Mathematics • Year 8 • Unit 4 • Lesson 14
Probability in the Real World
Apply the probability formula and complement rule to real situations: weather forecasts, lottery odds, board games, raffles, and class lucky-draws. Each problem demands both calculation and interpretation.
1. Word problems
Each problem uses the probability toolkit from Lesson 14. Show working — a single answer with no working earns half marks.
1.1 — Weather forecast. The weather app says P(rain) = 0.25 for tomorrow.
(a) Express this probability as a percentage.
(b) Use the complement rule to find P(no rain).
(c) Across the next 8 days with similar forecasts, how many days of rain should you expect on average? 3 marks
1.2 — Class raffle. A class of 25 students each buys ONE raffle ticket. There are 3 prizes (a chocolate, a movie ticket, a $20 voucher). Tickets are drawn separately without replacement.
(a) What is P(YOU win the first prize)?
(b) Assuming you didn't win the first prize, what is P(you win the second prize)?
(c) Are these two probabilities the same? Explain in one sentence. 3 marks
1.3 — Board game. In a board game, you roll two standard dice on each turn. Rolling double 6 lets you move twice. Rolling any double makes you take an extra turn.
(a) How many outcomes are in the sample space?
(b) How many of those outcomes are doubles? List them.
(c) Find P(rolling any double) AND P(rolling double 6). 3 marks
1.4 — Lucky dip. A bag of 50 lucky-dip tokens contains 5 winning tokens (marked "WIN!") and 45 losing tokens. Each token is replaced before the next draw.
(a) Find P(winning on one draw) as a fraction, decimal, and percentage.
(b) Find P(losing) using the complement rule.
(c) If 200 people each have one draw, how many wins should be expected? 4 marks
1.5 — Multiple choice exam. A multiple-choice question has 4 options (A, B, C, D), with exactly one correct answer. A student guesses at random.
(a) Find P(guess correctly) and P(guess incorrectly).
(b) The exam has 20 such questions. How many should the student expect to guess correctly?
(c) The exam pass mark is 12/20. Should the student rely on guessing alone? Justify briefly. 3 marks
2. Explain your thinking
This question is about communication, not just numbers. Use full sentences. 4 marks
2.1 A classmate says: "Since the weather has been sunny for 4 days in a row, the chance of rain tomorrow MUST be high." Explain (i) why this reasoning is wrong (often called the "gambler's fallacy"), (ii) the correct meaning of independent events using one example (such as a coin flip), (iii) why each new coin flip is still 50-50 even after 10 heads in a row, and (iv) one sentence describing what DOES affect the probability of rain tomorrow. Use the term independent events.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Weather forecast
(a) 0.25 = 25%.
(b) P(no rain) = 1 − 0.25 = 0.75 (75%).
(c) Expected rainy days = 8 × 0.25 = 2 days (on average over 8 such forecasts).
1.2 — Class raffle
(a) P(you win first prize) = 1 ÷ 25 = 1/25 = 4%.
(b) After first prize, 24 tickets remain. P(you win second prize, given you didn't win first) = 1 ÷ 24 = 1/24 ≈ 4.17%.
(c) Not quite the same — your chance is slightly higher for the second draw because the pool is smaller (24 tickets instead of 25).
1.3 — Board game
(a) Sample space = 6 × 6 = 36 outcomes.
(b) Doubles: (1,1), (2,2), (3,3), (4,4), (5,5), (6,6) — 6 doubles.
(c) P(any double) = 6 ÷ 36 = 1/6 ≈ 16.7%. P(double 6) = 1 ÷ 36 = 1/36 ≈ 2.8%.
1.4 — Lucky dip
(a) P(win) = 5 ÷ 50 = 1/10 = 0.10 = 10%.
(b) P(lose) = 1 − 0.10 = 0.90 = 90%.
(c) Expected wins = 200 × 0.10 = 20 wins.
1.5 — Multiple choice exam
(a) P(correct) = 1 ÷ 4 = 1/4 = 25%. P(incorrect) = 3/4 = 75%.
(b) Expected correct = 20 × 1/4 = 5 questions.
(c) No — the pass mark is 12, but random guessing only yields about 5 correct on average. Guessing alone gives a tiny chance of passing.
2.1 — Explain your thinking (sample response)
The reasoning is wrong because weather events on different days are not strictly random in the same way coin flips are, but the underlying logic the classmate is using is called the gambler's fallacy — the idea that past outcomes "balance out" future ones. In probability, independent events are events where one outcome doesn't affect the next, like coin flips: even after 10 heads in a row, the next flip is still P(heads) = 1/2 because the coin has no memory. Each new flip starts fresh — the probabilities don't "owe" anything to the past. For rain, what matters tomorrow is atmospheric conditions (pressure, fronts, humidity), not the streak of sunny days that just happened.
Marking: 1 mark for naming the flaw (gambler's fallacy / no memory); 1 mark for a coin-flip example showing independence; 1 mark for stating each flip remains 50-50; 1 mark for naming what really determines rain (atmospheric conditions) and using the term "independent events".