Probability Experiments (AQA GCSE Maths): Revision Notes

Probability experiments

What are probability experiments?

A probability experiment is any activity where you collect data about different outcomes that could happen. This could be something as simple as rolling a dice or spinning a spinner. The key is that you're testing what actually happens when you repeat the same action many times.

Fair vs biassed experiments

When working with probability experiments, it's important to understand whether your equipment is fair or biassed.

Fair equipment

With fair equipment, like a normal dice, each outcome has an equal chance of happening. For example, when you roll a fair six-sided dice, each number (1, 2, 3, 4, 5, 6) has exactly the same probability of being rolled - which is $\frac{1}{6}$. This means that if you roll the dice many times, you'd expect to get roughly the same number of each result.

Biassed equipment

However, if your dice or spinner is biassed (sometimes called "wonky"), then each outcome won't have an equal chance of happening. Some numbers might come up more often than others.

Using relative frequency to estimate probability

When you suspect that equipment might be biassed, you can use relative frequency to estimate the actual probability of each outcome. This involves doing the experiment many times and recording your results.

The relative frequency formula

The relative frequency of a result tells you how often that result happened compared to the total number of trials:

$\text{Relative frequency} = \frac{\text{Frequency}}{\text{Number of times you tried the experiment}}$

Working with relative frequency

Let's say you spin a spinner 100 times and record how many times each number comes up:

Score		Frequency
1		3
2		14
3		41
4		20
5		18
6		4

To find the relative frequency of each score, you divide each frequency by the total number of spins (100):

Here's the table written out clearly:

Score	Relative Frequency
1	3/100 = 0.03
2	14/10 = 0.14
3	41/100 = 041
4	20/100 = 0.2
5	18/100 = 0.18
6	4/100 = 0.04

Using relative frequency to estimate probability

The importance of repetition

The more times you repeat an experiment, the more accurate your estimate of the probability becomes. If you only rolled a dice 10 times, your results might be quite different from the true probability. But if you rolled it 1000 times, your relative frequencies would be much closer to the actual probabilities.

Checking if equipment is fair or biassed

You can use relative frequency to determine whether equipment is fair or biassed:

• If the equipment is fair, you'd expect all outcomes to have similar relative frequencies
• If the relative frequencies are very different from what you'd expect, the equipment is probably biassed

Recording results with frequency trees

When your experiment has two or more steps, you can record the results using frequency trees. These help you organise the data and calculate relative frequencies for different combinations of outcomes.

How frequency trees work

A frequency tree starts with the total number of trials and then branches out to show the different outcomes at each step. Each branch shows how many times that particular outcome occurred.

Calculating relative frequencies from frequency trees

Once you've completed your frequency tree, you can calculate the relative frequency of each final outcome by dividing the frequency of that outcome by the total number of trials.

Expected frequency calculations

Sometimes you want to predict how many times something will happen if you know the probability. This is called the expected frequency.

The expected frequency formula

$\text{Expected frequency} = \text{Probability} \times \text{Number of trials}$

Using expected frequency

When you don't know the probability

For instance, if previous experiments showed that a biassed dice had a relative frequency of 0.41 for rolling a 3, you could estimate that in 500 rolls, you'd expect about $0.41 \times 500 = 205$ threes.