Mathematics • Year 7 • Unit 4 • Lesson 16
Experimental Probability — Real World
Apply relative frequency to real situations: basketball shooting, a canteen survey, a school weather log, a free-throw simulation, and the bias check on a suspicious die.
1. Word problems
Show the formula line (frequency ÷ total trials) before stating each answer.
1.1 — Basketball training. Jordan takes 75 free-throw shots in a training session and makes 48 of them. (a) Find the experimental probability that Jordan makes a free throw. (b) Based on this experiment, estimate how many free throws Jordan would make in a game with 25 attempts. 3 marks
1.2 — Canteen survey. The canteen surveys 300 students about their lunch order: pasta 84, sushi 51, wraps 72, pies 45, sandwiches 48. (a) Find the experimental probability that a random student orders sushi. (b) Check that the five relative frequencies add to 1. (c) If 1500 students will order lunch this week, estimate how many will choose wraps. 4 marks
1.3 — School weather log. Over 120 school days, the science club records whether it rained at recess. It rained on 18 days. (a) Find the experimental probability of rain at recess on a school day. (b) Predict the number of rainy recess days in a 40-day term. (c) Why might this estimate be less reliable than one based on 1200 school days? 4 marks
1.4 — Free-throw simulation. A basketball player makes 1 in 3 free throws theoretically. You roll a die 120 times: 1 or 2 counts as a "make", 3–6 counts as a "miss". You record 44 makes in 120 rolls. (a) Find the experimental P(make). (b) What is the theoretical probability of a "make"? (c) How could you improve the simulation's accuracy? 4 marks
1.5 — Suspicious die. A street performer offers a game using "his special die". You roll it 300 times and record this frequency table: 1 → 40, 2 → 45, 3 → 48, 4 → 47, 5 → 42, 6 → 78. (a) Calculate the relative frequency for each face. (b) Which face would you accuse of being weighted, and what relative frequency would you expect for a fair die? 4 marks
2. Explain your thinking
Communication matters. Use full sentences. 4 marks
2.1 A Year 7 student says: "I flipped this coin 10 times and got 8 heads. The coin must be biased — P(heads) is really 0.8." Using the lesson, explain (i) why this conclusion is too quick, (ii) what number of trials would give a more trustworthy estimate, and (iii) what the law of large numbers predicts will happen as more flips are added.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Basketball training
(a) P(make) = 48 ÷ 75 = 16/25 = 0.64. (b) Expected makes in 25 attempts = 0.64 × 25 = 16 free throws.
1.2 — Canteen survey
(a) P(sushi) = 51 ÷ 300 = 17/100 = 0.17.
(b) Relative frequencies: pasta 84/300 = 0.28; sushi 0.17; wraps 72/300 = 0.24; pies 45/300 = 0.15; sandwiches 48/300 = 0.16. Sum = 0.28 + 0.17 + 0.24 + 0.15 + 0.16 = 1.00. ✓
(c) Expected wraps in 1500 = 0.24 × 1500 = 360 students.
1.3 — School weather log
(a) P(rain at recess) = 18 ÷ 120 = 3/20 = 0.15.
(b) Expected rainy days in 40 = 0.15 × 40 = 6 days.
(c) An estimate from 120 days is more affected by short-term weather patterns (a wet/dry term). With 1200 days the relative frequency would be much closer to the long-run truth — the law of large numbers.
1.4 — Free-throw simulation
(a) Experimental P(make) = 44 ÷ 120 = 11/30 ≈ 0.367.
(b) Theoretical P(make) = 2/6 = 1/3 ≈ 0.333 (rolls of 1 or 2 from 6 equally likely faces).
(c) Increase the number of die rolls (e.g. 1200 or 12 000) — more trials means the experimental value will settle closer to 1/3.
1.5 — Suspicious die
(a) Relative frequencies: 1: 40/300 ≈ 0.133; 2: 45/300 = 0.15; 3: 48/300 = 0.16; 4: 47/300 ≈ 0.157; 5: 42/300 = 0.14; 6: 78/300 = 0.26.
(b) Face 6 is the suspect. A fair die would give a relative frequency near 1/6 ≈ 0.167 for each face. 0.26 is far above this — much bigger than normal variation across 300 rolls.
2.1 — Explain your thinking (sample response)
(i) Ten flips is far too small. Random variation in tiny samples means it is quite easy to get 8 heads from a fair coin just by chance — this experimental value of 0.8 is not strong evidence of bias.
(ii) A much larger experiment — hundreds, or ideally a thousand or more flips — would give a relative frequency that we could really trust as an estimate of the true probability.
(iii) The law of large numbers predicts that as the number of flips grows, the relative frequency of heads will move closer and closer to the true theoretical value (0.5 if the coin is fair). If after thousands of flips it still hovers near 0.8, then bias becomes a reasonable conclusion.
Marking: 1 mark per part (i), (ii), (iii). 1 mark for clear sentences referencing the law of large numbers.