What you'll learn
This topic forms the foundation of probability in CIE IGCSE Mathematics, covering how to calculate the likelihood of single events occurring, identify all possible outcomes in a sample space, and recognise when outcomes are equally likely. These concepts appear regularly in Paper 2 and Extended Paper 4, typically worth 2-4 marks per question, and provide essential groundwork for more complex probability questions involving combined events and tree diagrams.
Key terms and definitions
Probability — A numerical measure of how likely an event is to occur, expressed as a fraction, decimal or percentage between 0 and 1 (or 0% and 100%).
Event — A specific outcome or collection of outcomes from a probability experiment, such as rolling a 4 on a die or drawing a red card.
Sample space — The complete set of all possible outcomes in a probability experiment, often denoted by the symbol S or Ω.
Equally likely outcomes — Outcomes that have the same probability of occurring, such as each face of a fair die or each card in a shuffled deck.
Fair — Describing an object or experiment where all outcomes are equally likely (e.g., a fair coin, fair die, unbiased spinner).
Random — A selection process where every outcome in the sample space has an equal chance of being chosen, with no predictable pattern.
Mutually exclusive — Events that cannot occur simultaneously; if one happens, the other cannot (e.g., rolling a 3 or a 5 on a single die roll).
Complementary events — Two events that cover all possible outcomes between them, where P(event) + P(not event) = 1.
Core concepts
The probability scale
Probability values always lie between 0 and 1 inclusive:
- P = 0 means the event is impossible (cannot happen)
- P = 0.5 means the event is as likely as not (50% chance)
- P = 1 means the event is certain (must happen)
CIE IGCSE Mathematics examiners frequently test whether students can place events correctly on a probability scale. Questions may ask you to mark specific probabilities on a line from 0 to 1, or to compare the likelihood of different events.
Common positions on the scale:
- Impossible: P = 0
- Unlikely: 0 < P < 0.5
- Even chance: P = 0.5
- Likely: 0.5 < P < 1
- Certain: P = 1
Calculating basic probability
For a single event where all outcomes are equally likely, the fundamental formula is:
P(event) = Number of favourable outcomes / Total number of possible outcomes
This can also be written as:
P(A) = n(A) / n(S)
where n(A) represents the number of outcomes in event A, and n(S) represents the total number of outcomes in the sample space.
Steps to calculate probability:
- Identify the sample space and list all possible outcomes
- Count the total number of outcomes n(S)
- Identify which outcomes satisfy the event you're interested in
- Count the number of favourable outcomes n(A)
- Divide favourable by total and simplify if possible
The answer can be left as a fraction (usually required in exam questions), or converted to a decimal or percentage if the question specifies.
Sample spaces and listing outcomes
A sample space contains every possible outcome that could occur in an experiment. CIE IGCSE Mathematics questions regularly require you to write out sample spaces systematically.
Single die roll: S = {1, 2, 3, 4, 5, 6} Total outcomes n(S) = 6
Flipping a coin: S = {H, T} where H = heads, T = tails Total outcomes n(S) = 2
Spinner with 5 equal sections numbered 1, 1, 2, 3, 4: S = {1, 1, 2, 3, 4} Total outcomes n(S) = 5 Note: The outcome 1 appears twice because there are two sections marked 1
Drawing a card from a standard deck: S contains 52 cards: 13 cards in each of 4 suits (hearts, diamonds, clubs, spades) Total outcomes n(S) = 52
When listing sample spaces systematically, use a logical order (numerical, alphabetical, or grouped by category) to ensure you don't miss any outcomes. For two-stage events, methods like tables or tree diagrams help identify all combinations, though these fall under combined events.
Equally likely vs. non-equally likely outcomes
Equally likely outcomes have identical probabilities. This occurs when:
- A fair die is rolled (each number has probability 1/6)
- A fair coin is flipped (heads and tails each have probability 1/2)
- A card is randomly selected from a shuffled deck (each card has probability 1/52)
- A name is randomly drawn from a hat containing n names (each has probability 1/n)
The basic probability formula P(event) = favourable/total only works when outcomes are equally likely.
