Mathematics • Year 8 • Unit 4 • Lesson 15

The Probability Scale in the Real World

Apply the probability scale to real situations: weather, sport, medical screening, exam predictions, and game design. Each problem demands calculation AND a placement decision.

Apply · Real-World Maths

1. Word problems

Each problem uses ideas from Lesson 15. Show working.

1.1 — Weather forecast. The weather app gives the following probabilities for tomorrow: P(rain) = 0.2, P(thunderstorm) = 0.05, P(sunshine) = 0.6, P(snow in Sydney in summer) = 0.

(a) Label each event on the scale (impossible / unlikely / even chance / likely / certain).
(b) Convert each probability to a percentage.
(c) Order the four events from least to most likely. 3 marks

Stuck? Anything between 0 and 0.5 is "unlikely". Sunshine at 0.6 is "likely".

1.2 — Sports prediction. The favourite has a 70% chance of winning the basketball final. Their opponents have a 30% chance.

(a) Convert both percentages to decimals.
(b) Label each event on the probability scale.
(c) Explain in one sentence why "favourite" doesn't mean "guaranteed to win". 3 marks

Stuck? "Likely" is 0.5 to 1 exclusive. P = 0.7 is likely, not certain.

1.3 — Medical screening. A diagnostic test for a rare condition has the following stats: P(false positive) = 0.05, P(true positive | patient has condition) = 0.98, P(patient has the condition in the general population) = 0.001.

(a) Label each of the three probabilities on the scale.
(b) Convert each to a percentage.
(c) Of the three, which is the smallest and which is the largest? 3 marks

Stuck? P = 0.001 is very unlikely (0.1%); P = 0.98 is likely (98%); P = 0.05 is unlikely (5%).

1.4 — Exam preparation. A student estimates: P(passing maths if I study 1 hour) = 0.4, P(passing maths if I study 5 hours) = 0.85, P(passing maths if I study 10 hours) = 0.95.

(a) Label each scenario on the scale.
(b) Convert each probability to a percentage.
(c) Why does the gap between 1-hour and 5-hour study (40% → 85%) feel bigger than the gap between 5-hour and 10-hour (85% → 95%)? 3 marks

Stuck? The gain from 1 hour to 5 hours is 45 percentage points; from 5 to 10 is only 10 — diminishing returns.

1.5 — Game design. A claw-machine arcade game is set so that P(winning a small prize) = 1/8 and P(winning a large prize) = 1/200.

(a) Convert both probabilities to percentages (3 d.p.).
(b) Label each event on the scale.
(c) Which is more likely, and how many times more likely? Show the ratio. 3 marks

Stuck? 1/8 = 0.125 = 12.5%. 1/200 = 0.005 = 0.5%. Ratio = 12.5 ÷ 0.5 = 25 times.

2. Explain your thinking

This question is about communication, not just numbers. Use full sentences. 4 marks

2.1 A weather presenter says: "There is a 99% chance of sunshine tomorrow — so you can guarantee a sunny day." Explain (i) why the word "guarantee" is mathematically incorrect, (ii) what the 1% leftover represents on the probability scale, (iii) why a 99% probability is "likely" but NOT "certain", and (iv) one sentence explaining when (if ever) a probability is high enough to use the word "certain". Use the term probability scale.

Stuck? Only P = 1 (i.e. 100%) is "certain". Anything less than 100% still leaves room for the unexpected.

How did this worksheet feel?

Got it Partly Lost

What I'll revisit before next class:

Answers — Do not peek before attempting

1.1 — Weather forecast

(a) Rain (0.2): unlikely. Thunderstorm (0.05): unlikely. Sunshine (0.6): likely. Snow in Sydney summer (0): impossible.
(b) Rain = 20%, thunderstorm = 5%, sunshine = 60%, snow = 0%.
(c) Least to most likely: Snow (0) < Thunderstorm (0.05) < Rain (0.2) < Sunshine (0.6).

1.2 — Sports prediction

(a) Favourite = 0.7. Opponents = 0.3.
(b) Favourite (0.7): likely. Opponents (0.3): unlikely.
(c) "Favourite" just means more likely to win than not — but 70% still leaves a 30% chance the favourite loses. Upsets happen because P < 1.

1.3 — Medical screening

(a) False positive (0.05): unlikely. True positive (0.98): likely (very close to certain). Prevalence (0.001): unlikely (very close to impossible).
(b) 5%, 98%, 0.1%.
(c) Smallest = prevalence (0.001 = 0.1%). Largest = true positive (0.98 = 98%).

1.4 — Exam preparation

(a) 1 hour (0.4): unlikely. 5 hours (0.85): likely. 10 hours (0.95): likely (close to certain).
(b) 40%, 85%, 95%.
(c) The first 4 extra hours of study gave a 45-percentage-point jump (40% → 85%), but the next 5 hours only gave 10 more (85% → 95%). This is diminishing returns — early study fills the biggest gaps; later study just polishes.

1.5 — Claw machine

(a) Small prize: 1/8 = 0.125 = 12.500%. Large prize: 1/200 = 0.005 = 0.500%.
(b) Both: unlikely (both below 0.5), but small prize is much more likely than large.
(c) Small prize is more likely. Ratio = 0.125 ÷ 0.005 = 25 times more likely.

2.1 — Explain your thinking (sample response)

"Guarantee" is wrong because a 99% probability sits on the probability scale between 0.5 and 1 — it is likely, but not certain. The remaining 1% represents the small probability of the event NOT happening — in this case, rain or cloud cover. A 99% probability means the event will almost always happen, but if you ran 100 such days, on average about 1 of them would still be cloudy. A probability is only "certain" when it equals exactly 1 (i.e. 100%) — anything strictly less than that, even 99.99%, leaves a non-zero chance of the alternative.

Marking: 1 mark for explaining 99% ≠ 100%; 1 mark for what the 1% represents; 1 mark for distinguishing "likely" from "certain" using the scale; 1 mark for stating that "certain" requires P = 1 exactly.