Mathematics • Year 10 • Unit 4 • Lesson 19

Conditional Probability in the Real World

Apply Lesson 19 to medical testing, traffic safety, school surveys and exam-style "given that" questions. Practise the HSC Note: always identify what is "given" before you calculate.

Apply · Real-World Maths

1. Word problems

Always state the "given" event first, then identify the reduced sample space.

1.1 — Medical test for a disease. In a screening for a rare condition, of 1000 people tested: 50 actually have the condition. The test correctly identifies 45 of them as positive. The test also returns a positive result for 90 of the 950 healthy people (a "false positive").

(a) Build a two-way table (Has disease Y/N × Test result + / −).
(b) Find P(test positive | has disease).
(c) Find P(has disease | test positive). (This is the question the patient actually cares about.)
(d) Compare (b) and (c) and explain in one sentence why they are so different.    4 marks

Stuck? Lesson 19 Think First Q2 — a test being "95% accurate" is not the same as "95% chance you actually have the disease". Most positives are false because most people don't have it.

1.2 — School elective survey. Of 100 Year 10 students: 60 study Visual Arts, 45 study Music, 25 study both.

(a) Find P(Music | Visual Arts).
(b) Find P(Visual Arts | Music).
(c) Are studying Visual Arts and studying Music independent? (Compare P(Music | Visual Arts) with P(Music).)    3 marks

1.3 — Driver age and crashes. A 12-month study of 500 drivers in a Sydney suburb records: 200 are aged 17-25 ("young"); the rest are over 25. 60 drivers were involved in a crash; 40 of those were young drivers.

(a) Build a two-way table (Age young/older × Crash Y/N).
(b) Find P(crash | young) and P(crash | older).
(c) Comment on whether being a young driver appears to increase the chance of a crash, using your two conditional probabilities.    3 marks

1.4 — Two-deck card game. A card is drawn from a standard 52-card deck.

(a) Find P(king | face card). (Face cards: J, Q, K — 12 total.)
(b) Find P(face card | king).
(c) Apply Lesson 19's HSC Note ("identify what is given before calculating") to explain why these two answers differ.    3 marks

1.5 — Sport and music school census. 60% of students play sport, 40% play music, 25% do both.

(a) Find P(music | sport).
(b) Find P(sport | music).
(c) Are playing sport and playing music independent? Show your test.    3 marks

2. Explain your thinking

Communication question. Use full sentences. 4 marks

2.1 You have just done Q1.1 (the medical test). Your friend reads the result and says: "If 90% of people who have the disease test positive, then I'm 90% sure I have it if my test comes back positive." Write a four-sentence reply that (i) names the misconception (confusing P(test+|disease) with P(disease|test+)), (ii) uses the numbers from Q1.1 to show the actual P(disease | test+) is much lower than 90%, (iii) explains the role of the base rate (how rare the disease is) in plain language, and (iv) finishes with a one-sentence rule of thumb that links the Lesson 19 HSC Note: "always identify what is given before calculating".

Stuck? In Q1.1 there are 135 positive tests in total, but only 45 of those people actually have the disease — that's just 33%. The "false positives" from a huge healthy population swamp the true positives.

How did this worksheet feel?

What I'll revisit before next class:

Answers — Do not peek before attempting

1.1 — Medical test

Two-way table:
| Test + | Test − | Total
Disease Y | 45 | 5 | 50
Disease N | 90 | 860 | 950
Total | 135 | 865 |1000 ✓
(b) P(test+ | disease) = 45/50 = 0.90 (= 90%).
(c) P(disease | test+) = 45/135 = 1/3 ≈ 0.333.
(d) These are very different because most positive tests come from the much larger healthy group (90 false positives vs 45 true positives). Conditioning on "test+" gives a population dominated by healthy people.

1.2 — Elective survey

(a) P(M | VA) = 25/60 = 5/12 ≈ 0.417.
(b) P(VA | M) = 25/45 = 5/9 ≈ 0.556.
(c) P(Music) = 45/100 = 0.45. P(M | VA) = 0.417 ≠ 0.45, so VA and Music are not independent (though they are close).

1.3 — Driver age and crashes

200 young, 300 older. 40 young in crash → 160 young no crash. 20 older in crash → 280 older no crash.
| Crash Y | Crash N | Total
Young | 40 | 160 | 200
Older | 20 | 280 | 300
Total | 60 | 440 | 500 ✓
(b) P(crash | young) = 40/200 = 0.20. P(crash | older) = 20/300 ≈ 0.067.
(c) The conditional probability of a crash given the driver is young is about 3 times that of older drivers (0.20 vs 0.067), suggesting young drivers in this sample are much more likely to crash.

1.4 — Card game

(a) P(king | face) = 4 / 12 = 1/3.
(b) P(face | king) = 4 / 4 = 1 (every king is a face card).
(c) The "given" changes everything. Given face card, the reduced sample is 12 cards (4 are kings → 1/3). Given king, the reduced sample is only 4 cards, and every one is a face card → certainty. HSC Note: always state what is given before calculating.

1.5 — Sport / music school

(a) P(music | sport) = 0.25 / 0.60 = 5/12 ≈ 0.417.
(b) P(sport | music) = 0.25 / 0.40 = 5/8 = 0.625.
(c) Independence test: P(music | sport) = 0.417 ≠ P(music) = 0.40. Not independent (very close, but P(music | sport) is slightly higher, suggesting sport players are a bit more likely than average to also play music).

2.1 — Explain your thinking (sample response)

Your friend has confused P(test+ | disease) (= 90% — the test's accuracy among the sick) with P(disease | test+) (the actual question a patient asks). From Q1.1, only 45 of 135 positive tests are true positives, so P(disease | test+) = 45/135 = 33%, not 90%. The reason is the base rate: the disease is rare (5% of the population), so even a small false-positive rate (90 out of 950 healthy people) produces far more false positives than the small total of true positives. Rule of thumb: always identify what is given before you calculate — P(A|B) and P(B|A) are not the same.

Marking: 1 mark per part (i)-(iv).

HSCScience • Mathematics • Year 10 • Unit 4 • Lesson 19 • Worksheet 2 (Apply)