1.3 The Probability Set Function

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Definition 1.3.1: Probability set function

Let

• $\mathcal{C}$ : Sample space
• $\mathcal{B}$ : Events ($\mathcal{B}\subset \mathcal{C}$)
• $P:\mathcal{B}\to \mathbb{R}$ (Function)

If

1. $P(A)\ge 0,\forall A\subset \mathcal{B}$
2. $P(\mathcal{C})=1$
3. If $\{A_n\}\in \mathcal{B}$ and $A_m\cap A_n=\varphi ,\forall m\ne n$ then

$$P\left(\bigcup _{n=1}^{\infty }{A}_n\right)=\sum _{n=1}^{\infty }P\left(A_n\right)$$

Then

• We say $P$ is a probability set function
• We call the return value of $P$ as the probability

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Theorems

Let $\mathcal{B}$ : Set of events

Then

• $P(A)=1-P(A^C),\forall A\in \mathcal{B}$
• $P(\varphi )=0$
• $A\subset \mathcal{B}⟹P(A)\le P(\mathcal{B})$
• $0\le P(A)\le 1,\forall A\in \mathcal{B}$
• $A,B\in \mathcal{C}⟹P(A\cup B)=P(A)+P(B)-P(A\cup B)$

Definition 1.3.2: Equilikely Case

Let

• $\mathcal{C}=\{x_1,\dots ,x_m\}$ : Finite sample space
• $p_i=1/m,\forall i=1,\dots ,m$
• $\#(A)$ : Number of elements in set $A$
• $P(A)$ : Probability set function

If $P(A)={\sum }_{x_i\in A}\frac{1}{m}=\frac{\#(A)}{m},\forall A\subset \mathcal{C}$

Then we say $P$ is a probability on $\mathcal{C}$, which we refer to as the equilikely case

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Theorem 1.3.6

Let $\{C_n\}$ : Sequence of events

If $\{C_n\}$ is nondecreasing

Then

$$\underset{n\to \infty }{\lim }P(C_n)=P\left(\underset{n\to \infty }{\lim }{C}_n\right)=P\left(\bigcup _{n=1}^{\infty }{C}_n\right)$$

If $\{C_n\}$ is decreasing

Then

$$\underset{n\to \infty }{\lim }P(C_n)=P\left(\underset{n\to \infty }{\lim }{C}_n\right)=P\left(\bigcap _{n=1}^{\infty }{C}_n\right)$$

Theorem 1.3.7: Boole’s inequality

Let $\{C_n\}$ : Arbitrary sequence of events

Then

$$P\left(\bigcup _{n=1}^{\infty }{C}_n\right)\le \sum _{n=1}^{\infty }P(C_n)$$