
She takes three counters at random from the bag without replacement.
Work out the probability that the three counters are all the same colour.
(4 marks)
11 The heights, h metres, of the 120 boys in an athletics club are recorded.
The table shows information about the heights of the boys.
Height
1.3 < h ≤ 1.4 1.4 < h ≤ 1.5 1.5 < h ≤ 1.6
1.6 < h ≤ 1.7
1.7 < h ≤ 1.8 1.8 < h ≤ 1.9
(h
metres)
Frequency
7
18
30
24
27
14
Three boys are chosen at random from the club.
Calculate the probability that one of the boys has a height greater than 1.8 m and the other two boys each
have a height of 1.4 m or less.
© 2026 Save My Exams, Ltd.
Get more and ace your exams at savemyexams.com
8
(4 marks)
12 (a) The Venn diagram shows information about the number of elements in sets A , B and ℰ .
n(A ∪ B ) = 23 Find the value of x .
x = ................................................
(3 marks)
(b) An element is chosen at random from ℰ .
Find the probability that this element is in (A ∪ B )' .
(2 marks)
© 2026 Save My Exams, Ltd.
Get more and ace your exams at savemyexams.com
9
13 (a)
Bag A contains 3 black balls and 2 white balls.
Bag B contains 1 black ball and 3 white balls.
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]
(4 marks)
(b) The balls are returned to their original bags.
Three balls are taken at random from bag A, without replacement.
i) Find the probability that they are all black.
[2]
ii) Find the probability that they are all white.
[1]
© 2026 Save My Exams, Ltd.
Get more and ace your exams at savemyexams.com
10
(3 marks)
(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 marks)
14 200 students estimate the total area, A m2 , of the windows in the classroom.
The table shows their results.
Area (A m2)
20 < A ≤ 60
60 < A ≤ 100
100 < A ≤ 150
150 < A ≤ 250
Frequency
32
64
80
24
Two students are chosen at random from those students that estimated the area of the windows to be more
than 100 m2 .
Find the probability that one of the two students estimates the area to be greater than 150 m2 and the other
student estimates the area to be 150 m2 or less.
(3 marks)
15 (a) The cumulative frequency graphs show information about the times taken by 100 male runners and by 100
© 2026 Save My Exams, Ltd.
Get more and ace your exams at savemyexams.com
11
female runners to finish the London marathon.
A male runner is chosen at random.
Find an estimate for the probability that this runner took less than 4 hours to finish the London marathon.
(2 marks)
(b) Use medians and interquartile ranges to compare the distribution of the times taken by the male runners with
the distribution of the times taken by the female runners.
(4 marks)
© 2026 Save My Exams, Ltd.
Get more and ace your exams at savemyexams.com
12
16 The cumulative frequency graph gives information about the weights, in grams, of 90 bags of sweets.
Roberto sells the bags of sweets to raise money for charity.
Bags with a weight greater than d grams are labelled large bags and sold for 3.75 euros each bag.
The total amount of money he receives by selling all the large bags is 93.75 euros.
Find the value of d .
© 2026 Save My Exams, Ltd.
Get more and ace your exams at savemyexams.com
13
(3 marks)
17 (a) The cumulative frequency diagram shows information about the time taken, t seconds, for a group of girls to
© 2026 Save My Exams, Ltd.
Get more and ace your exams at savemyexams.com
14
each solve a maths problem.
Use the cumulative frequency diagram to find an estimate for
i) the median,
© 2026 Save My Exams, Ltd.
Get more and ace your exams at savemyexams.com
15
.................................................. s [1]
ii) the interquartile range,
.................................................. s [2]
iii) the 20th percentile,
.................................................. s [1]
iv) the number of girls who took more than 66 seconds to solve the problem.
[2]
(6 marks)
(b) i) Use the cumulative frequency diagram to complete the frequency table.
Time (t seconds)
0 < t ≤ 20
Frequency
6
20 < t ≤ 40
40 < t ≤ 60
60 < t ≤ 80
80 < t ≤ 100
4
[2]
ii) Calculate an estimate of the mean time.
.................................................. s [4]
© 2026 Save My Exams, Ltd.
Get more and ace your exams at savemyexams.com
16
(6 marks)
(c) A group of boys solved the same problem. The boys had a median time of 60 seconds, a lower quartile of 46
seconds and an upper quartile of 66 seconds.
i) Write down the percentage of boys with a time of 66 seconds or less.
................................................ % [1]
ii) Howard says
"The boys' times vary more than the girls' times."
Explain why Howard is incorrect.
[2]
(3 marks)
18 The height, h cm, of each of 120 plants is measured. The frequency table shows this information.
Height, h cm
0 < h ≤ 10
10 < h ≤ 20
20 < h ≤ 30
30 < h ≤ 50
Frequency
2
18
62
38
A histogram is drawn to show the information in the frequency table.
