Question

Suppose S is a sample space and f (E) = n(E) for each event E of S. Prove that f is a probability
n(S) function by verifying that it obeys the three axioms.

Answer 1

I belive  f(E) = n(E) / n(S)

The three probability axioms are -

(1) P(E) 0 for all E S

(2) P(S) = 1

(3) If E1 E2 = , then P(E1 E2) = P(E1) + P(E2)

(1)

f (E) = n(E)/n(S)   0 because n(S) and n(E) cannot be negative and their ratio cannot be negative.

(2)

P(S) = f(S) = n(S) / n(S) = 1

(3)

If E1 E2 = , then E1 and E2 events are disjoint, then frequency of E1 or E2 happening should be n(E1) + n(E2)

P(E1 E2) = f (E1 + E2) = [n(E1) + n(E2)] / n(S) = n(E1) / n(S) + n(E2) / n(S) = P(E1) + P(E2)