Chapter 1 Probability Theory

1.1 Introduction

Experiment: Any process or procedure for which more than one outcome is possible.

Sample Space $S$: All the possible outcomes.

Probability $P(x_i)$ meet the requirement: $0\leq p_i\leq 1, i=1, 2, \dots, n$ and $p_1 + p_2 + \dots + p_n = 1$.

1.2 Events

Event $A$: An event is a subset of the sample space. The probability of an event is obtained by the probabilities of the outcomes contained within the event.

Complements of envents $A'$: Everything in the sample space not contained within event.

Elementary event: An event only contains one individual outcome.

1.3 Intersection of Events

Intersection $A\cap B$: Outcomes within both events $A$ and $B$.

Union $A\cup B$: Outcomes within event $A$ or event $B$.

$A\cap (B\cap C) = (A\cap B)\cap C$

$P(A\cup B) = P(A) + P(B) - P(A\cap B)$

$P(A\cup B) = P(A\cap B') + P(A'\cap B) + P(A\cap B)$

$(A\cup B)' = A' \cap B'$

$(A\cap B)' = A' \cup B'$

$(A\cup B\cup C) = A + B + C - (A\cap B) - (B\cap C) - (C\cap A) + (A\cap B\cap C)$

1.4 Conditional Probability

$P(A|B) = \frac{P(A\cap B)}{P(B)}, P(B)>0$

$P(A|B) = \frac{P(A\cap B)}{P(B)} \Longrightarrow P(A\cap B) = P(B)P(A|B)$

1.5 Probabilities of Event Intersections

$P(A|B\cap C) = \frac{P(A\cap B\cap C)}{P(B\cap C)} \Longrightarrow P(A\cap B\cap C) = P(B\cap C)P(A|B\cap C)$

$P(A_1 \cap \dots \cap A_n) = P(A_1)P(A_1|A_2)P(A_3|A_1\cap A_2)\dots P(A_n|A_1\cap \dots \cap A_{n-1})$

Independent: One event's occur would not affect another event.

1. $P(A|B) = P(A)$

2. $P(B|A) = P(B)$

3. $P(A\cap B) = P(A)P(B)$

1.6 Posterior Probability

If an event $B$ is contained within a sample space $A$, then we have $P(B) = \sum_{i=1}^{n}P(A_i)P(B|A_i)$

The Bayes' Theorem: