Probability synthesis

Objective. Combine complements, cases and expectations in unfamiliar problems.

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Learn: the key idea

Probability synthesis means combining several techniques — complements, tree diagrams, conditional rules and expected value — within one unfamiliar problem. The key strategy is to break the scenario into clear stages, decide whether events are independent or conditional, and use P(not A) = 1 − P(A) to simplify 'at least one' situations. For example, to find P(at least one six in two dice rolls), compute P(no six) = (5/6)² = 25/36, then subtract from 1 to get 11/36. Expectation ties outcomes to values via E(X) = Σ(x × P(x)), useful for decision-based questions like fair games. Always sketch the structure (tree or table) before calculating.

Worked example

A bag has 3 red and 2 blue counters. Two are drawn without replacement. Find the probability of at least one red counter, using probability rules.

1 $Find P(no red) = P(blue, blue) = (2/5) \times (1/4) = 2/20$
2 $Simplify P(no red) = 1/10$
3 $Apply complement: P(at least one red) = 1 - 1/10$
4 $Simplify to 9/10$

Answer: 9/10

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Common mistake. A frequent mistake is adding probabilities for 'at least one' events instead of using the complement, e.g. wrongly computing P(at least one six) as 1/6 + 1/6 = 1/3, which double-counts and ignores dependency between the two rolls.

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