GCSE Maths Practice: theoretical-vs-experimental-probability

Question 6 of 10

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

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\( \begin{array}{l}\text{A die is rolled 150 times, and 2, 4, 6 appear 40 times.} \\ \text{What is the experimental probability of rolling 2, 4, or 6?}\end{array} \)

Choose one option:

Always combine all favourable outcomes before forming the probability fraction.

Experimental Probability with Multiple Favourable Outcomes

Experimental probability is calculated using results collected from real experiments rather than from predictions. In Higher GCSE Maths, questions often increase in difficulty by requiring students to work with more than one favourable outcome, such as rolling certain numbers on a die.

In this type of question, it is important to recognise that several outcomes are counted together. Words such as or indicate that all listed outcomes should be included when calculating the probability.

The Core Method

To calculate experimental probability with multiple favourable outcomes, follow these steps:

  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 die is rolled 90 times. The numbers 1, 3, and 5 appear a total of 33 times. The experimental probability of rolling 1, 3, or 5 is:

\( \frac{33}{90} = \frac{11}{30} \)

This calculation combines all favourable outcomes before forming the probability.

Why This Is Higher Tier

Unlike basic experimental probability questions, this type requires students to recognise multiple favourable outcomes and combine results correctly. It also tests accuracy when simplifying fractions and converting between fractions and decimals.

Experimental vs Theoretical Probability

Theoretical probability is based on equally likely outcomes. For a fair die, even numbers make up half of all possible outcomes. Experimental probability, however, is based on observed results and may differ due to randomness.

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

Common Mistakes

  • Forgetting to include all favourable outcomes
  • Dividing by the wrong total number of trials
  • Failing to simplify the fraction fully
  • Assuming experimental probability must equal theoretical probability

Real-Life Applications

Experimental probability with combined outcomes is widely used in real life. Quality control teams count acceptable products. Sports analysts combine win and draw outcomes. Weather forecasts group multiple favourable conditions together.

In each case, decisions are based on observed data rather than assumptions.

Frequently Asked Questions

Should I always simplify the fraction?
Yes. GCSE Maths expects answers to be given in their simplest form unless stated otherwise.

Is a decimal answer acceptable?
Yes, as long as it is accurate and rounded appropriately.

Why are questions with multiple outcomes harder?
They require careful interpretation and correct combination of results.

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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