Mathematics • Year 8 • Unit 4 • Lesson 19
Experimental Probability in the Real World
Apply experimental probability, expected frequency, and the law of large numbers to real situations: medical test screening, manufacturing quality control, weather records, sports statistics, and simulation design.
1. Word problems
Each problem uses experimental probability ideas from Lesson 19 — frequency over trials, expected frequencies, theoretical vs experimental, and simulations. Show your working.
1.1 — Quality control. A factory makes light bulbs. Out of 500 bulbs tested, 12 are faulty.
(a) Calculate the experimental P(faulty).
(b) Out of a daily output of 10,000 bulbs, how many are expected to be faulty?
(c) The factory's target is at most 1.5% faulty. Are they meeting the target? 4 marks
1.2 — Weather records. Over the past 50 years, it has rained on Sydney's New Year's Eve 12 times.
(a) Calculate the experimental P(rain on NYE).
(b) Why is experimental probability (rather than theoretical) the right tool to estimate this?
(c) Over the next 30 NYEs, how many rainy New Year's Eves would you expect? 3 marks
1.3 — Basketball free-throw record. A WNBL player has made 320 free throws out of 400 attempts this season.
(a) Calculate her experimental P(free throw made).
(b) If she shoots 60 free throws next month, how many is she expected to make?
(c) Why is this a better estimate of her true success rate than a coach's 10-shot trial in training? 3 marks
1.4 — Game design. A board-game designer claims her custom 8-sided die is fair (P(any face) = 1/8). To test, you roll it 800 times. Face 7 comes up 75 times and face 8 comes up 125 times.
(a) State the expected frequency of each face for 800 rolls of a fair 8-sided die.
(b) Calculate the experimental P(7) and P(8). Compare to theoretical.
(c) Is there evidence the die is biased? Justify your answer using both values. 4 marks
1.5 — Simulation design. Suppose 30% of phone-screen-time apps successfully reduce screen time by 20%. You want to simulate, with a single die, the result of 60 students trying these apps (success / no success per student).
(a) Describe how to use the die so that P(success per roll) ≈ 0.3 (you do NOT need to use every face — explain what to do with the "leftover" outcomes).
(b) State the expected number of successes in 60 rolls.
(c) If you observed 22 successes in 60 simulated trials, is that consistent with the 30% claim? 3 marks
2. Explain your thinking
This question is about communication, not just answers. Use full sentences. 4 marks
2.1 A friend flips a coin 10 times and gets 8 heads. They announce: "This coin is biased — I have proof." In your own words explain (i) why 10 trials is far too few to claim bias, (ii) what the law of large numbers would predict if they flipped the coin 1000 times instead, and (iii) give one realistic example of how a small sample can mislead someone in everyday life (sports, gambling, online reviews, medical tests, etc.). Use the term law of large numbers somewhere in your answer.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Quality control
(a) P(faulty) ≈ 12/500 = 0.024 (= 2.4%).
(b) Expected faulty = 0.024 × 10,000 = 240 bulbs per day.
(c) Target 1.5% = 0.015. Their rate is 2.4% > 1.5% → NOT meeting the target (they exceed the target by 0.9 percentage points).
1.2 — NYE weather
(a) P(rain) ≈ 12/50 = 6/25 = 0.24.
(b) Whether it rains is not an "equally likely" outcome with countable favourable cases — we cannot calculate it theoretically. Long-run historical data gives the best estimate.
(c) Expected rainy NYEs in next 30 years = 0.24 × 30 = 7.2 ≈ 7 (round to nearest whole).
1.3 — WNBL free throws
(a) P(made) ≈ 320/400 = 4/5 = 0.80 (80%).
(b) Expected made in 60 attempts = 0.80 × 60 = 48.
(c) 400 attempts is a large sample so by the law of large numbers it gives a very reliable estimate of her true success rate. A 10-shot trial is too small and could land anywhere from say 6/10 to 10/10 just by chance.
1.4 — 8-sided die test
(a) Expected for each face = (1/8) × 800 = 100.
(b) P(7) ≈ 75/800 = 0.09375 (theoretical = 0.125). P(8) ≈ 125/800 = 0.15625 (theoretical = 0.125).
(c) Face 7 came up 25 below expected and face 8 came up 25 above expected. With a fair die we would not normally see deviations this large in 800 trials. There is evidence of bias — the die appears to favour 8 over 7. (A formal chi-square test would confirm but isn't needed for Year 8.)
1.5 — Simulation design
(a) Roll the die. Treat 1 and 2 as "success" (P = 2/6 ≈ 0.333, slightly above the 0.30 target — for a closer match, you could re-roll on a 1 with probability 1/10, but the simple 2/6 mapping is close enough for most simulations). Faces 3, 4, 5, 6 = "no success".
(b) Expected successes = 0.30 × 60 = 18.
(c) 22 is 4 above the expected 18 — well within normal variation for 60 trials. Yes, consistent with the 30% claim.
2.1 — Explain your thinking (sample response)
Ten flips is far too few to prove bias, because small samples are naturally noisy. Even a perfectly fair coin can produce 8 heads out of 10 just by chance — the law of large numbers tells us that experimental probability only gets close to the theoretical value after many trials. If the friend flipped the coin 1000 times instead, we would expect heads close to 500 (probably between about 470 and 530); 800 heads out of 1000 would indeed be strong evidence of bias. A real-world example: an online product review with only 5 star ratings all 5-star might look amazing, but those 5 reviews could just be the early lucky ones — a product with 1000 reviews and an average of 4.5 stars is a much more reliable indicator of quality.
Marking: 1 mark for explaining noise in small samples; 1 mark for what 1000-flip prediction would look like; 1 mark for a realistic everyday example; 1 mark for clear full-sentence answer using "law of large numbers".