GCSE Maths Practice: relative-frequency

Comparing Relative Frequencies for Consistency

At Higher GCSE level, probability questions often require you to compare results from different experiments and judge which one is most reliable or consistent. This goes beyond simple calculation and focuses on interpretation and reasoning. In this type of question, relative frequency is used as evidence to support a conclusion.

Theoretical vs Experimental Probability

Theoretical probability describes what should happen if an object is perfectly fair. For example, if a spinner has two equal sections, the theoretical probability of landing on a particular colour is one half. Experimental probability, or relative frequency, is calculated from observed results and may differ slightly due to random variation.

What Does “Most Consistent” Mean?

When a question asks which result is most consistent with a given probability, it means you should identify the experimental result that is closest to the theoretical value. Small differences are expected in experiments, but the smallest difference usually indicates the most reliable or consistent result.

How to Compare Results

To compare different experiments:

• Calculate the relative frequency for each experiment.
• Convert fractions to decimals if helpful.
• Compare each value with the theoretical probability.
• Choose the result with the smallest difference.

Worked Example 1

Three coins are tested. Coin X lands heads 48 times out of 100, Coin Y lands heads 26 times out of 50, and Coin Z lands heads 72 times out of 140. Each relative frequency is compared to the theoretical probability of one half to decide which coin behaves most fairly.

Worked Example 2

Three machines produce light bulbs. Machine A produces 92 working bulbs out of 180, Machine B produces 51 out of 100, and Machine C produces 44 out of 90. Relative frequencies are calculated and compared to judge which machine is most consistent with a target success rate.

Worked Example 3

Three students estimate the probability of rain based on past data. Each uses a different data set size. Their relative frequencies are compared to see which estimate best matches the expected probability.

Common Higher-Tier Mistakes

• Choosing the result with the largest sample instead of the closest value.
• Assuming a result must be exactly equal to be consistent.
• Ignoring small but important differences between values.
• Failing to convert fractions to decimals for easier comparison.

Why Exact Matches Are Rare

In real experiments, it is unusual for relative frequency to exactly equal theoretical probability. Random variation means results will differ slightly. Consistency is judged by closeness, not perfection.

Why Larger Samples Help

Although larger samples tend to give more reliable results, they do not automatically guarantee the most consistent outcome. The key factor in this type of question is how close the result is to the expected probability, not just how many trials were performed.

Frequently Asked Questions

Does the most consistent result prove fairness?
No. It only suggests that the result aligns closely with expectations.

Should I always compare decimals?
Yes. Decimals make it easier to judge closeness.

Can two results be equally consistent?
Yes, if they are equally close to the theoretical value.

Study Tip

In Higher GCSE questions, words like "most consistent", "closest to", or "best matches" mean you should compare relative frequencies carefully rather than look for an exact match.