GCSE Maths Practice: theoretical-vs-experimental-probability

Experimental Probability in Card Experiments

Experimental probability measures how often an event actually occurs when an experiment is repeated. Instead of predicting results using theory alone, it relies on real data collected from observations. In GCSE Maths, card experiments are a common way to test understanding of experimental probability because they involve clear outcomes and repeated trials.

The Key Formula

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

This formula always uses observed results and produces a value between 0 and 1. The probability can be written as a fraction, decimal, or percentage depending on what the question asks.

Worked Example

Suppose a card is drawn from a shuffled deck 40 times and a black card appears 18 times. The experimental probability of drawing a black card is:

\( \frac{18}{40} = \frac{9}{20} \)

This value comes directly from the results of the experiment rather than from the known composition of a deck.

Experimental vs Theoretical Probability

Theoretical probability is calculated using known information, such as the number of red and black cards in a deck. Experimental probability, however, is based entirely on recorded outcomes. Because experiments involve randomness, the experimental probability may differ from the theoretical probability.

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

Why Experimental Probability Can Vary

When the number of trials is small, results can be uneven due to chance. Increasing the number of trials usually makes the probability more reliable, as random variation has less effect.

For example, drawing cards 10 times may give unexpected results, while drawing cards hundreds of times usually produces a more balanced pattern.

Common Mistakes

• Using the total number of possible outcomes instead of observed results
• Assuming results must match theoretical probability
• Forgetting to simplify fractions
• Mixing up experimental and theoretical probability

Real-Life Applications

Experimental probability is widely used outside the classroom. Casinos analyse game outcomes to understand fairness. Scientists repeat experiments to check reliability. Businesses examine past sales data to predict future trends.

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

Frequently Asked Questions

Can experimental probability change?
Yes. If more trials are carried out, the probability may increase or decrease.

Does experimental probability have to be simplified?
Yes. In GCSE Maths, answers should always be simplified unless stated otherwise.

Why are card experiments used so often?
Because they clearly demonstrate randomness and repeated trials.

Study Tip

Whenever a question mentions outcomes that were observed, recorded, or happened out of a number of trials, immediately use the experimental probability formula: results ÷ trials.