GCSE Maths Practice: theoretical-vs-experimental-probability

Experimental Probability Using Dice with Larger Data Sets

Experimental probability is based on results collected from real experiments rather than predictions made in advance. In Higher GCSE Maths, students are expected to work confidently with larger numbers of trials and interpret how experimental results compare with theoretical expectations. Rolling a die many times is a clear and familiar way to explore this idea.

The Fundamental Formula

Experimental probability = number of times the event occurs ÷ total number of trials

This formula always uses observed data. The final probability can be written as a fraction, decimal, or percentage depending on the context of the question.

Worked Example

A die is rolled 180 times and the number 4 appears 29 times. The experimental probability of rolling a 4 is:

\( \frac{29}{180} \)

This result is taken directly from the experiment and does not rely on the assumption that each outcome must occur equally often.

Experimental vs Theoretical Probability

Theoretical probability is calculated using equally likely outcomes. For a fair die, each face has the same theoretical probability. Experimental probability, however, depends on what actually happens during the trials and may differ due to randomness.

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

Why Results Differ from Theory

Random variation means that even large experiments can produce results that differ from expectations. With smaller samples, variation can be quite noticeable. With larger samples, results usually become more stable, although perfect balance is never guaranteed.

This idea forms the basis of the law of large numbers, which is an important concept at Higher GCSE level.

Common Mistakes

• Assuming experimental probability must equal theoretical probability
• Failing to simplify fractions fully
• Using expected outcomes instead of observed data
• Rounding too early during calculations

Real-World Applications

Experimental probability using large data sets is common in real life. Engineers test components repeatedly to estimate failure rates. Scientists run experiments many times to check reliability. Game designers simulate dice rolls to ensure fairness.

In all cases, conclusions are based on observed evidence rather than assumptions.

Frequently Asked Questions

Does experimental probability become more accurate with more trials?
Yes. Larger samples usually give more stable results, though randomness never disappears completely.

Can experimental probability be written as a decimal?
Yes. Fractions, decimals, and percentages are all acceptable unless the question states otherwise.

Why is this considered a Higher-tier topic?
Because it involves interpreting larger data sets and understanding variation.

Study Tip

When working with large numbers of trials, always form the fraction first, simplify fully, and only then convert to a decimal if required.