GCSE Maths Practice: theoretical-vs-experimental-probability

Experimental Probability from Repeated Die Rolls

Experimental probability measures how often an outcome actually occurs when an experiment is carried out. Instead of relying on expectations or assumptions, it uses real data collected from repeated trials. Rolling a die is a common GCSE Maths example because it clearly shows how randomness affects results.

The Core Method

To calculate experimental probability, use the formula:

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

This calculation always uses observed results. The answer will be between 0 and 1 and may be written as a fraction, decimal, or percentage depending on the question.

Worked Example

A die is rolled 80 times and the number 5 appears 14 times. The experimental probability of rolling a 5 is:

\( \frac{14}{80} = \frac{7}{40} \)

This value comes directly from the experiment rather than from the assumption that each number is equally likely.

Experimental vs Theoretical Probability

Theoretical probability is calculated using known information about equally likely outcomes. For a fair die, each number has the same theoretical chance. Experimental probability, however, depends on the results collected during the experiment.

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

Why Results Can Differ

Random variation means that results can differ from expectations, especially with a small number of trials. Increasing the number of trials usually makes the probability more reliable, as chance has less influence.

This is why probability experiments often involve many repeated trials.

Common Mistakes

• Using the number of possible outcomes instead of observed results
• Assuming the probability must equal the theoretical value
• Forgetting to simplify the fraction
• Confusing experimental probability with theoretical probability

Real-Life Applications

Experimental probability is used widely outside the classroom. Scientists repeat experiments to test reliability. Engineers run repeated tests to estimate failure rates. Game developers simulate dice rolls to ensure games are fair.

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

Frequently Asked Questions

Can experimental probability change if more trials are added?
Yes. Adding more trials can change the result and usually improves accuracy.

Does experimental probability always equal theoretical probability?
No. It often gets closer with more trials, but differences can remain.

Why is experimental probability tested at GCSE?
It develops understanding of randomness, fractions, and data interpretation.

Study Tip

When a question mentions outcomes that were observed, recorded, or happened out of a number of trials, immediately apply the formula: observed outcomes ÷ total trials.