🎲 Experimental Probability

Cambridge Lower Secondary · Grade 7 · Probability & Statistics

Relative Frequency
Relative Frequency = Frequency ÷ Total Trials
After 40 flips, 22 heads → RF = 22 ÷ 40 = 0.55
Law of Large Numbers
More trials → closer to theoretical
10 flips: 0.7 · 100 flips: 0.53 · 1000 flips: 0.499
Expected Frequency
Expected = Probability × Number of Trials
P(6) = 1/6, 120 rolls → expect 20 sixes

Quick Demo: Flip this coin 10 times! 🪙

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What you'll learn:

  • The difference between theoretical and experimental probability
  • Calculating relative frequency from tallies and data tables
  • How the Law of Large Numbers makes results more reliable
  • Identifying bias in experiments (fair vs rigged)
  • Calculating expected frequency = probability × trials
  • Designing fair experiments and comparing results
Created by Mr Josh

📖 Learn: Experimental Probability

Part 1: Theoretical vs Experimental Probability

Theoretical probability is calculated using maths, assuming everything is perfectly fair.

📌 P(event) = Number of favourable outcomes ÷ Total possible outcomes
Example: Fair coin → P(Heads) = 1 ÷ 2 = 0.5 (or ½)
Example: Fair die → P(rolling a 3) = 1 ÷ 6 ≈ 0.167

Experimental probability is based on what actually happens when you carry out an experiment.

📌 Experimental P(event) = Frequency of event ÷ Total number of trials
Example: You flip a coin 50 times and get 27 heads → Exp. P(H) = 27 ÷ 50 = 0.54
💡 Experimental probability is never exactly the same as theoretical — but with more trials it gets closer!

Part 2: Relative Frequency

Relative frequency is another name for experimental probability. It tells you what fraction of trials gave a particular outcome.

📌 Relative Frequency = Frequency ÷ Total Trials
OutcomeTallyFrequencyRelative Frequency
Heads|||| |||| |||| |||1818 ÷ 40 = 0.45
Tails|||| |||| |||| |||||2222 ÷ 40 = 0.55
Total401.00
✅ All relative frequencies must always add up to 1 (or 100%).

Part 3: Law of Large Numbers

The more trials you carry out, the closer the experimental probability gets to the theoretical probability.

10 coin flips: might get 7 heads → relative frequency = 0.70 (far from 0.5)
100 coin flips: might get 53 heads → relative frequency = 0.53 (closer)
1000 coin flips: might get 499 heads → relative frequency = 0.499 (very close)
10 000 coin flips: will be extremely close to → 0.500
📈 This is called the Law of Large Numbers. Larger sample size = more reliable results!
⚠️ Even after 7 consecutive heads, the next flip is still 50/50. The coin has no memory!

Part 4: Bias and Fairness

An experiment is biased if outcomes are not equally likely when they should be. We can detect bias from experimental results.

A fair coin should give relative frequency ≈ 0.5 for heads
If after 200 trials, heads appears 148 times: RF = 148/200 = 0.74 → likely biased!
If after 200 trials, heads appears 99 times: RF = 99/200 = 0.495 → probably fair
🔍 The further the relative frequency is from the theoretical value, the more suspicious we should be — especially with a large number of trials.
💡 A few trials could give unusual results by chance. Many trials showing unusual results = likely biased.

Part 5: Expected Frequency

If we know the probability and plan to run an experiment, we can predict how many times an outcome will occur.

📌 Expected Frequency = Probability × Number of Trials
Fair die, rolled 120 times → Expected number of 6s = (1/6) × 120 = 20
Fair coin, flipped 300 times → Expected heads = 0.5 × 300 = 150
Bag has 3 red and 7 blue, pick one 200 times (replace each time) → Expected reds = 0.3 × 200 = 60
⚠️ Expected frequency is a prediction — the actual result will usually be close but not always exact.
Created by Mr Josh

💡 Worked Examples

Example 1: Calculating Relative Frequency

A spinner is spun 80 times. It lands on Red 24 times, Blue 32 times, Green 24 times. Calculate the relative frequency for each colour.

