What you'll learn
This revision guide covers the fundamental concepts of probability required for AQA GCSE Statistics. You'll learn how to calculate probabilities, understand probability scales, work with different types of events, and interpret probability in real-world contexts. These foundational skills underpin more advanced probability topics in your exam.
Key terms and definitions
Probability — A numerical measure of how likely an event is to occur, expressed as a value between 0 and 1 (or 0% to 100%)
Outcome — A possible result of a trial or experiment
Event — A set of one or more outcomes from a probability experiment
Sample space — The set of all possible outcomes of a probability experiment
Equally likely outcomes — Outcomes that have the same probability of occurring
Mutually exclusive events — Events that cannot happen at the same time
Exhaustive events — Events that cover all possible outcomes with no gaps
Relative frequency — An estimate of probability based on experimental data, calculated as (number of times event occurs) ÷ (total number of trials)
Core concepts
The probability scale
Probability is always expressed as a number between 0 and 1, where:
- Probability = 0 means the event is impossible (cannot happen)
- Probability = 1 means the event is certain (must happen)
- Probability = 0.5 means the event has an even chance of occurring
Probabilities can be expressed as:
- Fractions (e.g., 1/4)
- Decimals (e.g., 0.25)
- Percentages (e.g., 25%)
The closer a probability is to 1, the more likely the event. The closer to 0, the less likely.
Key principle: The sum of probabilities of all possible outcomes in a sample space always equals 1.
Calculating theoretical probability
Theoretical probability assumes all outcomes are equally likely and is calculated using:
P(event) = Number of favourable outcomes / Total number of possible outcomes
This formula applies when you can list all possible outcomes and they're equally likely.
Example contexts:
- Rolling fair dice
- Tossing fair coins
- Selecting items at random from a bag
- Picking cards from a well-shuffled deck
When calculating, always:
- Identify the total number of possible outcomes (the sample space)
- Count how many outcomes result in the event you're interested in
- Write as a fraction and simplify if possible
Experimental probability and relative frequency
When theoretical probability cannot be calculated (e.g., biased dice, real-world data), we use experimental probability based on trials.
Relative frequency = Frequency of event / Total number of trials
Important points:
- Relative frequency provides an estimate of probability
- More trials generally give a better estimate
- Results will vary between experiments due to random variation
- As the number of trials increases, relative frequency tends to approach the theoretical probability (Law of Large Numbers)
Example: If a drawing pin lands point-up 340 times in 500 throws, the relative frequency is 340/500 = 0.68
Listing outcomes systematically
For experiments with multiple stages, you must list all possible outcomes systematically to avoid missing any. Three main methods:
Sample space diagrams: A table or grid showing all possible combinations
- Useful for two events (e.g., rolling two dice)
- List outcomes of first event on one axis
- List outcomes of second event on the other axis
- Fill in all combinations
Lists: Writing outcomes in a systematic order
- Use notation like (First, Second)
- Work methodically through all possibilities
Tree diagrams: Branching diagrams showing stages
- Each branch represents one outcome
- Multiply probabilities along branches
- Add probabilities of different routes to the same result
Mutually exclusive events
Events are mutually exclusive if they cannot both happen at the same time.
Examples:
- Rolling a 3 or a 5 on a die (mutually exclusive)
- Selecting a red card or a black card from a deck (mutually exclusive)
- Rolling an even number or a 3 (NOT mutually exclusive — 2, 4, 6 are even; 3 is also possible)
Addition rule for mutually exclusive events:
P(A or B) = P(A) + P(B)
This only works when A and B cannot both occur.
Example: The probability of rolling a 2 or a 5 on a die = 1/6 + 1/6 = 2/6 = 1/3
Complementary events
The complement of an event A is "event A does not happen", written as A'.
Key relationship: P(A) + P(A') = 1
Therefore: P(A') = 1 - P(A)
This is useful when it's easier to calculate the probability something doesn't happen.
Example: If P(rain) = 0.3, then P(no rain) = 1 - 0.3 = 0.7
Common uses:
- P(at least one) = 1 - P(none)
- P(winning) = 1 - P(losing)
- P(passing) = 1 - P(failing)
Worked examples
Example 1: Theoretical probability with a sample space
Question: A fair six-sided die is rolled. Find the probability of: (a) Rolling a 4 [1 mark] (b) Rolling an even number [2 marks] (c) Not rolling a 6 [2 marks]
Solution:
(a) Sample space = {1, 2, 3, 4, 5, 6}, total outcomes = 6 Number of ways to roll a 4 = 1 P(4) = 1/6 ✓
(b) Even numbers are {2, 4, 6} Number of even outcomes = 3 P(even) = 3/6 = 1/2 ✓✓
(c) Method 1: Count favourable outcomes {1, 2, 3, 4, 5} P(not 6) = 5/6 ✓✓
Method 2: Use complement rule P(not 6) = 1 - P(6) = 1 - 1/6 = 5/6 ✓✓
Mark scheme notes: In part (a), 1 mark for correct answer. In part (b), 1 mark for identifying 3 favourable outcomes, 1 mark for correct simplified answer. In part (c), 1 mark for method shown, 1 mark for correct answer.
