Probability

What you'll learn

This revision guide covers all probability content in the Edexcel GCSE Mathematics specification, from Foundation to Higher tier. You'll learn how to calculate theoretical and experimental probabilities, use probability notation correctly, and apply advanced techniques including tree diagrams, Venn diagrams and conditional probability to solve multi-step problems.

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.

Sample space — the set of all possible outcomes of a trial or experiment.

Mutually exclusive events — events that cannot happen at the same time; if one occurs, the other cannot.

Independent events — events where the occurrence of one does not affect the probability of the other occurring.

Conditional probability — the probability of an event occurring given that another event has already occurred, denoted P(A|B).

Relative frequency — an estimate of probability based on experimental data, calculated as: number of times an event occurred ÷ total number of trials.

Expected frequency — the predicted number of times an event will occur, calculated as: probability × number of trials.

Exhaustive events — events that cover all possible outcomes in the sample space.

Core concepts

Basic probability calculations

Probability is calculated using the formula:

P(event) = number of favourable outcomes / total number of possible outcomes

All probabilities lie between 0 and 1 inclusive:

• P = 0 means the event is impossible
• P = 1 means the event is certain
• P = 0.5 means the event is equally likely to happen or not happen

The sum of probabilities of all possible outcomes in a sample space equals 1.

For any event A: P(A does not occur) = 1 - P(A occurs)

This is often written as P(A') = 1 - P(A), where A' represents "not A".

Example context: When rolling a fair six-sided die, each outcome (1, 2, 3, 4, 5, 6) has probability 1/6. The probability of not rolling a 6 is 1 - 1/6 = 5/6.

Listing outcomes systematically

For combined events, you must list all possible outcomes systematically to identify the sample space.

Two-way tables organise outcomes when two events occur together. For example, when flipping two coins:

	Head	Tail
Head	HH	HT
Tail	TH	TT

Sample space diagrams work similarly. When rolling two dice, create a grid showing all 36 possible outcomes.

List method — write outcomes systematically, for example when selecting two items from {A, B, C}: AA, AB, AC, BA, BB, BC, CA, CB, CC (if replacement) or AB, AC, BA, BC, CA, CB (if no replacement).

Combining independent events

When two or more independent events occur together, multiply their probabilities:

P(A and B) = P(A) × P(B)

This applies to:

• Events happening at the same time (flipping two coins)
• Events happening in sequence with replacement
• Naturally independent events (one person's result doesn't affect another's)

Example: The probability of rolling a 6 on a die AND flipping heads on a coin = 1/6 × 1/2 = 1/12.

For more than two independent events, continue multiplying: P(A and B and C) = P(A) × P(B) × P(C).

Mutually exclusive events and the addition rule

For mutually exclusive events that cannot occur simultaneously, add their probabilities:

P(A or B) = P(A) + P(B)

Example: When rolling a die, P(rolling a 2 or a 5) = 1/6 + 1/6 = 2/6 = 1/3.

This only works when events are mutually exclusive. You cannot roll both a 2 and a 5 on a single roll.

For events that are NOT mutually exclusive (can occur together), use:

P(A or B) = P(A) + P(B) - P(A and B)

You subtract P(A and B) to avoid counting the overlap twice.

Tree diagrams

Tree diagrams show all possible outcomes for successive events and calculate combined probabilities.

How to construct a tree diagram:

1. Draw branches for each outcome of the first event
2. Label each branch with its probability
3. From each first branch, draw branches for the second event
4. Label these with their probabilities
5. Continue for subsequent events if needed
6. Probabilities on each set of branches must sum to 1

How to use a tree diagram:

• Multiply along branches to find P(combined outcome)
• Add the probabilities of different routes to find P(at least one...) or P(exactly...)

Tree diagrams are particularly useful for:

• Events without replacement (probabilities change after first selection)
• Conditional probability problems
• "At least one" questions

Venn diagrams and probability

Venn diagrams use overlapping circles to represent events and their intersections.

Standard notation:

• ξ (xi) represents the universal set (sample space)
• ∩ means intersection (AND)
• ∪ means union (OR)
• A' means complement (NOT A)

For Venn diagrams with frequencies or numbers:

• Fill in the intersection first (A ∩ B)
• Work outwards to complete each region
• Calculate probabilities by dividing each region by the total

Example structure: If 50 students study Maths (M) and/or Physics (P):

• 35 study Maths
• 28 study Physics
• 18 study both

Work backwards: 18 in the intersection, then 35 - 18 = 17 study only Maths, 28 - 18 = 10 study only Physics.

P(M ∩ P) = 18/50, P(M ∪ P) = (17 + 18 + 10)/50 = 45/50.

Conditional probability

Conditional probability asks: "What is the probability of A happening, given that B has already happened?"

Written as P(A|B) and read as "probability of A given B".

Formula: P(A|B) = P(A ∩ B) / P(B)

On tree diagrams, conditional probability is shown by the second set of branches when probabilities change based on the first outcome.

Example: A bag contains 5 red and 3 blue counters. Two counters are selected without replacement.

P(second is red | first is red) = 4/7 (because one red has been removed, leaving 4 red and 3 blue).

When dealing with Venn diagrams, conditional probability restricts your sample space to just the event that has occurred.

