Counting Outcomes (AQA GCSE Maths): Revision Notes
Counting outcomes
Introduction to counting outcomes
When working with probability questions, the first step is often to identify all the possible outcomes that could occur. Having a complete list of outcomes makes calculating probabilities much more straightforward. There are two main approaches to counting outcomes: listing them all out systematically, or using mathematical rules to count them without listing.
Listing all outcomes
Sample space diagrams
A sample space diagram displays every possible outcome from a probability experiment. This can be presented as a simple list, but when dealing with two or more activities happening together, a two-way table often works better.
For example, when two activities are occurring simultaneously (such as tossing two coins or spinning two spinners), a two-way table helps organise all the possible combinations. The outcomes from one activity are listed along the top, while the outcomes from the other activity are listed down the side.
Two-way tables are particularly useful because they provide a visual representation that makes it easy to see all combinations and avoid missing any possible outcomes.
Working with two-way tables
Let's look at an example with two spinners. When both spinners are used together, we need to consider every possible combination of results. If one spinner shows the numbers 1, 2, and 3, and the other spinner shows 3, 4, and 5, we can create a table where:
- The first spinner's results go along the top
- The second spinner's results go down the side
- Each cell shows what happens when we add the two numbers together
Worked Example: Two Spinners
When spinning both spinners and adding the results, we get 9 total outcomes:
- Ways to get total = 6: (1,5), (2,4), (3,3) = 3 ways
- Total possible outcomes = 9
Therefore:
Even when some totals appear multiple times, each combination represents a separate outcome. This is important to remember when calculating probabilities - we count each individual outcome, not just the different totals.
To find the probability of a specific total, we divide the number of ways to achieve that total by the total number of possible outcomes.
The product rule to count outcomes
When to use the product rule
Sometimes listing all outcomes becomes impractical, especially when dealing with large numbers of possibilities or multiple activities. The product rule provides a mathematical shortcut for counting outcomes without having to list them all.
How the product rule works
The Product Rule: The number of ways to carry out a combination of activities equals the number of ways to carry out each activity multiplied together.
This rule is particularly useful when you're dealing with situations like rolling multiple dice, where listing all outcomes would be extremely time-consuming. The key principle is that when you have a combination of independent activities, the total number of possible outcomes equals the number of ways to carry out each individual activity multiplied together.
Example with dice
Consider rolling four fair six-sided dice. Each individual die can land in 6 different ways (showing 1, 2, 3, 4, 5, or 6).
Worked Example: Rolling Four Dice
Step 1: Calculate total possible outcomes
- Each die has 6 possible outcomes
- Total outcomes = different ways
Step 2: Calculate favourable outcomes (only even numbers)
- Each die has 3 even numbers (2, 4, 6)
- Ways to roll only even numbers = ways
Step 3: Calculate probability
Choosing the right method
Sample space diagrams work brilliantly when you can easily draw or list all outcomes. They give you a clear visual representation and make probability questions straightforward. However, when the number of outcomes becomes very large or when you're dealing with multiple activities, the product rule becomes much more efficient.
The product rule is particularly powerful because it allows you to count outcomes without having to write them all down, saving time and reducing the chance of missing possibilities.
Practice applications
Understanding these methods helps with various probability scenarios:
Practice Applications
- When three fair coins are tossed, you can either list all 8 possible outcomes or use the product rule:
- When ten fair coins are tossed, listing becomes impractical, but the product rule gives us possible outcomes
The key is recognising which method suits the specific problem you're solving.
Key Points to Remember:
- Start probability problems by identifying all possible outcomes - this forms the foundation for any probability calculation
- Sample space diagrams and two-way tables are excellent for visualising outcomes when the numbers are manageable
- The product rule multiplies the number of ways for each independent activity together to find the total number of outcomes
- Use the product rule when listing all outcomes would be too time-consuming or complex
- Always ensure you're counting individual outcomes, not just different totals or results