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.