Worked Examples: Data-Based Reasoning

This lesson brings together the statistical and probabilistic reasoning skills from the previous nine lessons. Each worked example is solved step by step, with explicit attention to the type of reasoning required, common traps, and time management. Use these examples for practice and revision.

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Example 1: Basic Probability

Time budget: 40 seconds

A box contains 8 white balls and 12 black balls. Two balls are drawn at random without replacement. What is the probability that both balls are black?

Step 1: P(1st black) = 12/20 = 3/5

Step 2: After removing one black ball: 11 black, 8 white remaining (19 total).

P(2nd black | 1st black) = 11/19

Step 3: P(both black) = (12/20) × (11/19) = 132/380 = 33/95 ≈ 0.347

Trap to avoid: Using 12/20 × 12/20 = 144/400 (which assumes replacement). The question says "without replacement," so the second draw has different probabilities.

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Example 2: "At Least One" Problem

Time budget: 45 seconds

A medical test has a 5% chance of giving a false negative (missing a real case). If three patients who all have the condition are tested independently, what is the probability that at least one test correctly identifies the condition?

Step 1: P(single test misses) = 0.05.

P(single test detects) = 0.95.

Actually, re-read: "probability that at least one test correctly identifies the condition." Using the complement: P(at least one detects) = 1 − P(none detect).

Step 2: P(none detect) = 0.05 × 0.05 × 0.05 = 0.000125

Step 3: P(at least one detects) = 1 − 0.000125 = 0.999875 (approximately 99.99%)

Trap to avoid: Calculating 0.95 × 0.95 × 0.95 = 0.857. This gives P(all three detect), not P(at least one detects). P(at least one) is much higher because it includes cases where 1, 2, or all 3 detect.

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Example 3: Expected Value Comparison

Time budget: 50 seconds

A hospital can invest in one of two pieces of equipment:

Equipment A: Cost £50,000. 60% chance of saving 200 bed-days per year (value: £100,000), 40% chance of saving only 50 bed-days per year (value: £25,000).

Equipment B: Cost £30,000. 80% chance of saving 100 bed-days per year (value: £50,000), 20% chance of saving 20 bed-days per year (value: £10,000).

Which equipment has the higher expected net benefit?

Equipment A:

$EV_A = 0.60 \times 100,000 + 0.40 \times 25,000 - 50,000$EVA​=0.60×100,000+0.40×25,000−50,000 $= 60,000 + 10,000 - 50,000 = \textbf{£20,000}$=60,000+10,000−50,000=£20,000

Equipment B:

$EV_B = 0.80 \times 50,000 + 0.20 \times 10,000 - 30,000$EVB​=0.80×50,000+0.20×10,000−30,000 $= 40,000 + 2,000 - 30,000 = \textbf{£12,000}$=40,000+2,000−30,000=£12,000

Answer: Equipment A has the higher expected net benefit (£20,000 vs £12,000).

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Example 4: Percentage Trap

Time budget: 40 seconds

In 2023, 40% of a hospital's 500 patients were treated within 4 hours. In 2024, 45% of the hospital's 400 patients were treated within 4 hours. A manager claims: "We treated more patients within 4 hours in 2024."

Is the manager correct?

2023: 40% of 500 = 200 patients

2024: 45% of 400 = 180 patients

The percentage increased (40% → 45%), but the absolute number decreased (200 → 180) because there were fewer patients overall.

Answer: The manager is incorrect. Fewer patients were treated within 4 hours in 2024, despite the higher percentage.

Lesson: A higher percentage does not necessarily mean a higher absolute number if the total has changed.

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Example 5: Weighted Average

Time budget: 45 seconds

Three wards report average patient stay durations: Ward A (50 patients, 3.0 days), Ward B (30 patients, 5.5 days), Ward C (20 patients, 8.0 days). What is the overall average stay?

$\text{Overall} = \frac{(50 \times 3.0) + (30 \times 5.5) + (20 \times 8.0)}{50 + 30 + 20}$Overall=50+30+20(50×3.0)+(30×5.5)+(20×8.0)​

$= \frac{150 + 165 + 160}{100} = \frac{475}{100} = \textbf{4.75 days}$=100150+165+160​=100475​=4.75 days

Note: The simple average of 3.0, 5.5, and 8.0 would be 5.5 — significantly different from the correct weighted average of 4.75.

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Example 6: Correlation vs Causation

Time budget: 40 seconds

A study found that regions with more fast-food restaurants per capita have higher rates of Type 2 diabetes. A newspaper headline reads: "Fast food causes diabetes epidemic."

Which of the following is the best evaluation?

A. The study proves that fast food causes diabetes. B. Regions with more fast food have unhealthier populations, confirming the causal link. C. There is a correlation between fast-food availability and diabetes rates, but confounding factors (income levels, access to healthcare, exercise facilities) could explain the association. D. The study is irrelevant because it does not use individual-level data.