GCSE Maths Practice: theoretical-vs-experimental-probability

Question 2 of 10

This Higher-level question tests experimental probability involving more than one favourable outcome.

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\( \begin{array}{l}\text{A bag contains red, blue, and green marbles.} \\ \text{In 100 draws, a red or blue marble is drawn 80 times.} \\ \text{What is the experimental probability of drawing a red or blue marble?}\end{array} \)

Choose one option:

Always combine all favourable outcomes before forming the probability fraction.

Experimental Probability with Multiple Favourable Outcomes

Experimental probability is calculated using data collected from real experiments rather than predictions. In Higher GCSE Maths, questions often increase in difficulty by asking for the probability of more than one favourable outcome occurring, such as drawing a red or blue marble.

The Core Method

To calculate experimental probability, follow these steps:

  1. Identify all favourable outcomes.
  2. Add the observed frequencies of those outcomes.
  3. Divide by the total number of trials.
  4. Simplify the fraction if required.

Experimental probability = favourable outcomes ÷ total trials

Worked Example

A bag contains different coloured counters. In 60 trials, red counters are drawn 18 times and yellow counters are drawn 22 times. The experimental probability of drawing red or yellow is:

\( \frac{18 + 22}{60} = \frac{40}{60} = \frac{2}{3} \)

This calculation combines multiple favourable outcomes before forming the probability.

Why This Is Higher Tier

This type of question requires more than applying a single formula. Students must recognise that multiple outcomes are favourable, combine frequencies correctly, and simplify accurately. These steps introduce additional reasoning compared to single-outcome experimental probability.

Experimental vs Theoretical Probability

Theoretical probability is calculated using known quantities, such as the number of marbles in a bag. Experimental probability uses observed data instead. In real experiments, these values may differ due to randomness.

As more trials are carried out, experimental probability often moves closer to the theoretical probability, but it may not match it exactly.

Common Mistakes

  • Forgetting to combine all favourable outcomes
  • Using theoretical counts instead of observed data
  • Failing to simplify the final fraction
  • Assuming experimental probability must match theory

Real-Life Applications

Experimental probability with combined outcomes is used in many real situations. Quality control teams count acceptable products. Weather forecasts combine multiple favourable conditions. Sports analysts evaluate win-or-draw outcomes.

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

Frequently Asked Questions

Do I always need to simplify the fraction?
Yes, unless the question explicitly allows an unsimplified answer.

Can experimental probability be written as a decimal?
Yes. Fractions, decimals, and percentages are all valid forms.

Why are multiple outcomes tested?
They assess deeper understanding of probability structure.

Study Tip

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

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