What you'll learn
Probability forms a significant component of the Edexcel GCSE Mathematics specification, assessed in both Foundation and Higher tier papers. This topic examines how to calculate and interpret the likelihood of events occurring, from simple single events through to complex combined probability scenarios. Mastery of probability is essential for Paper 1, Paper 2, and Paper 3, with questions typically worth 2-6 marks appearing consistently across all exam series.
Key terms and definitions
Probability — a numerical measure between 0 and 1 (or 0% and 100%) that quantifies the likelihood of an event occurring, where 0 means impossible and 1 means certain.
Sample space — the set of all possible outcomes of a probability experiment or trial.
Mutually exclusive events — two or more events that cannot occur simultaneously; if one happens, the others cannot.
Independent events — two or more events where the occurrence of one event does not affect the probability of the other event(s) occurring.
Relative frequency — an estimate of probability based on experimental data, calculated as the number of times an event occurs divided by the total number of trials.
Tree diagram — a visual representation showing all possible outcomes of two or more events in sequence, with probabilities written on each branch.
Venn diagram — a diagram using overlapping circles to represent sets and their relationships, useful for calculating probabilities involving unions and intersections.
Conditional probability — the probability of an event occurring given that another event has already occurred.
Core concepts
Calculating basic probability
The fundamental probability formula appears in virtually every Edexcel GCSE Mathematics paper:
Probability of an event = Number of favourable outcomes ÷ Total number of possible outcomes
This formula applies when all outcomes are equally likely. The result is expressed as a fraction, decimal, or percentage depending on the question requirements.
Key points for basic probability calculations:
- All probabilities lie on a scale from 0 to 1 inclusive
- An impossible event has probability 0
- A certain event has probability 1
- Probabilities can be expressed as fractions in simplest form, decimals, or percentages
- The sum of probabilities of all possible outcomes in a sample space equals 1
Probability scales and language
Edexcel papers frequently test understanding of probability vocabulary and scales. Questions may ask students to:
- Mark probabilities on a number line from 0 to 1
- Match events to probability values
- Use appropriate terminology: impossible, unlikely, even chance, likely, certain
- Understand that "equally likely" means outcomes have the same probability
Relative frequency and experimental probability
Relative frequency provides an estimate of theoretical probability through experimentation. The formula is:
Relative frequency = Frequency of event ÷ Total number of trials
As the number of trials increases, relative frequency typically becomes a more reliable estimate of theoretical probability. Edexcel papers commonly present data from experiments and require students to:
- Calculate relative frequency from given data
- Use relative frequency to estimate the number of times an event will occur in future trials
- Compare theoretical and experimental probability
- Explain why relative frequency may differ from theoretical probability
Expected outcomes and frequency
When probability and the number of trials are known, calculate expected frequency:
Expected frequency = Probability × Number of trials
This calculation appears regularly in Edexcel GCSE Mathematics papers, often in contexts involving:
- Predicting outcomes from large samples
- Quality control scenarios in manufacturing
- Genetic inheritance problems (Higher tier)
- Game theory and fairness questions
Mutually exclusive events and addition rule
Events are mutually exclusive when they cannot happen at the same time. For example, when rolling a single die, getting a 3 and getting a 5 are mutually exclusive.
For mutually exclusive events A and B:
P(A or B) = P(A) + P(B)
This extends to multiple mutually exclusive events. A critical application is finding complementary probability:
P(not A) = 1 − P(A)
Edexcel papers test this through:
- Finding missing probabilities from partially completed tables
- Calculating "at least one" probabilities by finding the complement
- Determining whether events are mutually exclusive from context
Independent events and multiplication rule
Events are independent when the outcome of one does not affect the probability of the other. For example, flipping a coin and rolling a die are independent.
For independent events A and B:
P(A and B) = P(A) × P(B)
This multiplication rule extends to more than two independent events. Common contexts in Edexcel papers include:
- Multiple coin tosses
- Drawing with replacement from bags or containers
- Successive independent trials of any kind
Tree diagrams
Tree diagrams systematically display all outcomes for two or more events in sequence. Each branch represents a possible outcome, with its probability written on the branch.
Constructing and using tree diagrams:
- Draw branches for all outcomes of the first event, labelling each branch with its outcome and probability
- From each first-stage branch, draw branches for all outcomes of the second event
- Continue for additional events if required
- Probabilities on branches leaving the same point must sum to 1
- To find the probability of a complete path, multiply probabilities along that path
- To find the probability of multiple paths, add the probabilities of those paths
Tree diagrams appear in both Foundation and Higher tier papers, with replacement and without replacement scenarios. Without replacement changes probabilities on second and subsequent branches because the sample space reduces.
Two-way tables and sample space diagrams
Two-way tables organise outcomes systematically, particularly useful when two variables interact. Sample space diagrams show all possible outcomes, commonly used for:
- Two dice problems
- Selecting items from two groups
- Combined selections requiring enumeration
From these diagrams, calculate probabilities by identifying favourable outcomes within the total sample space.
