Calculating Probability From a Sample Space

📋 Spec coverage: GCSE AQA Mathematics (8300) This article covers:

• AQA P7 (Probability) - constructing theoretical possibility spaces for single and combined experiments with equally likely outcomes, then using them to calculate probabilities.
• Tier: Foundation

To calculate probability from a sample space, count the outcomes that satisfy the event and divide by the total number of equally likely outcomes:

$$P(event)=\frac{number\; of\; favourable\; outcomes}{total\; number\; of\; possible\; outcomes}$$

What is a sample space?

A sample space is a list or diagram showing every possible outcome of an experiment. Outcomes must not be missed or repeated.

For example, when a fair coin is tossed once, the sample space is:

$$\{H,T\}$$

Here, $H$ means heads and $T$ means tails.

Worked example

A fair coin is tossed twice. Find the probability of getting exactly one head.

First, list the complete sample space systematically:

$$\{HH,HT,TH,TT\}$$

There are $4$ equally likely outcomes altogether.

The outcomes containing exactly one head are:

$$\{HT,TH\}$$

There are $2$ favourable outcomes, so:

$$P(exactly\; one\; head)=\frac{2}{4}=\frac{1}{2}$$

Therefore, the probability is $\frac{1}{2}$.

Systematic listing matters because combined outcomes have an order. In the example, $HT$ and $TH$ are different outcomes, even though both contain one head. Writing the first toss before the second toss makes every result visible and helps you check that the denominator is correct.

Exam method

1. Create the complete sample space using a list, grid or diagram.
2. Check that all outcomes are equally likely.
3. Count the outcomes matching the event.
4. Divide by the total number of outcomes.
5. Simplify the fraction if possible.

Common mistake

Do not count different results as equally likely unless the individual outcomes really are equally likely. For example, when rolling two dice, a total of $7$ can occur in more ways than a total of $2$. A complete sample space grid shows these differences clearly.

Completeness check: This covers constructing sample spaces for single and combined experiments and calculating theoretical probabilities from equally likely outcomes, completing the listed requirements of AQA P7.