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