Edexcel IGCSE

Probability

Probability measures how likely something is, on a scale from $0$ to $1$ . It is section 6.3 of the Edexcel International GCSE Mathematics A specification, and almost every question on it reduces to the same move: count how many outcomes there are altogether, count how many of them are the one you were asked about, and write the second over the first.

Edexcel IGCSE 4MA1Specification 6.372 approved practice questions

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Before you start

What you need to be able to do already

If any of these is shaky, fix it first — every worked example below leans on all three.

Write one quantity as a fraction of a total, and simplify itFractions
$18$ of the $30$ students in a class walk to school. What fraction is that, in its simplest form? If $\frac{3}{5}$ did not come quickly, fix fractions first: every worked example below ends in a simplification, and that is where the second mark sits.
Convert between a fraction, a decimal and a percentagePercentages
Write $\frac{2}{10}$ as a decimal, and then as a percentage. Questions here ask for whichever form they name, and a correct probability written in the wrong form still loses the accuracy mark.
Use ‘and’, ‘or’ and ‘not’ the way mathematics uses themSet language and notation
In a class, $10$ students play football, $14$ play cricket, and $6$ play both. How many play football or cricket? If you answered $24$ you counted those $6$ twice — it is $18$ . Every question below turns on reading one of these three words correctly.

The idea

What is probability?

The probability of an event is a number from $0$ to $1$ saying how likely that event is: $0$ means it cannot happen, $1$ means it is certain, and $0.5$ means it is exactly as likely to happen as not. When every outcome is equally likely — a fair dice, a counter taken at random from a bag — the probability of an event is just the fraction of the outcomes that make it happen. Nothing is memorised: you count, then you divide.

The notation, and how to read it

Every worked solution below is written in these five symbols.

SymbolReads asMeans
$P (A)$ the probability of $A$ A number from $0$ to $1$ giving how likely event $A$ is.
$P (A^{'})$ the probability of not $A$ The complement — every outcome in which $A$ does not happen. Always $P (A^{'}) = 1 - P (A)$ .
$P (A \cup B)$ the probability of $A$ or $B$ At least one of the two happens. If they are mutually exclusive this is $P (A) + P (B)$ ; otherwise subtract the overlap.
$P (A \cap B)$ the probability of $A$ and $B$ Both happen. If they are independent this is $P (A) \times P (B)$ .
$P (B ∣ A)$ the probability of $B$ given $A$ How likely $B$ is once you already know $A$ happened — the second branch of a without-replacement question.

The one idea underneath all of it

Every probability is a part written over a whole, and the whole is always $1$ . That one idea gives you the complement (what is left of the whole after $A$ : $P (A^{'}) = 1 - P (A)$ ), the addition rule for mutually exclusive events (separate parts of the same whole add), and the check that catches most slips: an answer above $1$ or below $0$ is not a probability, so the counting went wrong.

The smallest possible example

A bag holds three counters: two red and one blue. One counter is taken at random.

$P (red) = \frac{2}{3}$

There are three outcomes, and because the counter is taken at random all three are equally likely. Two of those three outcomes are red, so red occupies two of the three equal parts of the whole. Every other question on this page is this, with bigger numbers.

Key terms

The words a question will use

Exam wording is precise. These five carry the meaning a question depends on.

TermWhat it meansNotation
At random
Nobody is choosing and nothing is biased, so every outcome is equally likely. This phrase (or the word ‘fair’) is what licenses the counting method — without it, you cannot assume the outcomes are equally likely. $P = \frac{1}{n}$
Mutually exclusive
Two events that cannot both happen on the same trial: one roll of a dice cannot be both a $2$ and a $5$ . For these, and only these, you may add the probabilities. $P (A \cap B) = 0$
Independent
One event happening does not change how likely the other is — a coin has no memory of the last toss. For these, and only these, you may multiply the probabilities. $P (B ∣ A) = P (B)$
Without replacement
The first item is not put back, so for the second pick there is one fewer of that colour and one fewer in total. Both the top and the bottom of the fraction change; ‘with replacement’ means neither does. $\frac{r - 1}{n - 1}$
Relative frequency
A probability estimated from results that were actually collected, rather than from a fair, equally-likely model. It is what the words ‘estimate’ and ‘the results are shown’ signal, and it gets more reliable as the number of trials grows. $\frac{successes}{trials}$

The method

How to work out a probability

Five steps, in this order. The worked examples below point back to these numbers.

