GCSE Maths Practice: theoretical-vs-experimental-probability
This question tests how to calculate experimental probability using results from repeated die rolls.
Always calculate experimental probability using observed results, not expected outcomes.
Experimental Probability from Die Rolls
Experimental probability is based on results collected from carrying out an experiment. Instead of predicting what should happen using theory, it focuses on what actually happened. Rolling a die many times is a common GCSE Maths example because it clearly shows how randomness affects outcomes.
The Key Formula
Experimental probability = number of times the event occurs ÷ total number of trials
This formula always uses observed results. The final value will always lie between 0 and 1 and may be written as a fraction, decimal, or percentage.
Worked Example
A die is rolled 100 times and the number 2 appears 21 times. The experimental probability of rolling a 2 is:
\( \frac{21}{100} \)
This result comes directly from recorded outcomes rather than from the assumption that each number is equally likely.
Experimental vs Theoretical Probability
Theoretical probability assumes all outcomes are equally likely. For a fair die, each face has the same theoretical chance. Experimental probability, however, depends on observed results and may differ due to randomness.
As more trials are carried out, experimental probability often moves closer to the theoretical value, but it does not need to match it exactly.
Why Large Numbers Matter
With a small number of trials, results can vary significantly due to chance. Increasing the number of trials usually makes the probability more reliable, as random variation has less influence.
This is why probability experiments often involve many repetitions.
Common Mistakes
- Using the number of possible outcomes instead of observed results
- Failing to simplify fractions
- Assuming experimental probability must equal theoretical probability
- Confusing observed results with expected outcomes
Real-Life Applications
Experimental probability is widely used outside the classroom. Scientists repeat experiments to test reliability. Engineers run repeated tests to estimate failure rates. Game developers simulate dice rolls to check fairness.
In each case, decisions are based on collected data rather than assumptions.
Frequently Asked Questions
Can experimental probability change if more trials are added?
Yes. Adding more trials can change the probability and usually improves accuracy.
Does experimental probability have to be simplified?
Yes. GCSE Maths expects answers to be in their simplest form unless stated otherwise.
Why is experimental probability important?
It develops understanding of randomness, data handling, and fractions.
Study Tip
If a question mentions outcomes that were observed, recorded, or occurred out of a certain number of trials, immediately use the formula: observed outcomes ÷ total trials.
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