Non-equally likely outcomes occur when some outcomes have higher probabilities than others:
- A biased coin that lands on heads 60% of the time
- A spinner with unequal sections (e.g., red section takes up half the spinner, blue takes up one quarter)
- A bag containing 3 red balls and 7 blue balls (outcomes "red" and "blue" are not equally likely)
For non-equally likely outcomes at the level tested, you must:
- Count the individual items (e.g., 3 red balls and 7 blue balls means 10 items total, P(red) = 3/10)
- Use given probabilities directly
- Calculate areas or angles for geometric probability (spinners, target boards)
Complementary probability
The probability of an event NOT happening is called the complement of that event, often written as P(A') or P(not A).
Key rule: P(A) + P(not A) = 1
Therefore: P(not A) = 1 - P(A)
This relationship is extremely useful when it's easier to calculate the probability of an event not happening than happening directly.
Examples:
- If P(rain) = 0.3, then P(no rain) = 1 - 0.3 = 0.7
- If P(rolling a 6) = 1/6, then P(not rolling a 6) = 1 - 1/6 = 5/6
- If P(winning) = 2/5, then P(not winning) = 1 - 2/5 = 3/5
CIE examiners regularly set questions requiring you to find complementary probabilities, particularly in context (e.g., "What is the probability the train is not late?").
Expressing probability in different forms
CIE IGCSE Mathematics questions may require probability expressed as:
Fractions — The most common requirement. Always simplify fully unless the question states otherwise.
- P(even number on a die) = 3/6 = 1/2
Decimals — Convert the fraction by division. Give to the specified number of decimal places.
- P(even number on a die) = 1/2 = 0.5
Percentages — Multiply the decimal by 100 and add the % symbol.
- P(even number on a die) = 0.5 × 100 = 50%
Read the question carefully to determine which form is required. If not specified, fractions are safest and generally expected at IGCSE level.
Probability with simple contexts
CIE IGCSE Mathematics typically presents probability questions in realistic contexts:
Bags of coloured counters/balls: "A bag contains 5 red, 3 blue and 2 green counters. One counter is selected at random."
- Sample space has 10 items total
- P(red) = 5/10 = 1/2
- P(blue) = 3/10
- P(green) = 2/10 = 1/5
Spinners: "A fair spinner has 8 equal sections: 3 red, 3 blue, 1 yellow, 1 green."
- Sample space has 8 equally likely outcomes
- P(red) = 3/8
- P(not red) = 5/8
Playing cards: "A card is chosen at random from a standard deck of 52 cards."
- Remember: 4 suits, 13 cards per suit
- P(heart) = 13/52 = 1/4
- P(king) = 4/52 = 1/13
- P(red card) = 26/52 = 1/2
Random selection from a group: "A teacher randomly selects one student from a class of 30 students, where 18 are girls."
- P(girl) = 18/30 = 3/5
- P(boy) = 12/30 = 2/5
Worked examples
Example 1: Basic probability with a die
Question: A fair six-sided die is rolled once. (a) Write down the probability of rolling a 4. [1 mark] (b) Find the probability of rolling an even number. [2 marks] (c) Calculate the probability of not rolling a 1. [2 marks]
Solution:
(a) Sample space S = {1, 2, 3, 4, 5, 6}, so n(S) = 6 Only one favourable outcome (rolling a 4) P(rolling a 4) = 1/6 ✓
(b) Even numbers are 2, 4, and 6 Number of favourable outcomes = 3 ✓ P(even number) = 3/6 = 1/2 ✓
(c) Method 1 (complementary probability): P(not rolling a 1) = 1 - P(rolling a 1) ✓ = 1 - 1/6 = 5/6 ✓
Method 2 (direct counting):
Favourable outcomes are {2, 3, 4, 5, 6} = 5 outcomes ✓
P(not rolling a 1) = 5/6 ✓
Example 2: Probability with coloured balls
Question: A bag contains 4 red balls, 5 blue balls and 3 yellow balls. One ball is chosen at random. (a) How many balls are in the bag? [1 mark] (b) Find the probability that the ball is blue. [2 marks] (c) Find the probability that the ball is not yellow. [2 marks]
Solution:
(a) Total balls = 4 + 5 + 3 = 12 ✓
(b) Number of blue balls = 5 ✓ P(blue) = 5/12 ✓
(c) P(not yellow) = 1 - P(yellow) ✓ P(yellow) = 3/12 = 1/4 P(not yellow) = 1 - 1/4 = 3/4 ✓
Alternative: blue or red = 4 + 5 = 9 balls
P(not yellow) = 9/12 = 3/4 ✓
Example 3: Probability in context
Question: The table shows the number of students in a class who study different languages.