Step 1: Check total: 24 + 32 + 24 = 80 ✓
Step 2: RF(Red) = 24 ÷ 80 = 0.30
Step 3: RF(Blue) = 32 ÷ 80 = 0.40
Step 4: RF(Green) = 24 ÷ 80 = 0.30
Check: 0.30 + 0.40 + 0.30 = 1.00 ✓
💡 If the spinner were fair with 3 equal sections, theoretical probability = 1/3 ≈ 0.333 each. Red and Green are a bit low — Blue is getting more than its share!

Example 2: Expected Frequency

The probability of winning a prize in a game is 0.15. If 400 people play, how many would you expect to win?

Step 1: Use: Expected Frequency = Probability × Number of Trials
Step 2: Expected winners = 0.15 × 400 = 60
Answer: We would expect approximately 60 people to win a prize.
⚠️ This is a prediction — the actual number of winners might be 58 or 63, but should be close to 60.

Example 3: Is This Die Biased?

A die is rolled 300 times. The number 4 appears 87 times. Is the die likely to be biased?

Step 1: Find experimental probability of rolling 4:
Exp. P(4) = 87 ÷ 300 = 0.29
Step 2: Compare with theoretical: P(4) = 1/6 ≈ 0.167
Step 3: 0.29 is significantly greater than 0.167
Conclusion: Yes — with 300 trials, this result strongly suggests the die is biased towards 4.
🔑 Key: Always compare to the theoretical value AND comment on the number of trials. More trials = more convincing evidence.

Example 4: Designing a Fair Experiment

James wants to test if a coin is fair. Describe how he should design his experiment.

Step 1: Use a large number of trials — at least 100, ideally 500+
Step 2: Flip the coin in the same way each time (same height, same surface)
Step 3: Record results in a tally chart as you go
Step 4: Calculate relative frequency for heads and tails
Step 5: Compare each relative frequency to 0.5
Conclusion rule: If both are close to 0.5 after many trials → coin is likely fair. If one is significantly higher → probably biased.
🎯 A fair experiment is one where all outcomes have an equal chance and there are no external influences (no weighted coin, no trick throw).
Created by Mr Josh

🎲 Probability Simulator

🪙 Coin & Dice Simulator

Choose your experiment, then flip/roll. Watch relative frequency converge to the theoretical value!

Ready!

🕵️ Biased Spinner Detector

Watch 50 automatic spins. Then decide: is this spinner fair or rigged?

📊 Expected vs Actual Frequency

Enter the probability and number of trials — see the expected frequency, then run the real experiment!

🃏 Coloured Card Frequency Builder

You're drawing cards from a bag (with replacement). Click each colour each time you draw one. Watch the relative frequencies update live!

Created by Mr Josh

✏️ Exercise 1: Relative Frequency from Tables

Calculate the relative frequency for each outcome. Give answers as decimals to 2 d.p.

Created by Mr Josh

✏️ Exercise 2: Expected Frequency

Use: Expected Frequency = Probability × Number of Trials

Created by Mr Josh

✏️ Exercise 3: Theoretical vs Experimental

Compare theoretical and experimental probabilities. Enter answers as decimals to 2 d.p. where needed.

Created by Mr Josh

🔍 Exercise 4: Identifying Bias

Analyse each scenario. Decide: is the spinner/coin/die likely biased or fair? Type biased or fair.

Created by Mr Josh

🧪 Exercise 5: Design an Experiment & Mixed

Mixed questions including designing fair experiments.

