GCSE Maths Practice: relative-frequency

Using Relative Frequency to Judge Whether a Spinner Is Biased

At Higher GCSE level, probability questions often require you to interpret experimental results rather than simply perform calculations. One important application of relative frequency is deciding whether a spinner, die, or coin might be biased. This involves comparing what actually happened with what would be expected if the object were fair.

Theoretical Probability vs Experimental Results

Theoretical probability describes what should happen in an ideal situation. For example, if a spinner has four equal sections, each colour should appear with the same likelihood. Experimental probability, also known as relative frequency, is based on what actually happens when the spinner is used many times.

How to Decide Whether Bias May Be Present

To judge whether a result suggests bias, follow these steps:

• Calculate the relative frequency of the outcome from the experiment.
• Identify the theoretical probability for a fair spinner.
• Compare the two values.
• Decide whether the difference is small (likely due to chance) or very large (possible bias).

Worked Example 1

A spinner with five equal sections is spun 200 times. One colour appears 92 times. The relative frequency is compared with the expected probability for a fair spinner to decide whether the spinner may be biased.

Worked Example 2

A fair coin is flipped 1,000 times and lands heads 610 times. The observed relative frequency is compared with the expected probability of heads to judge whether the coin might be biased.

Worked Example 3

A dice is rolled 120 times and one number appears 18 times. The relative frequency is close to the expected value, suggesting that the result may simply be due to random variation rather than bias.

Common Higher-Tier Mistakes

• Stating that an object is definitely biased or definitely fair.
• Ignoring the size of the experiment.
• Failing to compare experimental and theoretical probabilities.
• Assuming any difference automatically proves bias.

Why the Wording Matters

In exam questions, answers are often phrased carefully using words like "may", "suggests", or "might indicate". This is because experimental data can never prove fairness or bias with complete certainty. It can only provide evidence that points in one direction.

Why Large Sample Sizes Are Important

Small experiments can produce unusual results purely by chance. As the number of trials increases, random variation becomes less significant and patterns become clearer. This is why conclusions drawn from larger experiments are generally more reliable.

Frequently Asked Questions

Does a large difference always mean bias?
No. However, a very large difference makes bias more likely than random variation.

Can an experiment prove fairness?
No. It can only suggest fairness or bias based on evidence.

Why is repeating the experiment helpful?
Repeating experiments increases reliability and helps confirm patterns.

Study Tip

In Higher GCSE probability questions, phrases like "what can you conclude" or "what does this suggest" mean you should avoid absolute statements and use careful, evidence-based reasoning.