Experimental probability and relative frequency

Relative frequency estimates probability from experimental data:

Relative frequency = number of successful trials / total number of trials

As the number of trials increases, relative frequency approaches the theoretical probability (Law of Large Numbers).

Expected frequency predicts future outcomes:

Expected frequency = probability × number of trials

Example: If P(winning) = 0.3 and you play 200 games, expected number of wins = 0.3 × 200 = 60.

Relative frequency is used when:

• Theoretical probability cannot be calculated
• Real-world data is available
• Testing whether a game/die is fair

Worked examples

Example 1: Tree diagram without replacement (Higher tier)

Question: A box contains 7 milk chocolates and 3 dark chocolates. Sarah picks two chocolates at random without replacement. Calculate the probability that she picks two chocolates of different types.

Solution:

First, draw a tree diagram:

First pick:

• P(milk) = 7/10
• P(dark) = 3/10

Second pick (probabilities change):

• After picking milk: P(milk) = 6/9, P(dark) = 3/9
• After picking dark: P(milk) = 7/9, P(dark) = 2/9

Different types means: (milk then dark) OR (dark then milk)

P(milk then dark) = 7/10 × 3/9 = 21/90

P(dark then milk) = 3/10 × 7/9 = 21/90

P(different types) = 21/90 + 21/90 = 42/90 = 7/15

[3 marks: 1 for correct tree diagram structure, 1 for correct multiplication, 1 for correct addition and simplification]

Example 2: Venn diagram problem (Higher tier)

Question: In a class of 30 students, 18 have a dog (D), 12 have a cat (C), and 5 have both a dog and a cat.

(a) Draw a Venn diagram to represent this information. [2 marks] (b) Find P(D ∪ C). [2 marks] (c) Find P(D|C). [2 marks]

Solution:

(a) Start with intersection: 5 have both

• Only dog: 18 - 5 = 13
• Only cat: 12 - 5 = 5
• Neither: 30 - (13 + 5 + 5) = 7

Venn diagram shows: Only D = 13, D ∩ C = 5, Only C = 5, Neither = 7

[2 marks: 1 for correct intersection value, 1 for all regions correct]

(b) P(D ∪ C) = students with dog or cat / total = (13 + 5 + 5)/30 = 23/30

[2 marks: 1 for correct numerator, 1 for correct final answer]

(c) P(D|C) = P(D ∩ C) / P(C) = (5/30) / (12/30) = 5/12

Alternative: Given student has a cat, 5 out of 12 cat owners also have a dog = 5/12

[2 marks: 1 for correct method, 1 for correct answer]

Example 3: Expected frequency (Foundation/Higher)

Question: The probability that a biased dice lands on 6 is 0.15. The dice is rolled 240 times. How many times would you expect it to land on 6? [2 marks]

Solution:

Expected frequency = probability × number of trials

Expected number of sixes = 0.15 × 240 = 36

[2 marks: 1 for correct method, 1 for correct answer]

Common mistakes and how to avoid them

• Adding instead of multiplying for combined events — Remember: multiply for AND, add for OR (when mutually exclusive). When a question asks for "both" or "all" events happening, multiply probabilities.

• Forgetting to change probabilities without replacement — On the second pick, the total number of items AND the number of each type changes. Always reduce both numerator and denominator appropriately.

• Not simplifying fractions or giving answers in the wrong form — Unless specified, give probability answers as simplified fractions. Decimals and percentages are acceptable if the question allows, but fractions are usually preferred.

• Double-counting in Venn diagrams — When completing a Venn diagram, always start with the intersection (overlap), then work outwards. Never add the intersection twice when calculating totals.

• Misreading tree diagram paths — To find P(specific outcome), multiply along one path. To find P(at least one) or P(exactly two), add multiple paths together. Read the question carefully to determine which paths to include.

• Confusing P(A|B) with P(B|A) — P(red|first was blue) is different from P(first was blue|red). Always identify which event has already occurred (this becomes your new total).

Exam technique for Probability

• Show all working clearly — Probability questions often award method marks. Write out calculations such as 3/10 × 2/9 = 6/90 = 1/15 rather than jumping to the answer.

• Command words matter — "Calculate" requires exact numerical answers with working. "Estimate" allows rounding. "Explain" or "Give a reason" requires a written justification, not just a number.

• Check probabilities are valid — Every probability answer must be between 0 and 1 (or 0% to 100%). If your answer is negative or greater than 1, you've made an error. Probabilities on branches leaving the same point must sum to 1.

• Use the mark allocation as a guide — A 3-mark probability question likely requires: identifying relevant probabilities (1 mark), performing a calculation (1 mark), and reaching a conclusion (1 mark). Don't write one-step answers for multi-mark questions.

Quick revision summary

Probability measures likelihood from 0 to 1. Use P(A') = 1 - P(A) for complementary events. Multiply probabilities for independent events (AND), add for mutually exclusive events (OR). Tree diagrams show successive events; multiply along branches, add different routes. Venn diagrams show overlapping events using set notation. Without replacement, probabilities change for subsequent selections. Conditional probability P(A|B) restricts the sample space to event B. Relative frequency estimates probability from experiments; expected frequency predicts future outcomes using probability × trials.