Created by Mr Josh

📝 Practice Questions

  1. A coin is flipped 60 times. Heads appears 33 times. What is the relative frequency of heads?
  2. A die is rolled 120 times. The number 5 appears 22 times. Calculate the relative frequency of rolling a 5.
  3. A spinner has 4 equal sections: Red, Blue, Green, Yellow. What is the theoretical probability of landing on Blue?
  4. In 200 spins, Blue appears 56 times. What is the experimental probability of Blue?
  5. Compare your answers to questions 3 and 4. Is the spinner likely to be fair? Explain briefly.
  6. A bag contains 5 red and 3 blue counters. One is picked and replaced. This is done 80 times. How many red counters would you expect?
  7. A drawing pin is dropped 100 times. It lands point-up 43 times. What is the relative frequency of point-up?
  8. Using the answer to question 7, how many times would you expect point-up in 300 drops?
  9. A fair die is rolled 600 times. How many times would you expect to roll an even number?
  10. A biased coin is flipped 500 times. Heads appears 380 times. What is the experimental probability of tails?
  11. A card is picked from a standard pack (52 cards) 260 times, with replacement. How many times would you expect to pick a heart?
  12. Explain why doing 10 coin flips is not enough to decide if a coin is biased.
  13. A spinner experiment gives these results after 200 spins: Red 48, Blue 52, Green 100. The theoretical probabilities are P(R) = P(B) = 0.25, P(G) = 0.5. Comment on whether the spinner seems fair.
  14. Two students both flip the same coin 50 times. Student A gets 28 heads, Student B gets 24 heads. Calculate each student's experimental probability of heads.
  15. Using question 14, which result is closer to the theoretical value of 0.5? What could the students do to get more reliable results?
  16. A six-sided die is rolled 90 times. Complete: Expected number of 1s = ___; Expected number of odd numbers = ___.
  17. A jar contains 2 green, 3 yellow and 5 purple sweets. One is picked and replaced each time. Out of 100 picks, how many of each colour would you expect?
  18. A spinner is spun 400 times and lands on Red 198 times. A student says "The spinner is definitely fair because it's close to 200." Explain one reason why they might be wrong to say "definitely".
  19. Describe one way to design a fair experiment when testing whether a die is biased.
  20. After 1000 coin flips, the relative frequency of heads is 0.503. After 10 flips, it was 0.8. Which result is more reliable evidence about whether the coin is fair? Why?
  1. 33 ÷ 60 = 0.55
  2. 22 ÷ 120 ≈ 0.18
  3. 0.25 (1 out of 4)
  4. 56 ÷ 200 = 0.28
  5. Theoretical is 0.25; experimental is 0.28 — fairly close, spinner is probably fair (small difference especially with 200 trials)
  6. P(red) = 5/8 = 0.625; Expected = 0.625 × 80 = 50
  7. 43 ÷ 100 = 0.43
  8. 0.43 × 300 = 129
  9. P(even) = 3/6 = 0.5; 0.5 × 600 = 300
  10. Tails = 500 − 380 = 120; Exp P(T) = 120 ÷ 500 = 0.24
  11. P(heart) = 13/52 = 0.25; Expected = 0.25 × 260 = 65
  12. 10 trials is too few — results vary hugely by chance. Need many more trials to draw reliable conclusions.
  13. RF: R = 0.24, B = 0.26, G = 0.50. All close to theoretical; spinner appears fair.
  14. Student A: 28/50 = 0.56; Student B: 24/50 = 0.48
  15. Student B (0.48) is closer to 0.5. They should pool results or do many more flips for better reliability.
  16. 1s: 90 × 1/6 = 15; Odd numbers: 90 × 3/6 = 45
  17. P(G)=0.2, P(Y)=0.3, P(P)=0.5 → Expected: Green=20, Yellow=30, Purple=50
  18. With only 400 trials, 198 is very close to 200 but small variations are normal — they can't say "definitely". Would need more evidence.
  19. Use a large number of trials (e.g. 600+), roll in the same way each time, record results carefully in a tally chart, compare relative frequencies to 1/6 each.
  20. The 1000-flip result is more reliable because more trials → closer to true probability (Law of Large Numbers). 10 flips can easily give extreme results by chance.
Created by Mr Josh

🏆 Challenge Questions

Multi-step and reasoning problems. Show your working!