Count how many outcomes there are altogether.This is the denominator. It is only valid when the outcomes are equally likely, so find the word ‘fair’ or ‘at random’ in the question before you use it. In a bag question the total is often not given — $12$ red and $8$ blue means a total of $20$ .
Count how many of those outcomes make the event happen.This is the numerator. Write the sample space out first — ${1, 2, 3, 4, 5, 6}$ — so this is a count and not a guess. ‘Greater than $4$ ’ is two outcomes, not four.
Write the probability as the favourable count over the total count.This unsimplified fraction is the method mark. Put $\frac{8}{20}$ on the page before you cancel it, so the mark is already earned if the cancelling then goes wrong.
Simplify, or convert, into the form the question asked for.‘In its simplest form’ means fully cancelled: $\frac{8}{20} = \frac{2}{5}$ . If the question wants a decimal or a percentage, convert now — this is the accuracy mark, and it is the step most often skipped.
Check the answer lies between $0$ and $1$ .A probability cannot be $1.4$ , and it cannot be $75$ . If it is, you divided the wrong way round or subtracted from $100$ instead of $1$ . Two seconds here catches the most common lost mark on this topic.

Use it when

The question tells you the outcomes are equally likely: a fair dice or coin, a counter taken at random, a name drawn from a hat. Then the probability really is just a count of outcomes over a count of outcomes.

Do not use it when

The outcomes are not equally likely, or the question hands you results from trials that were actually carried out. ‘A counter is taken and replaced $50$ times; red came up $18$ times — estimate the probability of red’ is a relative-frequency question: the answer is $\frac{18}{50} = \frac{9}{25}$ , taken from the data, not from counting colours in the bag. The same applies to a biased dice or a drawing pin.

Worked examples

6 IGCSE Maths practice questions on probability, fully worked

One per objective, so the range is covered rather than the same question repeated. Every step is written out, and the answer is shown — this is study material, not a test.

Example 1: a single event

A bag contains 12 red counters and 8 blue counters. A counter is taken at random. What is the probability that it is blue? Give your answer as a fraction in its simplest form, e.g. $\frac{3}{10}$ . [2 marks]

Understand the language of probability (outcomes, equal likelihood, events, random)

Worked solution

Step 1: There are 8 blue counters out of a total of $12 + 8 = 20$ counters.
Step 2: Probability of blue = number of blue counters ÷ total number of counters = $\frac{8}{20}$ .
Step 3 of the method
Step 3: Simplify the fraction by dividing numerator and denominator by 4: $\frac{8}{20} = \frac{2}{5}$ .
Step 4 of the method

Answer

\frac{2}{5}

Example 2: the same method, answered as a decimal

A bag contains 8 red counters and 2 blue counters. You pick one counter at random. What is the probability of picking a blue counter? [1 mark]

Understand and use the probability scale (P(certainty) = 1, P(impossibility) = 0)

Worked solution

Step 1: There are 10 counters in total (8 red + 2 blue).
Step 1 of the method
Step 2: The probability of picking a blue counter is the number of blue counters divided by the total number of counters.
Step 3 of the method
Step 3: That is 2 blue counters ÷ 10 total counters = 0.2.
Step 4 of the method

Answer0.2

Example 3

There are 20 red marbles and 5 blue marbles in a bag. Liam picks one marble at random. What is the probability it is blue? Give your answer as a fraction in its simplest form, e.g. $\frac{3}{10}$ . [2 marks]

Understand and use estimates or measures of probability from theoretical models

Worked solution

Step 1: Count the total number of marbles: 20 red + 5 blue = 25 marbles.
Step 2: In probability, the chance of an event is the number of favourable outcomes divided by the total number of outcomes.
Step 3: The favourable outcome is picking a blue marble. There are 5 blue marbles.
Step 4: Write the probability as a fraction: $\frac{5}{25}$ .
Step 5: Simplify the fraction by dividing the numerator and denominator by 5: $\frac{5}{25} = \frac{1}{5}$ .