Venn diagrams and set notation (Higher tier)
Higher tier papers use Venn diagrams to represent probabilities involving:
- Union (A ∪ B): elements in A or B or both
- Intersection (A ∩ B): elements in both A and B
- Complement (A'): elements not in A
Key formulae for Venn diagram probability:
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
This accounts for double-counting elements in the intersection. Edexcel Higher papers require students to:
- Complete Venn diagrams from given probability information
- Calculate probabilities of unions, intersections, and complements
- Determine whether events are independent using P(A ∩ B) = P(A) × P(B)
Conditional probability (Higher tier)
Conditional probability describes situations where the probability of an event depends on a previous outcome. Notation: P(B|A) means "probability of B given A has occurred."
From a tree diagram, conditional probability appears naturally on second and subsequent branches when sampling without replacement.
From a Venn diagram or two-way table:
P(B|A) = P(A ∩ B) ÷ P(A)
This represents the proportion of A outcomes that are also B outcomes.
Worked examples
Example 1: Basic probability with mutually exclusive events (Foundation/Higher)
A bag contains coloured counters: 5 red, 3 blue, and 2 green. A counter is selected at random.
(a) Find the probability the counter is red.
(b) Find the probability the counter is not blue.
Solution:
(a) Total counters = 5 + 3 + 2 = 10
P(red) = 5/10 = 1/2 [1 mark]
(b) Method 1: P(not blue) = 1 − P(blue) = 1 − 3/10 = 7/10 [2 marks: 1 for method, 1 for answer]
Method 2: P(not blue) = P(red or green) = 5/10 + 2/10 = 7/10
Example 2: Tree diagram without replacement (Higher)
A box contains 3 milk chocolates and 2 dark chocolates. Hannah selects two chocolates at random without replacement.
(a) Complete the probability tree diagram.
(b) Calculate the probability that Hannah selects two chocolates of the same type.
Solution:
(a) First selection:
- P(milk) = 3/5, P(dark) = 2/5
Second selection after milk first:
- P(milk|milk first) = 2/4 = 1/2
- P(dark|milk first) = 2/4 = 1/2
Second selection after dark first:
- P(milk|dark first) = 3/4
- P(dark|dark first) = 1/4 [2 marks for all correct probabilities]
(b) P(two same) = P(milk then milk) + P(dark then dark)
= (3/5 × 1/2) + (2/5 × 1/4)
= 3/10 + 2/20
= 6/20 + 2/20
= 8/20 = 2/5 [3 marks: 1 for identifying relevant paths, 1 for multiplication, 1 for addition and simplification]
Example 3: Relative frequency application (Foundation/Higher)
A biased coin is flipped 200 times. It lands on heads 130 times.
(a) Work out the relative frequency of heads.
(b) The coin is flipped 500 more times. Estimate how many times it will land on heads.
Solution:
(a) Relative frequency = 130/200 = 0.65 [1 mark]
(b) Expected heads = 0.65 × 500 = 325 [2 marks: 1 for method, 1 for answer]
Common mistakes and how to avoid them
Mistake: Adding probabilities when events occur together ("and" situations). Correction: Use multiplication for combined independent events: P(A and B) = P(A) × P(B). Addition is only for "or" with mutually exclusive events.
Mistake: Forgetting to adjust probabilities on tree diagram branches when sampling without replacement. Correction: Reduce the total and the relevant category count after each selection. If 3 red and 2 blue, and red is removed first, second branch has 2 red out of 4 total.
Mistake: Not simplifying fractions in final answers. Correction: Always express probability fractions in their simplest form unless the question specifies otherwise. Convert 6/10 to 3/5.
Mistake: Confusing mutually exclusive with independent. Correction: Mutually exclusive means events cannot both happen (non-overlapping). Independent means one event's occurrence doesn't affect the other's probability. Events cannot be both mutually exclusive and independent (except when one has probability zero).
Mistake: Finding only one path through a tree diagram when multiple paths satisfy the requirement. Correction: Identify all relevant paths before calculating. "At least one head" with two coins requires three paths: HH, HT, TH.
Mistake: Writing probabilities greater than 1 or negative. Correction: Check calculations immediately. If P > 1 or P < 0, an error has occurred. The probability scale is 0 ≤ P ≤ 1 always.
Exam technique for Probability
Command word "find" or "calculate": Show clear working for probability calculations. Write the formula structure, substitute values, and simplify. For 2-mark questions, method and answer typically each earn 1 mark.
Tree diagram questions: Draw diagrams neatly with clear labels. Write probabilities on every branch. Even if the diagram is partially given, check that probabilities from each point sum to 1. For "without replacement," clearly show how totals change on subsequent branches.
"Show that" questions: Present complete algebraic or numerical working that leads clearly to the given answer. The conclusion must be explicitly stated. These are method-heavy, with most marks for working.
Reading contexts carefully: Probability questions embed calculations in contexts (games, quality control, weather, genetics). Extract the mathematical structure: identify whether events are independent or dependent, whether sampling is with or without replacement, and what combination of events the question actually asks for ("and" requires multiplication; "or" with mutually exclusive requires addition).
Quick revision summary
Probability quantifies likelihood from 0 (impossible) to 1 (certain). Calculate using favourable outcomes divided by total outcomes when equally likely. For mutually exclusive events, add probabilities for "or" questions. For independent events, multiply probabilities for "and" questions. Tree diagrams show sequential events: multiply along paths, add between paths. Without replacement, adjust subsequent probabilities. Relative frequency estimates probability from experimental data. Expected frequency equals probability multiplied by number of trials. Higher tier includes conditional probability, Venn diagrams, and formal set notation.