GCSE Maths Practice: relative-frequency

Question 2 of 11

This question asks you to interpret experimental results and decide whether bias may be present.

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\( \begin{array}{l}\text{A die is rolled 300 times and lands on 6 a total} \\ \text{of 120 times. What can you conclude from this result?}\end{array} \)

Choose one option:

Avoid absolute conclusions. Use careful language when interpreting experimental probability.

Using Relative Frequency to Detect Bias

At Higher GCSE level, probability questions often require you to interpret results rather than simply calculate numbers. One common application of relative frequency is deciding whether an object such as a die, coin, or spinner might be biased. This involves comparing experimental results with what you would expect from a fair object.

Theoretical vs Experimental Probability

Theoretical probability describes what should happen if an object is perfectly fair. For example, when rolling a fair die, each number should appear with equal likelihood. Experimental probability, also known as relative frequency, is based on what actually happens when the experiment is carried out many times.

How to Judge Possible Bias

To decide whether bias may be present, follow these steps:

  • Calculate the relative frequency of the outcome.
  • Identify the theoretical probability for a fair object.
  • Compare the two values.
  • Decide whether the difference is small (likely random variation) or large (possible bias).

Worked Example 1

A spinner with five equal sections is spun 500 times. One section appears 190 times. The relative frequency is compared with the expected probability for a fair spinner. A large difference may suggest the spinner is not fair.

Worked Example 2

A coin is flipped 1,000 times and lands heads 560 times. The observed relative frequency is compared to the expected probability of a fair coin to decide whether the coin might be biased.

Worked Example 3

A dice is rolled 90 times and one number appears 22 times. The relative frequency is close to the expected value, so the result may simply be due to random variation rather than bias.

Common Higher-Tier Errors

  • 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 means bias.

Why Large Sample Sizes Matter

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 large data sets are usually more reliable.

Frequently Asked Questions

Does a difference always mean bias?
No. Some variation is expected even with a fair object, especially in smaller samples.

Why do exam answers often say “may be biased”?
Because experimental results suggest bias but do not prove it beyond doubt.

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

Study Tip

In Higher GCSE questions, look for phrases such as "what can you conclude" or "what does this suggest". These signal that you should interpret results carefully and avoid absolute statements.

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