Questions: Probability Axioms

Question 1

If P(A) = 0.3 and P(B) = 0.5 and events A and B are disjoint (mutually exclusive), what is P(A ∪ B)?

• A. 0.15
• B. 0.2
• C. 0.8
• D. 1.0

Answer: (C)

By the third axiom, for disjoint events P(A ∪ B) = P(A) + P(B) = 0.3 + 0.5 = 0.8. The answer 0.15 would be P(A) × P(B), which applies to independent events under a different rule, not disjoint events under the addition axiom.

Question 2

It is possible for a valid probability assignment to give P(A) = -0.1 for some event A.

• T. True
• F. False

Answer: False

The first axiom of probability requires that P(A) ≥ 0 for every event A. A negative probability has no meaningful interpretation — you cannot have less than no chance of something occurring. Any assignment that violates this axiom is not a valid probability measure.

Question 3

Why must the probability of the sample space equal exactly 1?

Model answer: The sample space contains all possible outcomes of the experiment, so some outcome must always occur. An event that is certain to happen has probability 1, and since exactly one outcome in the sample space must occur, P(S) = 1.

The second axiom encodes the idea of certainty. When you run an experiment, something happens — one of the outcomes in S occurs. The probability of 'something happens' must be 1 (certainty). Setting P(S) = 1 ensures the entire probability 'budget' is distributed across outcomes, which in turn means all probabilities must sum to at most 1 and individual probabilities cannot exceed 1.