Answer

\frac{1}{5}

Show 3 more worked examples

Example 4

The Venn diagram shows the number of students who play football (F) and hockey (H). There are 30 students in total. 12 play football, 8 play hockey, and 5 play both. Work out the probability that a student chosen at random plays football but NOT hockey. [2 marks]

Find probabilities from a Venn diagram

Worked solution

Step 1: First, find how many students play football but not hockey. That's the students in set F who are not in set H.
Step 2: Total football players = 12, and those who play both = 5. So football only = $12 - 5 = 7$ .
Step 3: The total number of students is 30. So the probability is 7 out of 30.
Step 4: Write that as a fraction: $\frac{7}{30}$ .

Answer

\frac{7}{30}

Example 5

Work out the probability that a fair coin lands on heads and a fair six-sided dice rolls a number greater than 4. Give your answer as a fraction in its simplest form, e.g. $\frac{1}{6}$ . [2 marks]

Understand the concepts of a sample space and an event, and how the probability of an event happening can be determined from the sample space

Worked solution

Step 1: A fair coin has two equally likely outcomes: heads or tails. The probability of heads is 1 out of 2, written as $\frac{1}{2}$ .
Step 2: A fair die has six faces numbered 1 to 6. Numbers greater than 4 are 5 and 6 — that's 2 outcomes out of 6. Write the probability as $\frac{2}{6}$ and simplify by dividing numerator and denominator by 2 to get $\frac{1}{3}$ .
Step 3: These two events are independent — the coin does not affect the die. For independent events, multiply the probabilities: $\frac{1}{2} \times \frac{1}{3} = \frac{1}{6}$ .

Answer

\frac{1}{6}

Example 6

A fair coin and a standard six-sided die are thrown together. Calculate the probability of getting a head on the coin and rolling a number greater than 4 on the die. Give your answer as a fraction in its simplest form, e.g. $\frac{3}{10}$ . [3 marks]

Probability with and without replacement

Worked solution

Step 1: A coin has two equally likely outcomes: heads and tails. The probability of heads is $\frac{1}{2}$ .
Step 2: A die has six equally likely outcomes (1 to 6). The numbers greater than 4 are 5 and 6, so the probability is $\frac{2}{6}$ , which simplifies to $\frac{1}{3}$ .
Step 3: These events are independent, so multiply the probabilities: $(\frac{1}{2}) \times (\frac{1}{3}) = \frac{1}{6}$ .

Answer

\frac{1}{6}

Common mistakes

Where marks actually get lost

Learnly tags 20 distinct misconceptions on this subtopic. These are errors that show up in real student work — not generic revision advice.

