Probability rules and axioms: sums and exclusivity

Probability • Probability

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State the non-negativity axiom.

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Every event has probability ≥ 0.

Key concepts

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Sample space and exhaustive sets

The sample space is the set of all possible outcomes of a chance experiment. An exhaustive set of outcomes contains every outcome that can occur, so one of them must occur on each trial. Because one outcome from an exhaustive set must occur, the probabilities of all outcomes in that set sum to one. This rule limits individual probabilities: no assignment may make the total exceed or fall below one.

Mutually exclusive events

Two events are mutually exclusive when they cannot occur at the same time. If one event happens, the other cannot happen in the same trial. For a set of mutually exclusive events that together cover all possibilities (an exhaustive partition), the probability of the union equals the sum of the probabilities of the events. Cause: no overlap. Effect: no double counting when adding probabilities.

Axiom 1 - Non-negativity

Probabilities take values greater than or equal to zero for every event. Negative probabilities are impossible because probability measures the long-run frequency or likelihood of occurrence. Limiting factor: probabilities must respect this lower bound when assigning or solving for unknown probabilities in a model.

Axiom 2 - Normalisation (sum of sample space equals 1)

The probability of the sample space equals one by definition, because the sample space contains all possible outcomes and one of them must occur. Cause: completeness of the sample space. Effect: the probabilities of an exhaustive set of outcomes must sum to one, which enables calculating missing probabilities and complements.

Axiom 3 - Additivity for mutually exclusive events

For two or more mutually exclusive events, the probability that one of them occurs equals the sum of their individual probabilities. Cause: events do not overlap. Effect: simple addition yields the probability of their union; addition fails if events are not mutually exclusive unless overlap is corrected for.

Complement rule and use of axioms

The complement of event A contains all outcomes in the sample space that are not in A. The normalisation axiom implies P(A) + P(A') = 1, so P(A') = 1 − P(A). Cause: A and A' form an exhaustive pair of mutually exclusive events. Effect: simple computation of ‘at least one’ or ‘not’ probabilities using the complement rule.

Key notes

Important points to keep in mind

Probabilities are always between 0 and 1 inclusive.

The sample space is exhaustive; its probability equals 1.

Mutually exclusive events have no overlap; their probabilities add directly.

Use P(A') = 1 − P(A) for complements to avoid complex counting.

For non-mutually exclusive events, subtract the intersection to avoid double counting.

Assignments of probabilities must respect non-negativity and normalisation.

Check that sums of probabilities for exhaustive partitions equal exactly 1.

If a calculated probability is negative or greater than 1, the model or arithmetic is incorrect.