Probability proof
Objective. Justify exhaustive cases and complementary strategies.
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Learn: the key idea
Probability proof justifies results about combined events by systematically listing exhaustive, mutually exclusive cases and showing they account for every possible outcome. The complement rule, P(not A) = 1 − P(A), is often the key tool, since it can be easier to find the probability an event fails to happen than that it succeeds. A full justification states why the listed cases cover everything and do not overlap, not just a final numeric answer. To prove P(at least one six in two rolls), find P(no six) = (5/6)² = 25/36, so P(at least one six) = 1 − 25/36 = 11/36. This complement strategy avoids listing many overlapping successful cases directly.
Worked example
Two fair coins are tossed. Prove, using the complement rule, the probability of getting at least one head.
- 1P(no heads at all) = P(tail, tail) = 1/2 × 1/2
- 2P(no heads) = 1/4
- 3P(at least one head) = 1 − P(no heads)
- 4P(at least one head) = 1 − 1/4 = 3/4
Answer: 3/4
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Start practisingCommon mistake. Students often try to add probabilities of overlapping 'at least one' cases directly, double-counting outcomes, instead of using the cleaner complement approach which avoids overlap entirely.
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