P5: Experimental vs Theoretical
Understand that empirical samples tend towards theoretical probability
📋 Key Concepts
Key Idea: As the number of trials increases, the relative frequency (experimental probability) gets closer to the theoretical probability.
Key Terms
| Term | Definition | How Found |
|---|---|---|
| Theoretical Probability | Expected probability based on equally likely outcomes | Calculation |
| Experimental Probability | Probability based on actual results | Trials/experiments |
| Relative Frequency | Same as experimental probability | Counting outcomes |
Law of Large Numbers:
As number of trials → ∞, relative frequency → theoretical probability
As number of trials → ∞, relative frequency → theoretical probability
📝 Theoretical vs Experimental
Theoretical: What we expect to happen based on maths.
Experimental: What actually happens when we try it.
Experimental: What actually happens when we try it.
Example 1
A fair coin should have P(Heads) = 0.5 (theoretical)
Experiment: Flip coin 10 times. Results: 7 heads, 3 tails
Experimental P(Heads) = 7/10 = 0.7
This differs from 0.5 due to small sample size.
Flip coin 1000 times. Results: 502 heads, 498 tails
Experimental P(Heads) = 502/1000 = 0.502
Now much closer to theoretical probability!
📝 Convergence
Pattern: With more trials, experimental probability becomes more stable and gets closer to theoretical probability.
Example 2
A dice is rolled multiple times. The number of sixes is recorded:
| Trials | Sixes | Relative Frequency |
|---|---|---|
| 10 | 3 | 3/10 = 0.300 |
| 100 | 19 | 19/100 = 0.190 |
| 1000 | 168 | 168/1000 = 0.168 |
| 10000 | 1662 | 1662/10000 = 0.1662 |
Theoretical probability = 1/6 ≈ 0.1667
As trials increase, relative frequency approaches 0.1667
📝 Estimating Probability
Use: When we don't know the theoretical probability (e.g., biased dice), we use experimental probability from many trials to estimate it.
Example 3
A biased dice is rolled 500 times. A 6 appears 120 times. Estimate P(6).
Solution:
Relative frequency of 6 = 120/500 = 0.24
Estimated P(6) = 0.24
Since the dice is biased, we cannot use theoretical probability.
Example 4
A spinner is spun 200 times. It lands on blue 85 times. Estimate how many times it would land on blue in 1000 spins.
Solution:
Experimental P(blue) = 85/200 = 0.425
Expected blue in 1000 spins = 0.425 × 1000 = 425
📝 Expected Frequency
Formula: Expected frequency = Probability × Number of trials
Example 5
A fair dice is rolled 300 times. How many times would you expect to roll a 6?
Solution:
P(6) = 1/6
Expected frequency = 1/6 × 300 = 50 times
This is an expectation - actual results will vary.
📝 Checking for Bias
Method: If experimental probability is consistently different from theoretical probability over many trials, the experiment may be biased.
Example 6
A coin is flipped 1000 times. Heads appears 600 times. Is the coin likely biased?
Solution:
P(Heads) experimental = 600/1000 = 0.6
P(Heads) theoretical = 0.5
With 1000 trials, we'd expect results close to 0.5
0.6 is quite far from 0.5 - coin is likely biased
🧮 Worked Examples — Exam Style
Experimental probability = successes ÷ trials. Expected frequency = probability × number of trials. Long-run relative frequency converges to the true probability.
A spinner lands on A 47 times in 200 spins. Estimate P(A) and how many times A would land in 500 spins.
P(A) ≈ 47/200 = 0.235 → expected in 500: 0.235 × 500 = 117.5 ≈ 118 times
M1 for 47/200, A1 for the estimate, A1 for 117.5/118. Expected values can be decimals — round sensibly and say 'approximately'.
A sweet packet contains 3 flavours. In 60 packets, 24 contained a blue sweet. The factory makes 5000 packets daily. Estimate daily blue packets.
P(blue) ≈ 24/60 = 0.4 → 0.4 × 5000 = 2000 packets
M1 for 0.4, A1 for 2000. Scaling estimates to production volumes is the classic applied version.
Exam wording to expect: "estimate the probability", "how many would you EXPECT", "do you think the coin is fair?".
❓ Practice Questions
Q1: A fair dice is rolled 60 times. How many times would you expect to roll an even number?
Q2: A spinner is spun 150 times. It lands on red 45 times. Estimate P(red).
Q3: A coin is flipped 100 times with 55 heads. Is this what you'd expect? Explain.
Q4: A bag contains coloured counters. In 40 draws, 16 were blue. Estimate how many blue counters there would be if there are 200 in total.
Q5: A biased dice has P(6) = 0.25. How many sixes would you expect in 400 rolls?
Q6: A weather station records rain on 84 of 365 days. Estimate P(rain) on a given day (2 d.p.)
Q7: P(double six) for a pair of dice is 1/36. Out of 720 rolls, how many double sixes would you expect?
