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Combined probability
Lesson 70 of 75
Review

Probability reasoning

Objective. Use complements and structured cases to justify solutions.

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  2. 2. Worked example
  3. 3. Practice
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Learn: the key idea

Probability reasoning uses complements and structured case analysis to justify solutions to problems that cannot be solved by a single direct calculation. The complement rule states that P(not A) = 1 − P(A), which is often the fastest way to find the probability of 'at least one' event occurring, by first finding the probability none occur. For example, finding the probability of rolling at least one six in two dice rolls is easier by calculating 1 minus the probability of no sixes at all: 1 − (5/6 × 5/6) = 1 − 25/36 = 11/36. Structured case analysis means systematically listing every distinct scenario that satisfies a condition before adding their probabilities together.

Worked example

A fair coin is flipped three times. Find the probability of getting at least one head, using the complement rule for this probability question.

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Illustrated card showing ?
  1. 1Find the probability of no heads at all, meaning all three flips are tails.
  2. 2P(all tails) = 1/2 × 1/2 × 1/2 = 1/8.
  3. 3Use the complement rule: P(at least one head) = 1 − P(no heads).
  4. 41 − 1/8 = 7/8.

Answer: 7/8

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Common mistake. Pupils sometimes try to calculate 'at least one' probabilities by adding individual probabilities directly, which overcounts overlapping outcomes. Use the complement rule, 1 minus the probability of none occurring, for 'at least one' questions.

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