  1. A die is suspected of being biased towards 6. It is rolled 300 times and 6 appears 73 times.
    (a) Calculate the experimental probability of rolling 6.
    (b) Calculate the theoretical probability.
    (c) Is there sufficient evidence of bias? Justify your answer.
  2. A bag contains n red and 4 blue marbles. One marble is picked and replaced 200 times. Red appears 150 times.
    (a) Find the experimental probability of red.
    (b) Use this to estimate n (find the total number of marbles and then n).
  3. Sophie and Marcus each test if the same coin is biased. Sophie flips 20 times (14 heads), Marcus flips 500 times (268 heads).
    (a) Calculate each person's experimental probability of heads.
    (b) Whose result is more reliable? Explain why.
    (c) Using Marcus's results, decide if the coin is likely biased.
  4. A spinner has 5 sections: A, B, C, D, E. After 500 spins:
    A: 95, B: 103, C: 98, D: 197, E: 7
    (a) Calculate the relative frequency of each outcome.
    (b) The theoretical probability of each section is 0.2. Which sections appear suspicious and why?
    (c) What is one explanation for the E result?
  5. A factory produces light bulbs. The probability one is defective is 0.04.
    (a) In a batch of 2500 bulbs, how many are expected to be defective?
    (b) They inspect 2500 bulbs and find 112 defective. Is the machine working correctly? Use relative frequency to justify your answer.
  6. An experiment drops a drawing pin 50, 100, 200, and 1000 times. Results of "point up": 22, 46, 98, 493.
    (a) Calculate the relative frequency for each trial count.
    (b) What does the pattern suggest about the true probability of "point up"?
    (c) Sketch a rough description of how the relative frequency changes as trials increase.
  7. A class of 30 students each flip a fair coin 10 times and record the number of heads. Explain why you would expect a wide range of results across the class, yet the class average (all 300 flips combined) should be close to 5 heads per student.
  8. Design a complete experiment to test whether a 6-sided die is fair. Your answer must include:
    • the number of trials and why
    • how to record data
    • what calculations to carry out
    • how to interpret the results to reach a conclusion
  1. (a) 73/300 ≈ 0.243  (b) 1/6 ≈ 0.167  (c) 0.243 is significantly higher than 0.167 over 300 trials — strong evidence of bias towards 6.
  2. (a) 150/200 = 0.75  (b) P(red) ≈ 0.75 → red/total = 0.75; total = n+4 and n/(n+4)=0.75 → n = 3(n+4)… solving: 4n = 3×4 → actually 0.75(n+4)=n → 0.75n+3=n → 3=0.25n → n=12. Total = 16 marbles.
  3. (a) Sophie: 14/20 = 0.70; Marcus: 268/500 = 0.536  (b) Marcus — more trials means more reliable result  (c) 0.536 is slightly above 0.5 over 500 trials — borderline, possibly slight bias towards heads but not conclusive; more trials recommended.
  4. (a) A=0.19, B=0.206, C=0.196, D=0.394, E=0.014  (b) D (0.394 vs 0.2) and E (0.014 vs 0.2) are very suspicious  (c) E section may be very narrow (almost no area on the spinner).
  5. (a) 0.04 × 2500 = 100 defective expected  (b) 112/2500 = 0.0448 vs 0.04 — slightly higher. The machine may be slightly out of specification; worth investigating with more data.
  6. (a) 22/50=0.44; 46/100=0.46; 98/200=0.49; 493/1000=0.493  (b) Pattern approaches ~0.49–0.50, suggesting P(point up) ≈ 0.49  (c) RF starts scattered and gradually levels off approaching ~0.49 as trials increase.
  7. With only 10 flips, chance variation is large (could get anything from 2 to 9 heads). But with all 300 flips pooled, the Law of Large Numbers applies — the overall relative frequency should be very close to 0.5.
  8. Ideal answer: Roll the die at least 600 times (100 per face). Record each outcome in a tally chart. Calculate RF for each face (frequency ÷ 600). Compare each RF to theoretical 1/6 ≈ 0.167. If any face deviates significantly (e.g. >0.22 or <0.12), that is evidence of bias. Use a large number to minimise random variation.
Created by Mr Josh