Example 2: Relative frequency
Question: A market researcher surveys 250 customers about their preferred brand.
| Brand | Frequency |
|---|---|
| A | 95 |
| B | 78 |
| C | 52 |
| D | 25 |
(a) Estimate the probability that a randomly chosen customer prefers Brand A. [2 marks] (b) If 1500 customers visit the shop tomorrow, estimate how many will prefer Brand C. [2 marks]
Solution:
(a) Total customers surveyed = 250 Frequency preferring Brand A = 95 P(Brand A) = 95/250 = 0.38 ✓✓
(b) Expected number = P(Brand C) × 1500 P(Brand C) = 52/250 = 0.208 ✓ Expected number = 0.208 × 1500 = 312 customers ✓
Mark scheme notes: In part (a), 1 mark for correct division, 1 mark for correct decimal answer. Accept equivalent forms (19/50 or 38%). In part (b), 1 mark for calculating P(Brand C) or showing method, 1 mark for correct final answer.
Example 3: Mutually exclusive events and complements
Question: A bag contains 5 red balls, 3 blue balls, 2 green balls, and 4 yellow balls. One ball is selected at random.
(a) Find P(red) [1 mark] (b) Find P(blue or green) [2 marks] (c) Find P(not yellow) [2 marks]
Solution:
(a) Total balls = 5 + 3 + 2 + 4 = 14 P(red) = 5/14 ✓
(b) Blue and green are mutually exclusive events P(blue or green) = P(blue) + P(green) ✓ = 3/14 + 2/14 = 5/14 ✓
(c) Method 1: Using complement P(not yellow) = 1 - P(yellow) ✓ = 1 - 4/14 = 10/14 = 5/7 ✓
Method 2: Direct counting Not yellow means red, blue or green P(not yellow) = (5 + 3 + 2)/14 = 10/14 = 5/7 ✓✓
Mark scheme notes: In part (a), 1 mark for correct answer. In part (b), 1 mark for adding probabilities, 1 mark for correct answer. In part (c), 1 mark for method (either approach), 1 mark for correct simplified answer.
Common mistakes and how to avoid them
Confusing probability with frequency: Probability is a fraction/decimal between 0 and 1, not the count of outcomes. Always divide the frequency by the total to get probability.
Not simplifying fractions: Answers like 6/12 should be simplified to 1/2. Examiners expect simplest form unless the question specifies otherwise.
Adding probabilities incorrectly: Only add probabilities for mutually exclusive events. Don't add probabilities for events that can overlap without adjusting for the overlap.
Forgetting the total in relative frequency: The denominator must be the total number of trials, not the total number of successful outcomes.
Mixing probability forms: Be consistent — if the question uses fractions, answer in fractions; if it uses decimals, use decimals. However, any equivalent form is usually acceptable.
Misinterpreting "at least one": P(at least one) = 1 - P(none). Students often try to calculate this directly rather than using the complement, which is much harder.
Exam technique for "Probability: Basic Probability"
"Calculate the probability..." requires a numerical answer between 0 and 1. Show your working: write the fraction as (favourable outcomes)/(total outcomes) then simplify. Usually worth 2 marks (1 for method, 1 for answer).
"Estimate the probability..." signals relative frequency. Calculate frequency ÷ total trials. The word "estimate" indicates experimental rather than theoretical probability.
Show all working: Even if you can do calculations mentally, write down intermediate steps. Method marks are awarded even if your final answer is wrong.
Check your answer is sensible: Probabilities must be between 0 and 1. If you get 1.5 or -0.2, you've made an error. If asked "which is more likely?", the higher probability is more likely.
Quick revision summary
Probability measures how likely events are, expressed between 0 (impossible) and 1 (certain). Calculate theoretical probability using favourable outcomes divided by total outcomes when outcomes are equally likely. Estimate probability using relative frequency (frequency ÷ trials) from experimental data. Mutually exclusive events cannot occur together; add their probabilities using P(A or B) = P(A) + P(B). Use the complement rule P(A') = 1 - P(A) for "not" questions. Always list outcomes systematically and simplify fractions in your final answers.