Mathematics • Year 9 • Unit 4 • Lesson 16
Probability — Mixed Challenge
Pull together theoretical probability, experimental probability and the law of large numbers. Choose the right tool, find a classmate's mistake, and design your own probability experiment.
1. Mixed problems — pick the right tool
Each question uses either theoretical or experimental probability (or both). Decide which before you start writing. 3 marks each
1.1 A bag contains 8 red, 5 blue and 7 green marbles. (a) Find the theoretical probability of drawing a blue. (b) Find the theoretical probability of NOT drawing a green.
1.2 A dart is thrown at a square dartboard 250 times. It hits the bullseye 12 times. (a) Find the experimental P(bullseye). (b) Estimate how many bullseyes you'd expect in 4 000 throws by the same player.
1.3 A fair 6-sided die is rolled 12 times. A student rolls 4 sixes and writes "experimental P(6) = 4/12 = 1/3, so the die is biased". Either agree or disagree, justifying with one sentence about sample size.
1.4 A spinner is spun 1 000 times. Section A appears 248 times. The spinner is labelled as having 4 equal sections. (a) Theoretical P(A)? (b) Experimental P(A)? (c) Does the data support the spinner being fair?
1.5 A factory tests 5 000 light bulbs. 35 are faulty. Find the experimental P(faulty). If the factory ships 80 000 bulbs in a week, how many faulty ones should they expect, and how many should they ship out?
1.6 Two events, "rain in Sydney on a random day" and "tossing a head on a fair coin", both have probability about 0.5. One is theoretical and one is experimental. State which is which and give a one-sentence reason for each.
2. Find the mistake
Read the student's working below for a coin-toss problem. Exactly one line contains a mistake. Spot it, explain why it's wrong, then write the corrected working. 3 marks
Problem the student is solving: A coin is tossed 200 times and lands on heads 116 times. The student is asked to find the experimental P(head), the expected count for a fair coin, and decide if the coin looks biased.
Line 1: Experimental P(head) = 116 ÷ 200 = 0.58
Line 2: Expected count for a fair coin = 200 × 0.5 = 100 heads
Line 3: "0.58 is bigger than 0.5, so by the law of large numbers, the coin MUST be biased."
Line 4: Conclusion: the coin is biased.
(a) Which line contains the mistake?
(b) Explain in one or two sentences why that line is wrong (use the actual definition of the law of large numbers from the lesson).
(c) Write the corrected reasoning the student should have used, and the corrected conclusion.
Stuck? Revisit lesson § "Common Misconceptions" — point 3 on guarantees, and § "Law of Large Numbers" — what the law actually says happens with more trials.3. Open-ended challenge — design your own experiment
This question has many valid answers — there is no single "right" experiment. 4 marks
3.1 Design a probability experiment a Year 9 student could run at home in 20 minutes to test whether a thumbtack lands point-up or point-down more often. (Theoretical probability is unknown — only experiment can give an answer.)
Your answer must include:
(i) The total number of trials you would run, and a brief reason that justifies your choice using the law of large numbers.
(ii) The exact formula you would use to calculate the experimental P(point-up).
(iii) One source of bias or error to avoid (e.g. the surface, the height, the way you toss).
(iv) What numerical evidence would convince you that point-up is genuinely more likely than point-down — for example, by what margin and over how many trials.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Coloured marbles
Total marbles = 8 + 5 + 7 = 20.
(a) P(blue) = 5/20 = 1/4 = 0.25.
(b) P(not green) = 1 − P(green) = 1 − 7/20 = 13/20 = 0.65.
1.2 — Dartboard
(a) Experimental P(bullseye) = 12 ÷ 250 = 0.048 (≈ 4.8%).
(b) Expected in 4 000 throws = 4 000 × 0.048 = 192 bullseyes.
1.3 — Die rolled 12 times
Disagree. 12 rolls is far too small a sample to draw a conclusion — random variation can easily give 4 sixes from a fair die. The law of large numbers requires many trials before experimental P stabilises near theoretical 1/6.
1.4 — 4-section spinner, 1 000 spins
(a) Theoretical P(A) = 1/4 = 0.25.
(b) Experimental P(A) = 248 ÷ 1 000 = 0.248.
(c) Yes — 0.248 is very close to 0.25, and with 1 000 spins this is exactly the sort of small random variation the law of large numbers permits. The data supports the spinner being fair.
1.5 — Light bulb factory
Experimental P(faulty) = 35 ÷ 5 000 = 0.007 (0.7%).
Expected faulty in 80 000 = 80 000 × 0.007 = 560 faulty.
Good bulbs to ship ≈ 80 000 − 560 = 79 440 good bulbs.
1.6 — Rain vs coin toss
"Tossing a head on a fair coin" is theoretical — we can list two equally likely outcomes and reason mathematically that P = 0.5.
"Rain in Sydney on a random day" is experimental — there is no clean list of equally likely outcomes; we estimate the probability from many years of observed weather data (relative frequency).
2 — Find the mistake
(a) The mistake is on Line 3.
(b) The student claims the law of large numbers guarantees the coin is biased because experimental P (0.58) doesn't equal theoretical P (0.5). But the law of large numbers says experimental probability approaches theoretical with more trials — it never guarantees exact equality. With only 200 trials, a gap of 0.08 (16 extra heads) is well within normal random variation.
(c) Corrected reasoning: experimental P(head) = 0.58 is higher than theoretical 0.5, but with only 200 tosses this gap could easily be random variation. The coin might be biased, but more trials (say, 2 000 or 10 000) are needed before we can decide. Conclusion: insufficient evidence to call the coin biased.
3 — Open-ended challenge (sample solution)
Trials: 200 drops. With 10 drops random variation dominates; 200 is enough that the law of large numbers makes the experimental P meaningful but still finishes in under 20 minutes.
Formula: Experimental P(point-up) = (number of point-up landings) ÷ 200.
Sources of bias to avoid: drop from the same height every time; use the same surface (a flat hard table, not carpet, which absorbs and tilts the tack); shake before each drop so you don't bias the orientation.
Convincing evidence: if 200 drops give point-up clearly more than 50% (say, over 60% — i.e. 120+ point-ups), this gap is too big for random variation alone and is strong evidence point-up is genuinely more likely. A second 200-trial run that agrees would seal it.
Marking: 1 mark per part (i)–(iv). Award full marks for any reasonable, lesson-consistent design where the trial count is at least 100 and the bias control is concrete.