Examiner traps

Forgetting to include 'neither'
What goes wrong: Students calculate the sum of the regions shown (e.g., $7 + 12 + 5 = 24$ ) and forget that the total must be 30, so they don't subtract from the total to find x.
What to do instead: Remember: total students = all regions including 'neither'. Add the given numbers, then subtract from the total to find the missing value.
Double counting the intersection
What goes wrong: Students add the numbers in the circles without accounting for the overlap, e.g., adding $7 + 12 + 5 + 12$ incorrectly.
What to do instead: In a Venn diagram, the numbers in each region are separate. Do not add the intersection twice. Add each region only once.
Getting a negative for 'neither'
What goes wrong: Students incorrectly add the regions and get a sum larger than the total, leading to a negative value for x.
What to do instead: If your sum of the given regions is more than the total, you have double-counted. Check that each region is counted once.
Misinterpreting 'only' and 'both'
What goes wrong: Students treat 'only football' as the total football players, ignoring the intersection, or treat 'both' as separate from the 'only' numbers.
What to do instead: 'Only football' means football but not rugby. 'Both' means both sports. Add all distinct regions: only F + only R + both + neither = total.
Treating without replacement as with replacement
What goes wrong: Students incorrectly assume the probability remains the same after the first draw, using $\frac{8}{14}$ for both picks instead of adjusting for the removed counter.
What to do instead: Remember: without replacement means the total decreases. After picking a red, there are 7 reds left and 13 total. Multiply $\frac{8}{14}$ by $\frac{7}{13}$ .
Ignoring order of selection
What goes wrong: Students may think that picking red then blue is different from blue then red, but for 'both red' order doesn't matter; they still multiply the probabilities correctly.
What to do instead: For 'both red', only one order works: first red then second red. So just multiply the probabilities for each draw without extra cases.

See 14 more examiner traps

Not simplifying the fraction
What goes wrong: Students leave the answer as an unsimplified fraction like $\frac{56}{182}$ instead of reducing to $\frac{4}{13}$ .
What to do instead: Always simplify your fraction. Divide numerator and denominator by the greatest common factor. For $\frac{56}{182}$ , divide by 14 to get $\frac{4}{13}$ .
Adding probabilities instead of multiplying
What goes wrong: Students add the probabilities of each event (e.g., $\frac{8}{14} + \frac{7}{13}$ ) instead of multiplying them for the combined event.
What to do instead: For both events to happen, multiply the probabilities. Adding is for 'or' situations. Here it's 'and', so multiply: $\frac{8}{14} \times \frac{7}{13}$ .
Forgetting to decrease denominator
What goes wrong: Students correctly reduce the numerator for the second draw but forget to reduce the denominator from 14 to 13.
What to do instead: After the first draw, the total number of counters decreases by one. So the second probability uses 13 as denominator, not 14.
Subtracting from 100 instead of 1
What goes wrong: Students incorrectly subtract the decimal from 100, e.g., $100 - 0.25 = 99.75$ , instead of subtracting from 1.
What to do instead: Remember probabilities are between 0 and 1. For complements, subtract the given probability from 1, not 100.
Adding instead of subtracting
What goes wrong: Students add the probability to itself or to 1, e.g., $0.25 + 0.25 = 0.5$ , instead of subtracting from 1.
What to do instead: The complement is what's left over. If probability is 0.25, the complement is 1 minus 0.25, not added.
Confusing decimals and fractions
What goes wrong: Students convert 0.25 to $\frac{1}{4}$ and then incorrectly calculate complement as $\frac{3}{4}$ but write as 0.34 or 0.75 as 75.
What to do instead: 0.25 as a fraction is $\frac{1}{4}$ , complement is $\frac{3}{4} = 0.75$ . Keep decimals: $1 - 0.25 = 0.75$ .
Ignoring decimal point
What goes wrong: Students treat 0.25 as 25 and compute $100 - 25 = 75$ , then write answer as 75 instead of 0.75.
What to do instead: Probabilities are decimals. 0.25 means 25 hundredths. Complement is $1 - 0.25 = 0.75$ , not 75.
Ignoring without replacement
What goes wrong: Students treat the second draw as independent with the same probability as the first, not adjusting for the reduced number of sweets.
What to do instead: When drawing without replacement, the total number of sweets decreases. Multiply probabilities: first green $(\frac{2}{5})$ , then green $(\frac{1}{4})$ .
Incorrect multiplication order
What goes wrong: Students multiply probabilities but use the wrong order or forget to multiply at all, e.g., adding instead of multiplying.
What to do instead: For 'both green', multiply the probability of first green by probability of second green given first was green.
Adding probabilities
What goes wrong: Students add probabilities of each event instead of multiplying them, thinking 'and' means add.
What to do instead: For two events both happening, multiply probabilities. Only add when it's 'or' for mutually exclusive events.
Denominator not decreased
What goes wrong: Students keep the denominator the same for the second draw, e.g., using 5 instead of 4.
What to do instead: After drawing one sweet, the total number decreases. For second draw, denominator is one less than first.
Omitting branches for second draw
What goes wrong: Students draw only one branch for the second draw instead of separate branches for each possible outcome, ignoring that probabilities change after the first pick.
What to do instead: Always draw two branches from each first outcome: one for red and one for blue. The second draw probabilities depend on what was taken first.
Incorrect denominator after first draw
What goes wrong: Students keep the total number of counters the same for the second draw instead of reducing it by one because the first counter is not replaced.
What to do instead: After taking one counter without replacement, the total number decreases by 1. For the second draw, use the new total.
Adding probabilities along branches
What goes wrong: Students add probabilities along the path instead of multiplying them to find the probability of a sequence of events.
What to do instead: To find the probability of a sequence, multiply the probabilities along the branches. Use addition only for different paths.

