IGCSE · Mathematics 0580 · Topic

Probability: IGCSE Mathematics questions and answers

10 free IGCSE Mathematics Probability questions written in the style of Cambridge Mathematics papers, each with its answer and mark scheme. 5 of them are A* challenge questions, built to catch the mistakes strong students still make.

This topic covers Introduction to probability, Relative & expected frequency, Combined events & tree diagrams. Work through the questions below, then generate a full Probability paper with a fresh set every time.

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Probability sub-topics

Introduction to probabilityRelative & expected frequencyCombined events & tree diagrams

Probability practice questions

Question 1Structured · 7 marksA* challenge

A bag contains 5 red counters and 3 blue counters. Two counters are taken at random from the bag, without replacement.

(a)

Find the probability that the two counters are different colours.

[3]
(b)

Find the probability that at least one of the counters is red.

[2]
(c)

Given that the two counters are the same colour, find the probability that they are both red.

[2]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)1528
58 × 37 M1
× 2 or add the two orders M1
1528 A1
3
(b)2528
1 − 38 × 27 M1
2528 A1
2
(c)1013
2056 ÷ 2656 M1
1013 A1
2
Question 2Structured · 12 marksA* challenge

Three fair six-sided dice are thrown.
The three scores can sometimes be arranged in an order that makes an arithmetic sequence (a sequence with a constant difference, which may be zero), for example 5, 1, 3 → 1, 3, 5.

(a)

Find the probability that the three scores can be arranged to make an arithmetic sequence.

[4]
(b)

Find the probability that the scores, in the order the dice are numbered, already make an arithmetic sequence.

[3]
(c)

Given that the three scores can be arranged to make an arithmetic sequence, find the probability that the three scores are all different.

[2]
(d)

Five fair dice are thrown. Find the probability that the five scores can be arranged to make an arithmetic sequence.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)736
all three equal: 6 outcomes B1
distinct sets {1,2,3}, {2,3,4}, {3,4,5}, {4,5,6}, {1,3,5}, {2,4,6} B1
6 sets × 6 orders = 36 M1
42216 = 736 A1
4
(b)112
6 constant sequences B1
each of the 6 sets in increasing or decreasing order: 12 M1
18216 = 112 A1
3
(c)67
3642 M1
67 A1
2
(d)411296
only difference 0 or 1 is possible (difference 2 would need 9) M1
6 + 2 × 5! = 246 outcomes M1
2467776 = 411296 A1
3
Question 3Structured · 4 marksA* challenge

The probability that it rains on any day is 0.4. The weather on different days is independent.

(a)

Find the probability that it rains on exactly one of two days.

[2]
(b)

Find the probability that it rains on at least one of three days.

[2]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)0.48
2 × 0.4 × 0.6 M1
0.48 A1
2
(b)0.784
1 − 0.63 M1
0.784 A1
2
Question 4Structured · 6 marksA* challenge

A spinner can land on red, blue, green or yellow.

ColourRedBlueGreenYellow
Probability0.10.15x2x
Table 4.1
(a)

Find the value of x.

[2]
(b)

The spinner is spun 500 times. Find the expected number of times it lands on yellow.

[2]
(c)

The spinner is spun twice. Find the probability that it lands on red both times or on blue both times.

[2]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)0.25
0.1 + 0.15 + x + 2x = 1 M1
0.25 A1
2
(b)250
500 × 0.5 M1
250 A1
2
(c)0.0325
0.12 + 0.152 M1
0.0325 A1
2
Question 5Structured · 3 marks

A bag contains 7 red, 6 blue and 5 green counters. One counter is taken at random.

(a)

Find the probability that it is red.

[1]
(b)

Find the probability that it is not green.

[1]
(c)

Find the probability that it is yellow.

[1]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)718
718 B1
1
(b)1318
1318 B1
1
(c)0
0 B1
1
Question 6Structured · 7 marks

A bag contains 7 red counters and 3 blue counters.
Amira takes a counter at random, notes its colour and does not replace it. She then takes a second counter at random.

First counterSecond counter710Red69Red39Blue310Blue......Red......Blue
Fig. 6.1
(a)

Complete the tree diagram.

[2]
(b)

Find the probability that both counters are red.

[2]
(c)

Find the probability that the counters are different colours.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)Lower branches: 7/9 and 2/9
7/9 B1
2/9 B1
2
(b)715
710 × 69 M1
715 A1
2
(c)715
710 × 39 M1
310 × 79 added M1
715 A1
3
Question 7Structured · 3 marks

The table shows the probabilities of a biased spinner landing on each colour.

ColourRedBlueGreenYellow
Probability0.200.250.35x
Table 7.1
(a)

Find x.

[2]
(b)

The spinner is spun 200 times. Find the expected number of times it lands on blue.

[1]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)0.20
1 − sum M1
0.20 A1
2
(b)50
50 B1
1
Question 8Structured · 3 marks

A drawing pin is dropped 100 times. It lands point up 32 times.

(a)

Find the relative frequency of landing point up.

[1]
(b)

The pin is dropped 600 more times. Estimate how many times it lands point up.

[1]
(c)

Explain how the estimate could be made more reliable.

[1]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)825
825 B1
1
(b)192
192 B1
1
(c)more trials
increase the number of trials B1
1
Question 9Structured · 5 marks

A box contains 5 red sweets and 3 green sweets. Sam takes a sweet at random, eats it, then takes another.

(a)

Draw a tree diagram and find the probability that both sweets are red.

[3]
(b)

Find the probability that the sweets are different colours.

[2]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)514
tree with correct branches B1
58 × 47 M1
514 A1
3
(b)1528
2 × 58 × 37 M1
1528 A1
2
Question 10Structured · 9 marksA* challenge

A bag contains 4 red counters and n blue counters. Two counters are taken at random, without replacement.
The probability that the two counters are the same colour is 49.

(a)

Form an equation in n.

[3]
(b)

Solve the equation to find n.

[3]
(c)

The bag is returned to its original contents. Three counters are now taken without replacement. Find the probability that exactly two of them are red.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)9(12 + n2 − n) = 4(n + 4)(n + 3) or equivalent
4n + 4 × 3n + 3 + nn + 4 × n − 1n + 3 M1
combined over the common denominator M1
equated and cross-multiplied correctly A1
3
(b)n = 5
expanded to a quadratic = 0 M1
factorise / formula M1
n = 5 (reject the other root) A1
3
(c)514
49 × 38 × 57 M1
× 3 for the arrangements M1
514 A1
3

Other IGCSE Mathematics topics

NumberAlgebra & graphsCoordinate geometryGeometryMensurationTrigonometryTransformations & vectorsStatistics

Frequently asked questions

What is in IGCSE Mathematics Probability?

In the 0580 syllabus, Probability covers Introduction to probability, Relative & expected frequency, Combined events & tree diagrams.

Where can I get Probability questions with mark schemes?

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Original Cambridge-style practice questions written for mentorriyaz.in. These are not Cambridge past paper questions, and this site is not affiliated with or endorsed by Cambridge University Press & Assessment. Updated 2026-11-04. Paper generator · mentorriyaz.in