GCSE Maths Practice: theoretical-vs-experimental-probability

Experimental Probability with Very Large Dice Samples

Experimental probability is determined using results collected from repeated trials rather than predictions made before an experiment. In Higher GCSE Maths, students are expected to interpret experimental probability using large data sets and understand how these results compare with theoretical expectations. Rolling a die hundreds of times provides a clear example of how randomness behaves over many trials.

The Core Formula

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

This formula always relies on observed data. The resulting probability can be expressed as a fraction, a decimal, or a percentage, depending on the requirements of the question.

Worked Example

A die is rolled 360 times and the number 3 appears 68 times. The experimental probability of rolling a 3 is:

\( \frac{68}{360} = \frac{17}{90} \)

This value comes directly from the experiment and does not assume that each face appears an equal number of times.

Experimental vs Theoretical Probability

Theoretical probability is calculated using equally likely outcomes. For a fair die, each face has a theoretical probability of one sixth. Experimental probability, however, is based on what actually happens during an experiment and may differ slightly because of randomness.

With very large numbers of trials, experimental probability often moves closer to theoretical probability, but it does not have to match it exactly.

Why Large Samples Are Important

In small experiments, random variation can cause large fluctuations in results. As the number of trials increases, these fluctuations usually become smaller, and the results become more stable. This principle is closely related to the law of large numbers, which explains why larger samples tend to give more reliable estimates.

However, even very large samples will still show some variation, as randomness can never be completely removed.

Common Mistakes

• Assuming experimental probability must equal theoretical probability
• Using expected outcomes instead of observed data
• Forgetting to simplify fractions fully
• Rounding too early when converting to decimals

Real-Life Applications

Experimental probability using large data sets is widely applied in real-world situations. Engineers repeatedly test components to estimate failure rates. Scientists perform experiments many times to check reliability. Game developers simulate large numbers of dice rolls to test fairness.

In all cases, decisions are based on collected evidence rather than assumptions.

Frequently Asked Questions

Does experimental probability become exact with more trials?
No. It usually becomes more stable, but randomness always remains.

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 large data sets and understanding variation.

Study Tip

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