Mathematics • Year 8 • Unit 4 • Lesson 19
Experimental Probability — Mixed Challenge
Pull together every idea from Lesson 19: experimental P, theoretical P, expected frequency, the law of large numbers, and designing simulations. Six mixed problems, one "find the mistake", and one open-ended challenge.
1. Mixed problems — choose the right move
Each question uses a different idea from Lesson 19. Show working. 3 marks each
1.1 A spinner has 5 equal sectors numbered 1–5. It is spun 150 times. Sector 3 appears 28 times. Calculate the experimental P(3) and compare to theoretical.
1.2 A coin is flipped 1000 times and Heads appears 487 times. (a) Calculate experimental P(H). (b) Compare to theoretical. (c) Why is this stronger evidence for fairness than 5 heads out of 10 flips?
1.3 If P(rain tomorrow) = 0.25, calculate the expected number of rainy days in (a) 30 days, (b) 365 days.
1.4 A trial of a new drug tested 200 patients; 144 showed improvement. (a) Calculate experimental P(improvement). (b) If the drug is given to 5,000 patients, how many would be expected to show improvement? (c) Why is experimental rather than theoretical probability the right tool here?
1.5 Two classes each roll a die 100 times. Class A gets P(6) ≈ 0.20; Class B gets P(6) ≈ 0.13. Both are different from theoretical 1/6 ≈ 0.167. Combine the two classes (200 trials total): assume Class A had 20 sixes and Class B had 13. Calculate the combined experimental P(6) and explain why it is more reliable.
1.6 Describe a simulation using a coin to estimate P(student is left-handed) if the true proportion is approximately 10%. Explain why a coin alone is not ideal and suggest a better random device.
2. Find the mistake
A student attempted this expected-frequency problem. Their working is shown below. Exactly one line contains a mistake. Spot it, explain why it's wrong, then re-do the working correctly. 3 marks
Problem: A spinner has 4 equal sectors. It will be spun 240 times. How many times is each sector expected to come up?
Line 1: Theoretical P(any one sector) = 1/4.
Line 2: Expected frequency = P + n = 1/4 + 240 = 240.25.
Line 3: So each sector is expected ≈ 240 times.
Line 4: In total, the four sectors give 240 × 4 = 960 spins.
(a) Which line contains the mistake?
(b) Explain in one or two sentences why that line is wrong.
(c) Write out the corrected working in full.
Stuck? Expected frequency = P × n (multiply), not P + n (add). The student used the wrong operator.3. Open-ended challenge — design and analyse your own experiment
This question has many valid answers. 4 marks
3.1 Your job: design a simple home or classroom probability experiment (e.g., flipping a paper cup, dropping a thumbtack, rolling a 2p coin). The outcome cannot be calculated theoretically — you must run trials.
Your design must include:
(i) Describe the experiment and identify all possible outcomes.
(ii) Predict what you think will happen and why.
(iii) Specify how many trials you will run (state a minimum of 30) and explain why running more is better.
(iv) Sketch the frequency table you would use to record results, then write the formula you would use to calculate the experimental probability of each outcome.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Spinner sector 3
Experimental P(3) = 28/150 = 14/75 ≈ 0.187. Theoretical P(3) = 1/5 = 0.200. Close to theoretical; difference (~0.013) is well within normal variation for 150 spins.
1.2 — Coin 1000 flips
(a) P(H) ≈ 487/1000 = 0.487.
(b) Theoretical = 0.500. Very close — difference is 0.013.
(c) 1000 trials is a large sample, so by the law of large numbers the estimate is much more reliable. With only 10 flips, 5 H is also "fair-looking" but could easily appear from a biased coin; 487/1000 strongly suggests the coin really is close to fair.
1.3 — Expected rainy days
(a) 0.25 × 30 = 7.5 ≈ 8 days.
(b) 0.25 × 365 = 91.25 ≈ 91 days.
1.4 — Drug trial
(a) P(improvement) ≈ 144/200 = 0.72 (72%).
(b) Expected improved in 5,000 = 0.72 × 5000 = 3,600 patients.
(c) The drug's effect on individuals is not a "theoretical" probability calculable from equally likely outcomes — only real patient trials can estimate the true response rate. Experimental P from data is the only way.
1.5 — Combined classes
Combined: P(6) = (20 + 13) / 200 = 33/200 = 0.165. This is very close to the theoretical 1/6 ≈ 0.167. The combined estimate uses 200 trials so by the law of large numbers it is much more reliable than either single class's 100-trial estimate.
1.6 — Simulation for 10% rate
A coin alone is not ideal because a coin gives 50/50, which is nowhere near 10%. A better device is a 10-sector spinner where one sector represents "left-handed" (P = 1/10 = 10%). Alternatively use a single digit 0–9 from a random-number generator or two digits 01–10 from a random table, treating one designated outcome as "left-handed".
2 — Find the mistake
(a) The mistake is on Line 2.
(b) Expected frequency is P MULTIPLIED by n, not P + n. The student wrote a plus sign instead of a multiplication. Adding a probability and a count gives nonsense (you can't add a fraction to a number of trials).
(c) Corrected working:
Theoretical P(any sector) = 1/4.
Expected frequency = P × n = (1/4) × 240 = 60 per sector.
Check: 60 × 4 = 240 total spins ✓.
3 — Open-ended design (sample solution)
(i) Experiment: Drop a thumbtack from a height of 30 cm onto a desk. Outcomes: {point-up, point-down/sideways}. We cannot calculate this theoretically because the tack is not symmetric.
(ii) Prediction: I predict the tack will land point-down more often than point-up because the heavier metal end tends to point down. My guess is around 60% down, 40% up.
(iii) Number of trials: I will drop the tack 100 times. By the law of large numbers, 100 trials gives a much more reliable estimate than 10 because random variation averages out — small samples can give wildly different results just by chance.
(iv) Frequency table:
| Outcome | Tallies | Frequency |
|---|---|---|
| Point-up | ______ | ______ |
| Point-down | ______ | ______ |
Formula: P(outcome) ≈ frequency of outcome ÷ 100.
Marking: 1 mark for clear experiment with outcomes; 1 mark for a sensible prediction; 1 mark for justifying ≥ 30 trials with reference to the law of large numbers; 1 mark for the frequency-table sketch AND the formula.