Experimental probability (Edexcel GCSE Statistics): Revision Notes
Experimental probability
What is experimental probability?
Experimental probability helps us find estimates of how likely something is to happen by actually carrying out experiments and recording the results. Unlike theoretical probability (which we can work out using maths), experimental probability is based on real data from trials we've conducted.
Each time we repeat an experiment or collect a response from a survey, this is called a trial. The more trials we carry out, the more reliable our estimate becomes.
The key difference is that theoretical probability uses mathematical calculations to predict outcomes, while experimental probability uses actual data collected from real experiments to estimate likelihood.
The experimental probability formula
The formula for experimental probability is:
This gives us a decimal between 0 and 1, which we can also express as a fraction or percentage. Remember that probabilities can never be negative or greater than 1.
Why repeat experiments?
The key principle with experimental probability is that the more times we repeat the experiment, the more accurate our estimate becomes. This is because:
- Small samples can be misleading due to random variation
- Larger samples average out these random fluctuations
- We get closer to the true probability as our sample size increases
Sample Size Matters!
Never draw conclusions from just a few trials. The reliability of your experimental probability estimate depends heavily on having enough data points to minimise the impact of random variation.
Comparing experimental and predicted results
We can use experimental probability to test whether something is fair or biassed by comparing our experimental results with what we would theoretically expect.
For example, if we spin a fair 4-sided spinner 100 times, we would predict each side to come up about 25 times. If our experimental results are very different from this prediction, it might suggest the spinner is biassed.
When testing for bias, look for significant differences between experimental and expected results. Small variations are normal due to random chance, but large consistent differences often indicate bias.
Worked example: Testing a spinner for bias
Worked Example: Testing a Spinner for Bias
Problem: Gary has a 4-sided spinner with sides labelled A, B, C and D. He spins it 200 times and records these results:
| Side | A | B | C | D |
|---|---|---|---|---|
| Frequency | 52 | 15 | 54 | 79 |
(a) If the spinner is fair, how many times would we expect it to land on side D?
Solution:
- Total number of spins = 200
- If fair, each side should be equally likely
- Expected frequency for each side =
- Expected number of times to land on D = 50
(b) Compare the experimental results with the expected results.
Solution: Let's compare each side:
- Side A: Got 52, expected 50 (slightly higher than expected)
- Side B: Got 15, expected 50 (much lower than expected)
- Side C: Got 54, expected 50 (slightly higher than expected)
- Side D: Got 79, expected 50 (much higher than expected)
Conclusion: The spinner appears to be biassed in favour of side D and against side B, as these show the biggest differences from what we'd expect.
Testing a coin for bias
Worked Example: Testing a Coin for Bias
Experiment: Testing whether a coin is biassed by flipping it multiple times.
| Outcome | Number of heads | Number of throws | Experimental probability |
|---|---|---|---|
| Result | 34 | 50 |
This gives an experimental probability of 0.68 for getting a head. Since we'd expect 0.5 for a fair coin, this suggests the coin might be biassed towards heads.
Important points about sample size
When comparing different experimental probability estimates, remember that larger sample sizes generally give more accurate results. This is because:
- Random variation has less impact on larger samples
- The estimate becomes more stable as we collect more data
- We can be more confident in conclusions drawn from larger datasets
Sample Size Comparison
For instance, if three people test the same dice with 60, 90, and 150 rolls respectively, the person with 150 rolls will likely have the most accurate estimate of the true probability.
Key Points to Remember:
- Experimental probability =
- More trials lead to more accurate estimates
- Compare experimental results with theoretical predictions to detect bias
- Large differences between experimental and expected results suggest bias
- Always consider sample size when evaluating how reliable an experimental probability estimate is