Probability axioms
Objective. Derive probability laws from the axioms.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
The probability axioms are the three foundational rules from which every other probability law is derived: P(A) ≥ 0 for any event A, P(S) = 1 for the entire sample space S, and for mutually exclusive events A and B, P(A ∪ B) = P(A) + P(B). From these, further results follow logically rather than by assumption: for instance, P(Aᶜ) = 1 − P(A) because A and Aᶜ partition S and are mutually exclusive. Similarly, the general addition rule P(A ∪ B) = P(A) + P(B) − P(A ∩ B) corrects for double-counting the overlap when A and B are not mutually exclusive. For a fair die, P(even) = P({2,4,6}) = 3/6 = 1/2 follows directly from axiom three applied to the three mutually exclusive outcomes.
Worked example
Using the probability axioms, find P(A ∪ B) given P(A) = 0.4, P(B) = 0.5 and P(A ∩ B) = 0.2.
- 1Recognise that A and B are not mutually exclusive since P(A ∩ B) ≠ 0.
- 2Apply the general addition rule: P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
- 3Substitute the given values: P(A ∪ B) = 0.4 + 0.5 − 0.2.
- 4Simplify the sum to find the final probability.
Answer: P(A ∪ B) = 0.7
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Practice
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Start practisingCommon mistake. Students often add P(A) + P(B) to find P(A ∪ B) even when A and B overlap, forgetting to subtract P(A ∩ B); this third axiom only applies directly when the events are mutually exclusive.
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