GCSE Maths Practice: theoretical-vs-experimental-probability

Experimental Probability with a Large Number of Trials

Experimental probability is based on results collected from real experiments rather than on predictions. When the number of trials is large, experimental probability becomes especially important because it demonstrates how results begin to stabilise. Coin flips are commonly used in GCSE Maths to illustrate this idea clearly.

The Core 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 what the question asks.

Worked Example

A coin is flipped 800 times and lands on heads 396 times. The experimental probability of landing heads is:

\( \frac{396}{800} = \frac{99}{200} \)

This value is found entirely from the results of the experiment and does not assume that the coin behaves perfectly.

Why Large Numbers Matter

With a small number of trials, experimental probability can vary significantly due to chance. As the number of trials increases, results usually become more consistent. This idea is linked to the law of large numbers, which states that experimental results tend to move closer to the theoretical probability as the number of trials increases.

This does not mean the results will ever be perfectly equal, but large samples reduce the impact of random variation.

Experimental vs Theoretical Probability

Theoretical probability predicts outcomes using equally likely events. For a fair coin, the theoretical probability of heads is one half. Experimental probability, however, uses observed results and may differ slightly, even with many trials.

Understanding the difference between these two types of probability is essential at Higher GCSE level.

Common Mistakes

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

Real-Life Applications

Experimental probability with large data sets is used in many real-world situations. Polling companies analyse thousands of responses to estimate opinions. Scientists rely on repeated trials to test reliability. Engineers analyse large samples to predict failure rates.

In all cases, decisions are based on observed data 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 forms.

Why is this a Higher-tier topic?
Because it involves understanding convergence, accuracy, and large data sets.

Study Tip

When a question involves a very large number of trials, focus on forming the correct fraction first and simplifying accurately before converting to a decimal if needed.