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Basic Probability and Sample Spaces
Basic Probability and Sample Spaces
This lesson introduces the fundamental ideas of probability as required by the AQA GCSE Mathematics specification. You will learn how to describe the likelihood of events using the probability scale, calculate simple probabilities from equally likely outcomes, list outcomes systematically, and use sample space diagrams when two events are combined.
The Probability Scale
Probability is a measure of how likely an event is to happen. It is always expressed as a number between 0 and 1 (inclusive).
| Probability value | Meaning |
|---|---|
| 0 | The event is impossible — it can never happen |
| 0.5 | The event is even chance — equally likely to happen or not |
| 1 | The event is certain — it will definitely happen |
Probabilities can be written as fractions, decimals, or percentages.
| Fraction | Decimal | Percentage | Likelihood |
|---|---|---|---|
| 0 | 0 | 0% | Impossible |
| 1/4 | 0.25 | 25% | Unlikely |
| 1/2 | 0.5 | 50% | Even chance |
| 3/4 | 0.75 | 75% | Likely |
| 1 | 1.0 | 100% | Certain |
Exam Tip: A probability can never be less than 0 or greater than 1. If your answer falls outside this range, check your working — you have made an error.
Calculating Simple Probabilities
When all outcomes are equally likely, the probability of an event is:
P(event) = number of favourable outcomes / total number of possible outcomes
Worked Example 1
A fair six-sided die is rolled once. What is the probability of rolling a 4?
- Number of favourable outcomes = 1 (just the number 4)
- Total number of outcomes = 6 (the numbers 1, 2, 3, 4, 5, 6)
P(4) = 1/6
Worked Example 2
A bag contains 3 red balls, 5 blue balls, and 2 green balls. A ball is chosen at random. What is the probability that the ball is blue?
- Total number of balls = 3 + 5 + 2 = 10
- Number of blue balls = 5
P(blue) = 5/10 = 1/2
Exam Tip: Always simplify your fractions where possible. Examiners expect answers in their simplest form unless the question says otherwise.
Listing Outcomes
When solving probability problems, it is important to list outcomes systematically so that none are missed. This means working through possibilities in a logical order.
Worked Example 3
A coin is flipped and a die is rolled. List all the possible outcomes.
Organising by coin result first:
H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6
There are 12 possible outcomes in total.
This can be confirmed by the counting principle: 2 (coin outcomes) x 6 (die outcomes) = 12.
Sample Space Diagrams
A sample space diagram (also called a two-way table or possibility space) is a grid that shows all possible outcomes when two events are combined. It is especially useful for finding probabilities involving two dice, two spinners, or a combination of events.
Worked Example 4 — Two Dice
Two fair six-sided dice are rolled and their scores are added together. Find the probability of getting a total of 7.
| 1 | 2 | 3 | 4 | 5 | 6 | |
|---|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| 5 | 6 | 7 | 8 | 9 | 10 | 11 |
| 6 | 7 | 8 | 9 | 10 | 11 | 12 |
- Total number of outcomes = 6 x 6 = 36
- Outcomes that give a total of 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) = 6 outcomes
P(total of 7) = 6/36 = 1/6
Worked Example 5 — Two Spinners
Spinner A has sections labelled 1, 2, 3. Spinner B has sections labelled 1, 2, 3, 4. Both spinners are spun and the results are multiplied.
| 1 | 2 | 3 | 4 | |
|---|---|---|---|---|
| 1 | 1 | 2 | 3 | 4 |
| 2 | 2 | 4 | 6 | 8 |
| 3 | 3 | 6 | 9 | 12 |
Total outcomes = 3 x 4 = 12
P(product is 6) = 2/12 = 1/6 (the outcomes (2,3) and (3,2))
P(product is even) = count all even products in the table, then divide by 12.
Even products: 2, 4, 2, 4, 6, 8, 6, 12 = 8 outcomes
P(product is even) = 8/12 = 2/3
Exam Tip: When drawing a sample space diagram in the exam, always label the rows and columns clearly. This makes it easy for the examiner to see your working and award method marks even if you make an arithmetic slip.
The Complement of an Event
The complement of an event A is the event "A does not happen", written as A' (or sometimes as "not A").
P(A') = 1 - P(A)
Worked Example 6
The probability that it rains tomorrow is 0.3. What is the probability that it does not rain?
P(not rain) = 1 - 0.3 = 0.7
Exam Tip: The complement rule is one of the most useful tools in probability. If it is hard to calculate P(event) directly, try calculating P(not event) and subtracting from 1 instead.
Summary
- Probability is measured on a scale from 0 (impossible) to 1 (certain).
- For equally likely outcomes: P(event) = favourable outcomes / total outcomes.
- Always list outcomes systematically to avoid missing any.
- A sample space diagram is a grid showing all outcomes for two combined events.
- The complement rule states that P(not A) = 1 - P(A).
- Probabilities can be expressed as fractions, decimals, or percentages.
Exam Tip: In multiple-mark questions, always show your sample space diagram or outcome list — you will earn method marks even if your final answer is incorrect.