Exam technique

How the marks are actually awarded

Method marks are given for working a marker can follow. This is what to put on the page.

Where the marks are: Questions on this section are typically worth $1$ or $2$ marks. On a two-mark question the first mark is for the method — a correct fraction of favourable outcomes over total outcomes — and the second is for the accurate final answer in the form requested. Write the unsimplified fraction down: $\frac{18}{50}$ visible on the page earns the method mark even if the cancelling afterwards is wrong. A bare correct answer with no working scores full marks, but a bare wrong answer with no working scores nothing.
Command words to expect: Work outCalculateFind the probability thatWrite downList all the possible outcomesEstimateGive your answer as a fraction in its simplest formExplain why
Accuracy and rounding: A probability may be given as a fraction, a decimal or a percentage, and any of the three is accepted unless the question names one. It must not be given as a ratio or in ‘out of’ wording: $3 : 4$ and ‘3 out of 4’ are not accepted as probabilities even when $\frac{3}{4}$ would have been right. ‘In its simplest form’ means fully cancelled — $\frac{9}{25}$ , not $\frac{18}{50}$ . Where a decimal does not terminate, give $3$ significant figures unless told otherwise, and never round a probability up to $1$ or down to $0$ .
Calculator: A calculator is allowed in every paper of this qualification, on both Foundation and Higher tiers. That makes converting $\frac{18}{50}$ to $0.36$ a keystroke — but it will not tell you whether the question wanted a simplified fraction, so decide the required form before you type.

Check yourself

You should now be able to:

Place an event on the probability scale, and say what $0$ , $0.5$ and $1$ mean.
Work out the probability of a single event in a fair, equally-likely set-up, and give it as a fraction in its simplest form.
List the outcomes for one event, and for two successive events, systematically enough that none is missed or repeated.
Use $P (A^{'}) = 1 - P (A)$ for a ‘not’ question, subtracting from $1$ rather than from $100$ .
Read a two-set Venn diagram, including the region outside both circles, and find $P (A \cup B)$ without double-counting the overlap.
Add probabilities for mutually exclusive events, and multiply along the branches of a tree diagram for events in sequence.
Reduce the denominator for a second pick made without replacement, and leave it unchanged when the item is replaced.
Estimate a probability from collected data as a relative frequency, and say why more trials make that estimate more reliable.
Work out an expected frequency by multiplying a probability by the number of trials.

Specification coverage

The 14 things 4MA1 asks you to do

The assessable objectives for statistics and probability — probability, in the order the specification lists them.

