What is the theoretical probability of getting heads on a fair coin?
Two equal outcomes.
1 favourable outcome out of 2 → 1/2.
Theoretical vs Experimental Probability
Theoretical probability is calculated using known outcomes, while experimental probability is based on results from trials. It builds on basic probability and connects directly to relative frequency.
Overview
Theoretical probability is based on all possible equally likely outcomes.
Experimental probability is based on actual results from an experiment.
The two values can be different, especially when the number of trials is small.
With many trials, the experimental probability often gets closer to the theoretical probability.
What you should understand after this topic
- Understand the difference between theoretical and experimental probability
- Calculate theoretical and experimental probability
- Understand why they do not always match exactly
- Understand why larger sample sizes matter
- Compare expected and actual results
Theoretical probability
\( \frac{\text{favourable outcomes}}{\text{total possible outcomes}} \)
Experimental probability
\( \frac{\text{times event happened}}{\text{number of trials}} \)
Key Definitions
Theoretical Probability
A probability based on what should happen from known possible outcomes.
Experimental Probability
A probability based on actual experiment results.
Relative Frequency
Another name for experimental probability.
Trial
One repeat of an experiment.
Outcome
A result of an experiment.
Sample Size
The number of trials carried out.
Key Rules
Theoretical formula
\( \frac{\text{favourable}}{\text{possible}} \)
Experimental formula
\( \frac{\text{frequency}}{\text{trials}} \)
Small samples
Can give results quite different from theory.
Large samples
Usually give results closer to theoretical probability.
Quick Comparison
Theoretical
Use when predicting what should happen.
Experimental
Use when analysing what actually happened.
Fair dice example
Theoretical probability of rolling a 6 is \( \frac{1}{6} \).
Experiment example
If a 6 appears 9 times in 40 rolls, experimental probability is \( \frac{9}{40} \).
How to Solve
Step 1: Understand theoretical probability
Theoretical probability is based on what should happen when outcomes are equally likely.
\( P(\text{event}) = \frac{\text{favourable outcomes}}{\text{total possible outcomes}} \)
For a fair coin, \(P(\text{heads}) = \frac{1}{2}\).
Step 2: Understand experimental probability
Experimental probability is based on what actually happens in an experiment.
\( \text{Experimental probability} = \frac{\text{number of times event happens}}{\text{total trials}} \)
If heads appears 18 times in 40 flips, experimental probability is \(\frac{18}{40} = 0.45\).
This builds on basic probability.
This is also known as relative frequency.
Step 3: Compare the two types
Theoretical
What should happen.
Experimental
What actually happened.
Step 4: Understand why results differ
Experimental probability can vary because real results do not always match the expected result exactly.
Small numbers of trials can give unreliable results.
Exam tip: More trials usually give a better estimate.
Step 5: Use experimental probability to predict
You can use experimental probability to estimate future outcomes.
\( \text{Expected number} = \text{experimental probability} \times \text{number of trials} \)
Step 6: Know when to use each type
Use theoretical probability
When all possible outcomes are known and equally likely.
Use experimental probability
When real experiment results are given.
Step 7: Exam method summary
See relative frequency for experimental probability from data.
- Decide whether the question gives expected outcomes or experiment results.
- Use theoretical probability for equally likely outcomes.
- Use experimental probability for real data.
- For predictions, multiply probability by the number of future trials.
- Comment that more trials usually improve reliability.
Example Questions
Exam Checklist
Common Mistakes
These are common mistakes students make when working with theoretical and experimental probability in GCSE Maths.
Mixing up the formulas
Incorrect
A student uses the wrong formula for the situation.
Correct
Theoretical probability is based on equally likely outcomes: \(\frac{\text{favourable outcomes}}{\text{total outcomes}}\). Experimental probability is based on data: \(\frac{\text{number of successes}}{\text{number of trials}}\).
Using the wrong type of probability
Incorrect
A student uses experimental data when theoretical probability is required, or vice versa.
Correct
Read the question carefully to determine whether you should use known outcomes (theoretical) or collected data (experimental).
Expecting exact agreement
Incorrect
A student assumes both probabilities must be the same.
Correct
Experimental probability is an estimate and may differ from theoretical probability, especially with small samples.
Ignoring sample size
Incorrect
A student treats all experimental results as equally reliable.
Correct
Larger sample sizes give more reliable experimental probabilities, as results tend to get closer to the theoretical value.
Not identifying equally likely outcomes
Incorrect
A student applies theoretical probability when outcomes are not equally likely.
Correct
Theoretical probability assumes equally likely outcomes. If this is not true, the method may not be appropriate.
Try It Yourself
Practise comparing theoretical and experimental probabilities.
Foundation
Foundation Practice
Compare theoretical and experimental probabilities.
A coin is flipped 10 times and lands on heads 7 times. What is the experimental probability of heads?
Use relative frequency.
7 ÷ 10 = 7/10.
Why might experimental probability differ from theoretical probability?
Think randomness.
Experimental results vary due to chance.
A die is rolled 20 times and a 6 appears 4 times. Find the experimental probability.
4 ÷ 20.
4 ÷ 20 = 1/5.
What is the theoretical probability of rolling a 6 on a fair die?
6 outcomes total.
1 out of 6 → 1/6.
A spinner lands on red 12 times out of 30 spins. Find the experimental probability.
Simplify 12/30.
12 ÷ 30 = 2/5.
Which is always exact?
Based on theory.
Theoretical probability is exact.
A coin is flipped 50 times and lands heads 28 times. Find the experimental probability.
Simplify 28/50.
28 ÷ 50 = 14/25.
What happens as the number of trials increases?
Think large numbers.
More trials → better estimate.
A die is rolled 60 times and a 3 appears 15 times. Find the experimental probability.
Simplify 15/60.
15 ÷ 60 = 1/4.
Higher
Higher Practice
Analyse and compare theoretical and experimental probabilities.
A coin is flipped 100 times and lands heads 55 times. Which statement is correct?
Compare to 0.5.
55/100 ≈ 0.5 → close estimate.
A die is rolled 200 times and a 6 appears 40 times. Find the experimental probability.
Simplify 40/200.
40 ÷ 200 = 1/5.
A spinner is fair with 4 equal sections. Experimental probability of A is 0.35. What does this suggest?
Theoretical is 0.25.
Difference likely due to small sample.
A coin is flipped 400 times and lands heads 210 times. Estimate probability of heads.
Simplify 210/400.
210 ÷ 400 = 21/40.
Which factor improves reliability of experimental probability?
Think accuracy.
More trials → better estimate.
A machine produces 500 items. 25 are faulty. Estimate probability of fault.
Simplify 25/500.
25 ÷ 500 = 1/20.
If experimental probability is very different from theoretical, what might be true?
Think causes.
Could be small sample or bias.
A die is rolled 600 times and a 6 appears 90 times. Estimate probability of rolling a 6.
Simplify 90/600.
90 ÷ 600 = 3/20.
A student claims experimental probability is always correct. Why is this wrong?
Think randomness.
Experimental probability is an estimate.
A spinner lands on blue 48 times out of 120 spins. Estimate probability of blue.
Simplify 48/120.
48 ÷ 120 = 2/5.
Games
Practise this topic with interactive games.
Games coming soon.
Frequently Asked Questions
What is theoretical probability?
Based on expected outcomes.
What is experimental probability?
Based on actual results.
Why compare them?
To understand real-world variation.