Experimental Probability – Grade 8 Maths

Relative Frequency

freq ÷ total trials

Theoretical vs Experimental

Expected vs actual results

Bias and Fairness

Detecting unfair spinners/dice

Large Samples

More trials → closer to theory

Estimating Probability

Use relative frequency as estimate

1. Relative Frequency

Experimental probability (also called relative frequency) is calculated from actual trials rather than theory.

Relative Frequency = Frequency of outcome ÷ Total number of trials

A coin is flipped 40 times. Heads appears 23 times.
Relative frequency of heads = 23/40 = 0.575

Relative frequency is an estimate of the true probability. It changes each time you repeat the experiment.

2. Theoretical vs Experimental Probability

	Theoretical	Experimental
Based on	Mathematical reasoning	Actual trials/experiments
Example (fair coin)	P(heads) = 0.5 exactly	Flip 10 times → might get 6 heads → 0.6
Accuracy	Always exact (if fair)	Varies — depends on sample size
When used	Fair dice, cards, spinners	Biased objects, real-world events

For a fair die: P(6) = 1/6 ≈ 0.167 (theoretical)
Roll 60 times, get 8 sixes: relative freq = 8/60 ≈ 0.133 (experimental)

3. Detecting Bias

A spinner or die is biased (unfair) if some outcomes are more likely than others. Experimental probability helps detect bias.

A spinner has 4 equal sectors (A, B, C, D). Theoretical P(each) = 0.25.
After 200 spins: A=62, B=58, C=47, D=33.
Relative freq of D = 33/200 = 0.165 — much lower than 0.25.
Evidence the spinner may be biased against D.

The more trials you run, the stronger the evidence of bias. A small difference with few trials could just be chance.

4. Large Samples Converge to Theory

As the number of trials increases, the experimental probability gets closer to the theoretical probability.

Flipping a fair coin: theoretical P(H) = 0.5
10 flips: might get 3H → 0.3
100 flips: might get 48H → 0.48
1000 flips: likely to get ~500H → ≈0.5

This is called the Law of Large Numbers — random variation averages out over many trials.

In exam questions: "Explain why more trials would give a better estimate." Answer: More trials reduce the effect of random variation, making the relative frequency closer to the true probability.

5. Using Relative Frequency to Estimate Probability

When we don't know the true probability (e.g. a biased coin, traffic lights, weather), we use relative frequency as our best estimate.

Estimated P(event) = Relative frequency = Favourable trials ÷ Total trials

A drawing pin is dropped 50 times. It lands point-up 32 times.
Estimated P(point up) = 32/50 = 0.64
If dropped 200 times: Expected point-up = 0.64 × 200 = 128

Use this estimated probability to predict future outcomes, just like theoretical probability.

Example 1 — Calculate Relative Frequency

A die is rolled 120 times. Results: 1→18, 2→22, 3→19, 4→24, 5→17, 6→20.

Relative freq of 4: 24/120 = 0.2

Theoretical P(4): 1/6 ≈ 0.167

Comparison: 0.2 vs 0.167 — some difference, but with only 120 trials this is not unusual for a fair die.

Relative freq of getting ≤ 3: (18+22+19)/120 = 59/120 ≈ 0.492

Example 2 — Predicting with Relative Frequency

In 80 trials, a biased spinner lands on red 28 times.

Estimated P(red) = 28/80 = 0.35

Predict red in 200 spins: 0.35 × 200 = 70 times

P(not red) = 1 − 0.35 = 0.65. Predict non-red: 0.65 × 200 = 130 times

Example 3 — Comparing with Theory

A fair coin is flipped and the cumulative relative frequency of heads is recorded:

Trials	Heads	Rel. Freq.
10	3	0.30
50	22	0.44
100	48	0.48
500	251	0.502

As trials increase, relative frequency approaches theoretical value of 0.5.

Example 4 — Detecting Bias

A spinner has 3 equal sectors (Red, Blue, Green). After 150 spins: R=62, B=54, G=34.

Theoretical P(each) = 1/3 ≈ 0.333

Rel. freq Green = 34/150 ≈ 0.227 — notably below 0.333

Rel. freq Red = 62/150 ≈ 0.413 — notably above 0.333

Conclusion: The spinner appears biased — Red is more likely and Green less likely than expected.

🎲 Relative Frequency Calculator

Successes: Total trials:

📊 Frequency Table Analyser

Enter outcome frequencies (comma-separated). Theoretical probability per outcome = 1/n.

Frequencies:

🪙 Coin Flip Simulator

Number of flips:

Exercise 1 — Relative Frequency

Give answers as decimals (2 d.p.).

1. 15 heads in 40 flips. Relative frequency?

2. 36 sixes in 120 rolls. Relative frequency?

3. 7 red in 35 draws. Relative frequency?

4. 48 heads in 200 flips. Relative frequency?

5. 90 successes in 300 trials. Relative frequency?

