Mathematics • Year 10 • Unit 4 • Lesson 15
Probability in the Real World
Apply Lesson 15's probability rules to real Australian contexts: weather forecasts, sport, the canteen, raffles and games of chance. Practise expressing probabilities as fractions, decimals and percentages — and using the complement to save time.
1. Word problems
For each scenario, identify the sample space, count favourable outcomes, and give the answer as a fraction AND a decimal (and a percentage where asked). Always check 0 ≤ P ≤ 1.
1.1 — School canteen raffle. The Year 10 raffle drum holds 40 tickets: 12 belong to Year 7, 10 to Year 8, 9 to Year 9, and the rest to Year 10. One ticket is drawn.
(a) How many Year 10 tickets are in the drum?
(b) Find P(Year 10) as a fraction and a decimal.
(c) Use the complement rule to find P(not Year 10). 3 marks
1.2 — BOM weather forecast. The Bureau of Meteorology says "60 % chance of rain tomorrow."
(a) Express this as a fraction and a decimal.
(b) Use the complement rule to find P(no rain).
(c) Does "60 % chance of rain" mean it will rain for 60 % of the day? Explain using the lesson's definition of probability. 3 marks
1.3 — NRL kick-off coin toss. An NRL coin toss decides who gets the kick-off. Two referees disagree:
Ref A says "P(heads) = 1/2 because there are 2 outcomes."
Ref B says "the probability of getting heads 3 times in a row is 3 × 1/2 = 1.5."
(a) Which referee is wrong, and using Lesson 15's "probability between 0 and 1" rule, why?
(b) Correctly calculate P(3 heads in a row). 3 marks
1.4 — Sydney Lotto. A Lotto game has 45 balls numbered 1–45. The first ball is drawn.
(a) Find P(the first ball is your number, 23).
(b) Find P(the first ball is an even number).
(c) Find P(the first ball is greater than 40), as a fraction.
(d) Use the complement rule to find P(the first ball is 40 or less). 4 marks
1.5 — Trivia night categories. A trivia question category is chosen at random from 6 options: Sport, Music, Geography, Science, Movies, History.
(a) Find P(the category starts with the letter S).
(b) Find P(the category is NOT Geography) using the complement rule.
(c) Find P(the category is one of {Music, Movies}) as a fraction. 3 marks
2. Explain your thinking
This question is about justifying with the lesson definitions, not just calculating. Use full sentences. 4 marks
2.1 A friend says: "If I flip a fair coin 10 times and get 7 heads, the coin must be biased — the probability of heads is supposed to be 0.5, but I'm getting 0.7." Using Lesson 15's idea of theoretical probability vs short-run results, write a four-sentence reply that (i) restates the theoretical P(heads), (ii) explains why short-run results often differ from the theoretical value, (iii) refers to either "sample space" or "complement", and (iv) finishes with one rule of thumb for thinking about randomness.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Canteen raffle
(a) Year 10 tickets = 40 − (12 + 10 + 9) = 40 − 31 = 9.
(b) P(Year 10) = 9/40 = 0.225.
(c) P(not Year 10) = 1 − 9/40 = 31/40 = 0.775.
1.2 — Weather forecast
(a) 60 % = 60/100 = 3/5 = 0.6.
(b) P(no rain) = 1 − 0.6 = 0.4 = 40 %.
(c) No. "60 % chance" is the probability of rain occurring at all, not the fraction of the day it will rain. Lesson 15: probability measures "how likely an event is to occur" (0 = impossible, 1 = certain), not duration.
1.3 — NRL coin toss
(a) Ref B is wrong. Probability cannot exceed 1, but Ref B got 1.5. The lesson says probability is "from 0 (impossible) to 1 (certain)".
(b) P(3 heads in a row) = 1/2 × 1/2 × 1/2 = 1/8 = 0.125 (multiply for independent events, do NOT add).
1.4 — Sydney Lotto
(a) P(23) = 1/45 ≈ 0.022.
(b) Even numbers 1–45: 2, 4, ..., 44 → 22 evens. P = 22/45.
(c) Greater than 40: {41, 42, 43, 44, 45} → 5 favourable. P = 5/45 = 1/9.
(d) P(≤40) = 1 − 5/45 = 40/45 = 8/9.
1.5 — Trivia categories
(a) S-categories: Sport, Science → 2. P = 2/6 = 1/3.
(b) P(not Geography) = 1 − 1/6 = 5/6.
(c) {Music, Movies} → 2 favourable. P = 2/6 = 1/3.
2.1 — Explain your thinking (sample)
The theoretical P(heads) for a fair coin is 1/2 = 0.5, based on the sample space {H, T} where each outcome is equally likely (Lesson 15 definition). In the short run, however, random samples often differ from the theoretical value — getting 7 heads out of 10 from a fair coin is normal variation, not evidence of bias. Over much larger samples (say 1000 flips), the proportion would settle close to 0.5; the sample space and the calculated probability don't change just because a short experiment didn't match them. Rule of thumb: don't conclude a coin (or anything else) is biased from a small sample — short-run results scatter around the theoretical probability before settling close to it over many trials.
Marking: 1 mark for stating theoretical P(H) = 0.5, 1 for explaining short-run variation, 1 for correct use of "sample space" or "complement", 1 for a clear rule of thumb.