✅ Answers
- P(even) = 3/6 = 1/2. Expected = 1/2 × 60 = 30 times
- P(red) = 45/150 = 3/10 = 0.3
- 55/100 = 0.55 is close to 0.5, so this is expected variation for a fair coin
- 16/40 = 0.4. Expected blue in 200 = 0.4 × 200 = 80 blue counters
- Expected sixes = 0.25 × 400 = 100 sixes
📋 Model Answers — Full Working
Every mark comes from the working, not just the answer. These model answers show the lines a mark scheme expects to see.
Q1. P(even) = 1/2 → 60 × 1/2 = 30
Q2. 45/150 = 0.3
Q3. 0.55 vs 0.5 → yes, within normal variation — 5% off with n = 100 is unremarkable
Q4. 16/40 = 0.4 → 0.4 × 200 = 80 blue
Q5. 0.25 × 400 = 100 sixes
Q6. 84/365 = 0.23
Q7. (1/36) × 720 = 20
📌 Key Facts to Memorise
- Experimental P = successes ÷ trials (a relative frequency)
- Expected frequency = P × trials — can be a decimal before sensible rounding
- More trials → estimate improves (law of large numbers, in spirit)
- Fairness checks: compare experimental to theoretical across MANY trials
- Bias suspected when relative frequency stays far from theory as trials grow
🗺️ In the Exam
Experimental probability: 2–4 marks on Papers 1/2. The estimate-then-scale two-step is the standard format; the fairness evaluation is the AO3 layer.
Links to: probability scale (P3), fairness (P2), trees (P8).
🧭 Board Focus — AQA
Spec 8300. Three papers, each 80 marks and 1h30: Paper 1 non-calculator, Papers 2 and 3 calculator. Foundation/Higher tiers. Roughly a third of the marks sit under AO2 (reasoning) and AO3 (problem-solving), so show every line of working — method marks survive arithmetic slips.
Always check: specification details change — confirm paper structure and set texts against the current specification for your exam year.
🎯 Exam Tips
- More trials = more accurate estimate of true probability
- Expected frequency = probability × number of trials
- Small samples can give misleading results - need many trials
- Use experimental probability when you can't calculate theoretical
- If experimental ≈ theoretical over many trials, the object is likely fair
🧠 Problem-Solving Strategies
Problem-Solving
When comparing experimental and theoretical probability: (1) Calculate theoretical probability first as a benchmark, (2) Calculate relative frequency from the data, (3) More trials = relative frequency gets closer to theoretical, (4) Expected frequency = probability × number of trials, (5) If experimental probability is very different from theoretical over many trials, suspect bias.
Multi-Step Problem
A dice is rolled 300 times. The number 6 appears 75 times. (a) Find the experimental P(6). (b) Is the dice fair? (c) How many sixes would you expect from 300 rolls of a fair dice?
Solution: (a) Experimental P(6) = 75/300 = 0.25. (b) Theoretical P(6) = 1/6 ≈ 0.167. Expected sixes from 300 = 1/6 × 300 = 50. We got 75, which is much higher than 50, so the dice is likely biased. (c) Expected = 50 sixes.
⚠️ Common Errors
Watch Out!
1. Wrong: Saying a coin is biased after only 10 flips because you got 7 heads Correct: With small samples, results vary a lot — you need many trials (hundreds) before concluding bias
2. Wrong: Calculating expected frequency as probability + number of trials instead of multiplying Correct: Expected frequency = probability × number of trials (multiply, not add)
3. Wrong: Saying "the relative frequency will eventually equal the theoretical probability" Correct: It tends towards (gets closer to) the theoretical probability, but rarely equals it exactly
✍️ 6-Mark Exam Question
Extended Answer
6 marks: A company tests a biased coin. In 50 flips, heads appears 35 times. (a) Find the experimental probability of heads. (b) Use this to estimate the number of heads in 200 flips. (c) Another test of 500 flips gives 310 heads. Calculate the new experimental probability and comment on which estimate is more reliable.
(a) Experimental P(H) = 35/50 = 0.7.
(b) Expected heads in 200 = 0.7 × 200 = 140.
(c) New experimental P(H) = 310/500 = 0.62.
The second estimate (0.62) is more reliable because it is based on 500 trials rather than 50. With more trials, the relative frequency gives a better estimate of the true probability. The true probability is likely between 0.62 and 0.7, and closer to 0.62.
Mark scheme: M1 for 35/50, A1 for 0.7, M1 for 0.7×200, A1 for 140, M1 for 310/500=0.62, A1 for stating second estimate more reliable with reason (more trials)
📊 AO3: Reason & Interpret
Reasoning and Interpretation
Aisha rolls a fair dice 60 times and gets 15 sixes. Ben rolls the same dice 600 times and gets 108 sixes.
(a) Calculate the relative frequency of sixes for each person.
(b) Whose result is closer to the theoretical probability? Explain why.
(c) Aisha says "The dice must be biased because I got way more sixes than expected." Evaluate her claim.
Answers: (a) Aisha: 15/60 = 0.25. Ben: 108/600 = 0.18. (b) Ben's result (0.18) is closer to theoretical P(6) = 1/6 ≈ 0.167 because he did more trials. (c) Aisha's claim is not well supported. With only 60 rolls, getting 15 sixes (expected 10) is not unusual variation. Ben's much larger sample shows 0.18, much closer to 0.167, suggesting the dice is likely fair.
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