The height of the bar representing the interval 10 < h ≤ 20 is 7.2 cm.
Calculate the height of the bar representing the interval 30 < h ≤ 50 .
............................................ cm
(2 marks)
19 The table shows information about the times, t seconds, taken by each of 100 students to solve a puzzle.
© 2026 Save My Exams, Ltd.
Get more and ace your exams at savemyexams.com
17
Time (t seconds)
0 < t ≤ 10
10 < t ≤ 15
15 < t ≤ 20
20 < t ≤ 40
40 < t ≤ 75
9
18
22
30
21
20 < t ≤ 40
40 < t ≤ 75
Frequency
Emmanuel draws a histogram to show this information.
The table shows the heights, in cm, of some of the bars for this histogram.
Complete the table.
Time (t seconds)
0 < t ≤ 10
10 < t ≤ 15
15 < t ≤ 20
Height of bar (cm)
3.6
14. 4
17. 6
(3 marks)
20 The table gives some information about the birds Paula sees in her garden one day.
Bird
Frequency
Magpie
15
Thrush
10
Startling
20
Sparrow
27
© 2026 Save My Exams, Ltd.
Get more and ace your exams at savemyexams.com
18
Complete the accurate pie chart.
(3 marks)
21 (a) Here are the heights in centimetres of 20 men.
165
164
176
179
188
178
183
172
180
190
167
159
156
176
173
168
169
182
167
192
Show this information in an ordered stem and leaf diagram.
(3 marks)
(b) Work out the percentage of these men with a height greater than 184 cm.
(2 marks)
© 2026 Save My Exams, Ltd.
Get more and ace your exams at savemyexams.com
19
22 A student measures the height, h cm, of each of 400 plants.
The cumulative frequency diagram shows the results.
Use the diagram to find an estimate for
i) the median
.......................................... cm[1]
ii) the interquartile range
.......................................... cm[2]
iii) the 80th percentile
.......................................... cm[2]
iv) the number of plants with a height greater than 60cm.
[2]
(7 marks)
© 2026 Save My Exams, Ltd.
Get more and ace your exams at savemyexams.com
20
23 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.
Maria takes two packets at random without replacement.
Find the probability that she takes two packets of different flavoured chips.
(4 marks)
24 (a) Two ordinary fair dice are rolled.
Complete the tree diagram.
(1 mark)
(b) Work out the probability that both dice land on a number less than 3.
(1 mark)
© 2026 Save My Exams, Ltd.
Get more and ace your exams at savemyexams.com
21
(c) Work out the probability that exactly one of the dice lands on a number less than 3.
(2 marks)
© 2026 Save My Exams, Ltd.
Get more and ace your exams at savemyexams.com
22
25 (a) A darts team is going to play a match on Saturday and on Sunday. The probability that the team will win on
Saturday is 0.45
If they win on Saturday, the probability that they will win on Sunday is 0.67 If they do not win on Saturday, the
probability that they will win on Sunday is 0.35
Complete the probability tree diagram.
(2 marks)
(b) Find the probability that the team will win exactly one of the two matches.
(3 marks)
26 Talika has a bag which contains 10 red balls and 8 blue balls. Talika takes three balls at random from her bag,
without replacement.
Calculate the probability that the three balls are the same colour.
© 2026 Save My Exams, Ltd.
Get more and ace your exams at savemyexams.com
23
(4 marks)
27 (a) Machine A and machine B make bottles.
The probability that a bottle made by machine A is faulty is 0.02 The probability that a bottle made by machine
B is faulty is 0.05
Complete the probability tree diagram.
(2 marks)
(b) Shazia takes at random one bottle made by machine A and one bottle made by machine B.
Work out the probability that at least one of these bottles is faulty.
(3 marks)
© 2026 Save My Exams, Ltd.
Get more and ace your exams at savemyexams.com
24
28 (a)
ℰ = {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.
Complete the Venn diagram to show this information.
(2 marks)
(b) Two students who play baseball are chosen at random.
Find the probability that they both also play football.
(3 marks)
© 2026 Save My Exams, Ltd.
Get more and ace your exams at savemyexams.com
25
29 (a) Items made at a factory have to pass two checks.
90% pass the first check.
The items that fail are scrapped. 99% of the items that pass the first check pass the second check.
The items that fail are scrapped.
Complete the tree diagram.
(2 marks)
(b) An item is chosen at random before the checks.
Work out the probability that the item is scrapped.
(3 marks)
30
Morgan picks two of these letters, at random, without replacement.
Find the probability that he picks two letters that are the same.
© 2026 Save My Exams, Ltd.
Get more and ace your exams at savemyexams.com
26
(3 marks)
© 2026 Save My Exams, Ltd.
Get more and ace your exams at savemyexams.com
27