GCSE Maths Practice: theoretical-vs-experimental-probability

Question 10 of 10

This Higher-level question tests experimental probability involving multiple favourable spinner outcomes.

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\( \begin{array}{l}\text{A spinner has 6 equal sections: red, blue, green, yellow, purple, black.} \\ \text{In 60 spins, it lands on red or blue a total of 40 times.} \\ \text{What is the experimental probability of landing on red or blue?}\end{array} \)

Choose one option:

Always combine all favourable outcomes before forming the probability fraction.

Experimental Probability with Combined Spinner Outcomes

Experimental probability is calculated using results collected from an experiment rather than from predictions made in advance. In Higher GCSE Maths, questions often increase in difficulty by requiring students to work with more than one favourable outcome, such as when a spinner lands on one colour or another.

Spinners are particularly useful for demonstrating this idea because they clearly show how outcomes are grouped and counted. When the word or appears in a probability question, it indicates that multiple outcomes should be combined before calculating the probability.

The Core Method

To calculate experimental probability with multiple favourable outcomes:

  1. Identify all outcomes that are considered favourable.
  2. Add together the observed frequencies of those outcomes.
  3. Divide by the total number of trials.
  4. Simplify the fraction or convert it to a decimal if required.

Experimental probability = favourable outcomes ÷ total number of trials

Worked Example

A spinner with eight equal sections is spun 90 times. It lands on red or green a total of 54 times. The experimental probability of landing on red or green is:

\( \frac{54}{90} = \frac{3}{5} \)

This calculation combines the observed outcomes before forming the probability.

Experimental vs Theoretical Probability

Theoretical probability is calculated using the number of possible outcomes. For a spinner with six equal sections, each colour has the same theoretical chance. Experimental probability, however, is calculated using observed results and may differ because of randomness.

As the number of spins increases, experimental probability often moves closer to theoretical probability, but it does not need to match it exactly.

Why This Is Higher Tier

This type of question requires careful interpretation. Students must recognise that more than one outcome is favourable, combine results correctly, and then simplify or convert the probability accurately. These additional steps increase the level of reasoning compared to single-outcome questions.

Common Mistakes

  • Using only one favourable outcome instead of combining them
  • Dividing by the wrong total number of trials
  • Confusing experimental probability with theoretical probability
  • Rounding decimals too early

Real-Life Applications

Combined experimental probability is widely used in real situations. Sports analysts combine wins and draws. Quality control teams group acceptable results. Weather forecasts often combine multiple favourable conditions.

In all cases, conclusions are based on collected data rather than assumptions.

Study Tip

Whenever a probability question includes the word or, always combine all favourable outcomes before dividing by the total number of trials.

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