Question

With knowledge of axioms of probability, show that P (A) ≤ 1, for any event A

Answer 1

Hi! Using the axioms of probability, we can show that P(A) ≤ 1 for any event A as follows:

1. Non-negativity axiom: P(A) ≥ 0 for any event A.
2. Normalization axiom: P(S) = 1, where S is the sample space.

Now, let's consider the relationship between event A and its complement, A'. By the third axiom, the addition rule, we have:

P(A) + P(A') = P(S), since A and A' together make up the entire sample space.

Substituting the normalization axiom, we get:

P(A) + P(A') = 1.

Since P(A') ≥ 0 (by the non-negativity axiom), we can conclude that:

P(A) ≤ 1 for any event A.