8.2 Three Definitions

8.2.1 Classical

\(P (A) = \frac{n(A)}{n(S)}\)

8.2.2 Relative frequency

\[\lim_{n(S) \to \infty} \frac{n(A)}{n(S)}\]

8.2.3 Axiomatic

Three axioms

Say, S is sample space and A is an event

• \(0 \le P (A) \le 1\) (NOT \(P(A) \ge 0\))
• At least one of S will occur. P (S) = 1; Certain event.
• \(P(A_1 U A_2 U ... U A_n)=P(A_1) + P(A_2) + ... + P(A_n)\) or
  
• \[P\left(\cup _{i=1}^{\infty }E_{i}\right)=\sum _{i=1}^{\infty }P(E_{i})\]