Listing Outcomes and Expected Frequency (Edexcel GCSE Maths): Revision Notes
Listing outcomes and expected frequency
Understanding outcomes in probability
When tackling probability problems, the first step is often to identify all the possible results that could occur. These possible results are called outcomes. Creating a complete list of outcomes makes the rest of any probability question much more manageable.
Having a systematic approach to listing outcomes is essential for solving probability problems correctly. This foundational step prevents you from missing important possibilities that could affect your final answer.
Sample space diagrams for combined events
When you have two activities happening together (like tossing two coins, rolling a die and spinning a spinner, or spinning two spinners), a sample space diagram provides an excellent way to display all possible outcomes systematically.
Working with two spinners
Let's examine how sample space diagrams work with a practical example involving two triangular spinners.
| + | 3 | 4 | 5 |
|---|---|---|---|
| 1 | 4 | 5 | 6 |
| 2 | 5 | 6 | 7 |
| 3 | 6 | 7 | 8 |
Worked Example: Creating a Sample Space Diagram
When creating a sample space diagram, you follow these steps:
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Set up the grid: Place all possible outcomes from one spinner along the top of your table, and all outcomes from the other spinner down the side.
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Fill in the combined results: Add the scores from both spinners together to find each possible total. Work through each cell systematically.
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Count the outcomes: Even though some totals might be repeated, each cell represents a separate outcome. In the spinner example, there are 9 possible outcomes altogether.
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Calculate probabilities: To find the probability of any specific total, count how many ways that total can occur, then divide by the total number of possible outcomes.
For instance, to score a total of 6, there are 3 different ways this can happen out of 9 total possibilities. Therefore, the probability equals , which simplifies to .
Expected frequency calculations
Probability becomes particularly useful when you want to estimate how often something will happen over many trials. This is called expected frequency.
The expected frequency formula
The relationship between probability and expected frequency is straightforward:
Expected frequency = Probability × Number of trials
This formula allows you to predict approximately how many times an event will occur if you repeat an experiment many times.
Applying expected frequency to games
Consider a practical example involving a dice game:
Worked Example: Dice Game Expected Frequency
To solve expected frequency problems:
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Calculate the basic probability first: Determine how likely the event is to happen in a single trial. For winning by rolling a 5 or 6 on a fair six-sided die, there are 2 favourable outcomes out of 6 possible outcomes, giving a probability of .
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Apply the expected frequency formula: Multiply the probability by the number of trials. If someone plays 180 games, the expected number of wins = wins.
Problem-solving approach
When facing probability questions, remember that creating a systematic list often provides the clearest path to the solution. Sample space diagrams are essentially organised lists that help you avoid missing any possible outcomes.
This systematic approach works whether you're dealing with coins, dice, spinners, or any other probability scenario involving multiple events. The key is to be methodical and thorough in your approach.
Key Points to Remember:
- Start with outcomes: Always begin by listing all possible results that could happen
- Use sample space diagrams: These systematic grids help you organise outcomes when two activities occur together
- Expected frequency formula: Expected frequency = Probability × Number of trials
- Work step by step: First find the probability, then multiply by the number of trials for expected frequency
- When in doubt, make a list: A systematic approach prevents you from missing important outcomes