GCSE Maths Practice: relative-frequency

Question 1 of 11

This question uses experimental results to estimate probability using relative frequency.

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\( \begin{array}{l}\text{A die is rolled 250 times. The number 6 appears} \\ \text{50 times. Which of the following is the best estimate} \\ \text{of the probability of rolling a 6?}\end{array} \)

Choose one option:

When a question asks for a best estimate, always calculate relative frequency and use a decimal.

Estimating Probability Using Relative Frequency

At Higher GCSE level, probability questions often move beyond simply calculating relative frequency. Instead, you are expected to use experimental results to estimate the probability of an event occurring. This is known as experimental probability and is based on observed data rather than theory.

Relative Frequency as an Estimate

Relative frequency gives us an estimate of probability by looking at how often an event actually occurs in a large number of trials. While theoretical probability describes what should happen in an ideal situation, experimental probability reflects real-world results. When the number of trials is large, relative frequency usually gives a good approximation of the true probability.

Method for Estimating Probability

The method is always the same:

  • Count how many times the event occurred.
  • Divide by the total number of trials.
  • Write the result as a decimal.
  • Use this value as the estimated probability.

Worked Example 1

A spinner is spun 400 times and lands on blue 92 times. The estimated probability of landing on blue is found by dividing the number of blue outcomes by the total number of spins and writing the answer as a decimal.

Worked Example 2

A factory tests 1,200 light bulbs and finds that 36 are faulty. The relative frequency of a faulty bulb is calculated and used as an estimate of the probability that a randomly selected bulb will be faulty.

Worked Example 3

A basketball player takes 250 shots in training and scores 138 times. The relative frequency of scoring is used to estimate the probability that the player will score on their next shot.

Common Higher-Tier Mistakes

  • Using theoretical probability when the question asks for an estimate.
  • Leaving the answer as a fraction instead of a decimal.
  • Choosing an option that looks familiar rather than calculating.
  • Ignoring the phrase "best estimate".

Why Large Numbers of Trials Matter

With small numbers of trials, results can vary widely and give unreliable estimates. As the number of trials increases, random variation has less effect, and the relative frequency usually settles closer to the true probability. This idea is fundamental in statistics and is used in science, economics, and data analysis.

Frequently Asked Questions

Why is this only an estimate?
Because it is based on observed results, which may vary if the experiment is repeated.

Should I always convert to a decimal?
Yes, if the question asks for a probability estimate or provides decimal answer options.

Is relative frequency more reliable than theory?
It depends on the context. For real-world data, relative frequency is often more useful.

Study Tip

In Higher GCSE questions, phrases like "best estimate", "experimental probability", or "based on these results" are strong clues that you should calculate relative frequency and give your answer as a decimal.

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