Understand the language of probability (outcomes, equal likelihood, events, random)
Understand and use the probability scale (P(certainty) = 1, P(impossibility) = 0)
Understand and use estimates or measures of probability from theoretical models
Find probabilities from a Venn diagram
Understand the concepts of a sample space and an event, and how the probability of an event happening can be determined from the sample space
Probability with and without replacement

Show 8 more objectives

List all the outcomes for single events and for two successive events in a systematic way
Estimate probabilities from previously collected data
Calculate the probability of the complement of an event happening
Use the addition rule of probability for mutually exclusive events
Understand and use the term 'expected frequency'
Draw and use tree diagrams
Use simple conditional probability when combining events
Apply probability to simple problems

See who is stuck on probability before you teach it.

Create a class, share the 8-character code, and set probability as practice. Your students work through all 72 approved practice questions — not just the 6 on this page — and you see which objectives the class has not got yet. Free for individual students and teachers.

How Learnly works for teachers

Set up your class — free Practise on your own

What you need to be able to do already

Write one quantity as a fraction of a total, and simplify itFractions

Convert between a fraction, a decimal and a percentagePercentages

Use ‘and’, ‘or’ and ‘not’ the way mathematics uses themSet language and notation

What is probability?

The notation, and how to read it

The words a question will use

At random

Mutually exclusive

Independent

Without replacement

Relative frequency

How to work out a probability

6 IGCSE Maths practice questions on probability, fully worked

A bag contains 12 red counters and 8 blue counters. A counter is taken at random. What is the probability that it is blue? Give your answer as a fraction in its simplest form, e.g. 103​. [2 marks]

A bag contains 8 red counters and 2 blue counters. You pick one counter at random. What is the probability of picking a blue counter? [1 mark]

There are 20 red marbles and 5 blue marbles in a bag. Liam picks one marble at random. What is the probability it is blue? Give your answer as a fraction in its simplest form, e.g. 103​. [2 marks]

The Venn diagram shows the number of students who play football (F) and hockey (H). There are 30 students in total. 12 play football, 8 play hockey, and 5 play both. Work out the probability that a student chosen at random plays football but NOT hockey. [2 marks]

Work out the probability that a fair coin lands on heads and a fair six-sided dice rolls a number greater than 4. Give your answer as a fraction in its simplest form, e.g. 61​. [2 marks]

A fair coin and a standard six-sided die are thrown together. Calculate the probability of getting a head on the coin and rolling a number greater than 4 on the die. Give your answer as a fraction in its simplest form, e.g. 103​. [3 marks]

Where marks actually get lost

Forgetting to include 'neither'

Double counting the intersection

Getting a negative for 'neither'

Misinterpreting 'only' and 'both'

Treating without replacement as with replacement

Ignoring order of selection

Not simplifying the fraction

Adding probabilities instead of multiplying

Forgetting to decrease denominator

Subtracting from 100 instead of 1

Adding instead of subtracting

Confusing decimals and fractions

Ignoring decimal point

Ignoring without replacement

Incorrect multiplication order

Adding probabilities

Denominator not decreased

Omitting branches for second draw

Incorrect denominator after first draw

Adding probabilities along branches

How the marks are actually awarded

You should now be able to:

The 14 things 4MA1 asks you to do

See who is stuck on probability before you teach it.

A bag contains 12 red counters and 8 blue counters. A counter is taken at random. What is the probability that it is blue? Give your answer as a fraction in its simplest form, e.g. $\frac{3}{10}$ . [2 marks]

There are 20 red marbles and 5 blue marbles in a bag. Liam picks one marble at random. What is the probability it is blue? Give your answer as a fraction in its simplest form, e.g. $\frac{3}{10}$ . [2 marks]

Work out the probability that a fair coin lands on heads and a fair six-sided dice rolls a number greater than 4. Give your answer as a fraction in its simplest form, e.g. $\frac{1}{6}$ . [2 marks]

A fair coin and a standard six-sided die are thrown together. Calculate the probability of getting a head on the coin and rolling a number greater than 4 on the die. Give your answer as a fraction in its simplest form, e.g. $\frac{3}{10}$ . [3 marks]