| Language | Number of students |
|---|---|
| French | 12 |
| Spanish | 8 |
| German | 5 |
One student is chosen at random from the class. (a) Find the probability that the student studies Spanish. [2 marks] (b) Find the probability that the student does not study German. [2 marks]
Solution:
(a) Total number of students = 12 + 8 + 5 = 25 ✓ P(Spanish) = 8/25 ✓
(b) P(not German) = 1 - P(German) ✓ P(German) = 5/25 = 1/5 P(not German) = 1 - 1/5 = 4/5 ✓
Or: Students not studying German = 12 + 8 = 20
P(not German) = 20/25 = 4/5 ✓
Common mistakes and how to avoid them
• Confusing probability with number of outcomes — Students write "3" instead of "3/6" when finding the probability of rolling an even number. Always express probability as a fraction, decimal or percentage, not as a simple count. The answer must be between 0 and 1.
• Not simplifying fractions — Leaving answers as 6/12 instead of 1/2 loses marks in CIE papers. Always simplify fractions to their lowest terms unless the question explicitly states otherwise. Divide numerator and denominator by their highest common factor.
• Using the wrong total — When finding P(not red), students sometimes use only the number of non-red items as the denominator rather than all items. The denominator must always be the total number of outcomes in the sample space, regardless of which event you're calculating.
• Forgetting complementary probability sums to 1 — When checking answers, verify that P(A) + P(not A) = 1. If you get P(red) = 3/10 and P(not red) = 6/10, you've made an error because 3/10 + 6/10 ≠ 1.
• Applying equally likely formula to non-equally likely situations — If a spinner has a large red section and a small blue section, you cannot say P(red) = 1/2 just because there are two colours. Only use the favourable/total formula when outcomes are genuinely equally likely. For spinners, consider angles or areas.
• Misreading the question — Questions asking for "the probability of choosing a red or blue counter" require you to add the probabilities, but this extends beyond single events. For single event questions, carefully identify exactly which outcome or set of outcomes is required.
Exam technique for basic probability
• Command words matter — "Write down" (1 mark) means the answer should be obvious or require minimal working; "Find" or "Calculate" (2 marks) requires clear working shown. Always write out the formula P = favourable/total even for simple questions.
• Show your method — Even when using a calculator, write the fraction before the simplified answer: "P(red) = 6/18 = 1/3". This ensures method marks if you make an arithmetic error. CIE mark schemes frequently award M1 for correct method, A1 for correct answer.
• State the sample space size — Writing "Total = 12" or "n(S) = 12" as a first step demonstrates clear understanding and helps you avoid denominator errors. This is especially valuable in multi-part questions where the total is used repeatedly.
• Check probabilities are valid — Before writing your final answer, verify it lies between 0 and 1. If you calculate P(blue) = 7/5, you've made an error because probabilities cannot exceed 1. Similarly, negative probabilities are impossible.
Quick revision summary
Probability measures how likely an event is, expressed as a value between 0 (impossible) and 1 (certain). For equally likely outcomes, P(event) = number of favourable outcomes / total number of possible outcomes. The sample space contains all possible outcomes. Complementary events satisfy P(A) + P(not A) = 1. Always simplify fractions and show working by writing the unsimplified fraction first. Check your denominator is the total number of outcomes in the sample space, and verify your probability lies between 0 and 1.