## 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
• $P\left(\cup _{i=1}^{\infty }E_{i}\right)=\sum _{i=1}^{\infty }P(E_{i})$