6. 3 in 25. Relative frequency?

7. 42 in 60. Relative frequency?

8. 11 in 44. Relative frequency?

9. 250 in 1000. Relative frequency?

10. 13 in 52. Relative frequency?

Exercise 2 — Predicting with Relative Frequency

Use estimated probability × new total.

1. Rel. freq = 0.4, 200 more trials. Expected successes?

2. 18 successes in 60. Expected successes in 120?

3. 25 red in 100. Expected red in 300?

4. 12 in 40. Expected in 200?

5. Rel. freq = 0.35, 500 trials. Expected?

6. 9 in 30. Expected in 150?

7. Rel. freq = 0.6, 250 trials. Expected?

8. 70 in 200. Expected in 500?

9. 44 in 80. Expected in 600?

10. 3 in 15. Expected in 100?

Exercise 3 — Theoretical vs Experimental

Compare results. Give theoretical values as decimals (3 d.p. where needed).

1. Fair die, 60 rolls. Theoretical P(3)?

2. Fair coin, theoretical P(tails)?

3. Bag: 3R, 3B, 3G, 3Y (12 total). Theoretical P(red)?

4. Spinner with 5 equal sectors. Theoretical P(any one)?

5. 40 rolls of fair die, expected number of 6s?

6. 60 rolls, expected even numbers?

7. 80 flips fair coin. Expected heads?

8. 120 draws from 3R,3B,3G bag (12 total). Expected red?

9. 300 rolls. Expected P(≤2)?

10. 500 trials. Theoretical P = 0.4. Expected?

Exercise 4 — Finding Trials or Successes

1. Rel. freq = 0.4, successes = 20. Total trials?

2. Rel. freq = 0.25, total = 80. Number of successes?

3. 36 successes, rel. freq = 0.6. Total trials?

4. Rel. freq = 0.3, successes = 45. Total?

5. 14 successes, rel. freq = 0.7. Total?

6. Rel. freq = 0.15, total = 200. Successes?

7. Rel. freq = 0.8, successes = 120. Total?

8. 90 successes, total = 360. Rel. freq?

9. Rel. freq = 0.45, total = 400. Successes?

10. 35 successes, total = 140. Rel. freq?

Exercise 5 — Mixed

1. Relative frequency = 0.55. In 80 trials, expected successes?

2. 30 successes in 75. Rel. freq?

3. Rel. freq = 0.2. In 450 trials, expected?

4. 100 successes, rel. freq = 0.5. Trials?

5. Fair die. Theoretical P(prime: 2,3,5)?

6. 150 trials, 45 successes. Predict successes in 600.

7. Rel. freq = 0.72, total = 250. Non-successes?

8. 33 in 110. Expected in 330?

9. Successes = 180, trials = 450. Rel. freq?

10. Rel. freq = 0.85, n = 60. Expected?

🏋️ Practice — 20 Questions

1. 28 in 80. Rel. freq?

2. Rel. freq = 0.3, 150 trials. Expected?

3. 54 in 90. Expected in 300?

4. Rel. freq = 0.6, successes = 48. Trials?

5. Fair die 180 rolls. Expected multiples of 2?

6. 45 in 150. Rel. freq?

7. Rel. freq = 0.45, n=200. Expected?

8. 110 in 400. Rel. freq?

9. Rel. freq = 0.25, expected = 50. Trials?

10. 36 in 120. Expected in 500?

11. Fair coin 400 flips. Expected tails?

12. 64 successes, 160 trials. Rel. freq?

13. Rel. freq = 0.7, 350 trials. Successes?

14. 5 in 25. Expected in 400?

15. Rel. freq = 0.15, n = 300. Expected?

16. 88 in 320. Rel. freq?

17. Rel. freq = 0.9, successes = 270. Trials?

18. 24 in 60. Expected in 750?

19. Rel. freq = 0.05, n = 600. Expected?

20. 48 in 64. Rel. freq?

🏆 Challenge — 8 Questions

1. A biased spinner lands on Red 42 times in 140 trials. How many times would you expect Red in 350 trials?

2. A fair die is rolled 240 times. The actual count of 6s is 52. How many more 6s than expected were rolled?

3. A drawing pin is dropped and relative frequency of point-up is 0.62 after 50 trials. After 50 more trials, the combined total shows 68 point-up out of 100. What is the new relative frequency?

4. Experimental probability of event A = 3/7. How many successes in 490 trials?

5. A coin is flipped 500 times. 285 heads. Is this evidence of bias? Enter 1 for biased, 2 for not clearly biased.

6. Relative frequency of event X is 0.35 after n trials. There were 63 successes. How many trials were there?

7. A four-sector spinner (A,B,C,D) is spun 200 times: A=58, B=62, C=43, D=37. What is the relative frequency of (C or D)?

8. The theoretical probability of winning is 0.4. After 150 trials, you won 54 times. How many fewer wins than expected?

🧪 Experimental Probability