Mathematics • Year 10 • Unit 4 • Lesson 18
Theoretical vs Experimental Probability — Real-World Maths
Apply Lesson 18 to real settings: testing a "fair" coin in PE, a school canteen survey, drug-trial data, a basketball free-throw record, and a class lottery. Use the law of large numbers to judge whether observed data matches a fair model.
1. Word problems
Show working. State whether you are using theoretical or experimental probability — and which formula.
1.1 — PE coin-toss test. Mr Singh suspects a coin is biased. The class flips it 100 times: 63 heads, 37 tails.
(a) Find the experimental P(heads).
(b) State the theoretical P(heads) for a fair coin.
(c) Use the law of large numbers to argue whether 100 flips is enough evidence to call the coin "biased". 3 marks
1.2 — Canteen-line survey. Over 5 weeks (200 lunchtimes recorded), the canteen logs 50 lunchtimes where it ran out of pies before all students were served.
(a) Find the experimental P(running out of pies on a given day).
(b) If the canteen runs for another 80 days, how many of those days would you predict it will run out of pies, assuming conditions don't change? 3 marks
1.3 — Drug-trial data. A new acne cream is tested on 400 teenagers. After 8 weeks, 312 report "clearer skin".
(a) Find the experimental P(clearer skin) from the trial.
(b) The drug company advertises "8 out of 10 teens see clearer skin". Is this claim supported by the trial data? Explain in one sentence. 3 marks
1.4 — Free-throw record. A WNBL player has taken 480 free throws this season and made 384.
(a) Find the experimental P(make a free throw).
(b) Treat the free throws as independent. Using Lesson 18's Exam Tip (P(A|B) = P(A) is the test for independence), explain in one sentence why treating free throws as independent is a reasonable assumption.
(c) Use the multiplication rule for independence to find P(two free throws in a row). 3 marks
1.5 — Class lottery. A class lottery sells 50 tickets for a prize. There is exactly one winning ticket. The lottery is run once a week for 30 weeks (with a fresh draw each week and 50 fresh tickets).
(a) Theoretical P(any ticket wins) in one week = ______.
(b) Over 30 weeks, how many wins would you expect for a student who buys 1 ticket every week?
(c) After 30 weeks, the student records 2 wins. Explain — using the law of large numbers — why this is consistent with a fair draw. 3 marks
2. Explain your thinking
Communication question. Use full sentences. 4 marks
2.1 A classmate, Ramy, plays a "guess the card" game online. He notices that after 20 rounds he has won 14 (70%), but the website states the theoretical P(win) per round is only 0.40. Ramy is convinced the website is lying. Write a four-sentence reply that (i) identifies which Lesson 18 idea explains Ramy's result (law of large numbers vs short-run variation), (ii) calculates the expected number of wins in 20 rounds at the stated P = 0.40, (iii) suggests what Ramy could do to test the claim more reliably, and (iv) finishes with a one-sentence warning about trusting short-run experimental results.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Coin-toss test
(a) Exp P(heads) = 63/100 = 0.63.
(b) Theor P(heads) = 0.5.
(c) The difference is 0.13. 100 trials is enough to suggest something is off (the expected number of heads is 50, observed is 63), but to be confident the coin is biased we want a much larger trial — the law of large numbers says experimental values get closer to theoretical only as trials grow. A short trial can still be misleading.
1.2 — Canteen out-of-pies
(a) Exp P = 50/200 = 1/4 = 0.25.
(b) Expected = 80 × 0.25 = 20 days.
1.3 — Drug-trial
(a) Exp P(clearer) = 312/400 = 0.78.
(b) The ad claims 8/10 = 0.80, which is close to but slightly higher than the trial's 0.78. The claim slightly overstates the trial result; "about 4 in 5" would be more accurate.
1.4 — Free throws
(a) Exp P = 384/480 = 0.8 = 4/5.
(b) Independence is reasonable because one free throw does not change the rules of the next (the basket and player are reset). Whether truly independent depends on psychology (Lesson 18 Exam Tip: check P(A|B) = P(A)).
(c) Assuming independence, P(two in a row) = 0.8 × 0.8 = 0.64 = 16/25.
1.5 — Class lottery
(a) Theor P(win) per week = 1/50 = 0.02.
(b) Expected wins in 30 weeks = 30 × 0.02 = 0.6 wins.
(c) Observing 2 wins is higher than expected (0.6) but still very plausible in 30 trials with a 1/50 chance. The law of large numbers only says the proportion approaches 1/50 over many more weeks; in 30 weeks short-run variation is large.
2.1 — Ramy's "guess the card" game (sample response)
Ramy is seeing short-run variation — Lesson 18's law of large numbers only kicks in after many trials. In 20 rounds, the expected number of wins at P = 0.40 is 20 × 0.40 = 8 wins; Ramy got 14, which is unusual but possible. To test the website's claim more reliably, he should play many more rounds (say 500 or 1000) and check whether the long-run proportion of wins settles near 0.40. Warning: never trust experimental probability based on a small number of trials — short streaks can be very misleading even when the underlying model is correct.
Marking: 1 mark per part (i)-(iv).