GCSE Maths Practice: theoretical-vs-experimental-probability
This Higher-level question tests experimental probability using repeated rolls of a die.
Always calculate experimental probability using observed results rather than expected outcomes.
Experimental Probability with Large Dice Experiments
Experimental probability is based on results collected from repeated trials rather than on predictions made before an experiment. In Higher GCSE Maths, students are expected to interpret results from larger data sets and understand how experimental probability behaves as the number of trials increases. Dice experiments are a useful way to explore this because each outcome is equally likely in theory, yet results often vary in practice.
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 die is rolled 250 times and the number 2 appears 52 times. The experimental probability of rolling a 2 is:
\( \frac{52}{250} = \frac{26}{125} \)
This value comes directly from the experiment and does not assume that each face appears exactly the same number of times.
Experimental vs Theoretical Probability
Theoretical probability is calculated using equally likely outcomes. For a fair die, each face has the same theoretical chance. Experimental probability, however, is calculated using observed results and may differ because of randomness.
As the number of trials increases, experimental probability often moves closer to the theoretical probability. However, even with many trials, the results do not need to match the theoretical value exactly.
Why Larger Samples Matter
With small numbers of trials, random variation can have a large effect on results. As more trials are carried out, results usually become more stable and reliable. This idea is closely linked to the law of large numbers, which is an important concept at Higher GCSE level.
Larger samples reduce the impact of chance but never remove it entirely.
Common Mistakes
- Assuming experimental probability must equal theoretical probability
- Forgetting to simplify fractions fully
- Using expected outcomes instead of observed data
- Rounding too early during calculations
Real-Life Applications
Experimental probability using large data sets is common in real life. Engineers test components repeatedly to estimate failure rates. Scientists run experiments many times to check reliability. Game designers simulate dice rolls to ensure fairness in games.
In all cases, conclusions are based on observed 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 specifies otherwise.
Why is this considered a Higher-tier question?
Because it involves interpreting large data sets and understanding variation.
Study Tip
When working with a large number of trials, always form the fraction first, simplify fully, and only then convert to a decimal if required.
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