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title: 機率TP3
tags: [Theory of Probability, 必修, NTNU, CSIE]

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title: 機率TP3
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# Theory of Probability：<br> Conditional Probability, Total Probability Theorem, Bayes Rule
NTNU 機率論

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###### tags: `NTNU` `CSIE` `必修`  `Theory of Probability`

## Conditional Probability
- Conditional probability provides us with a way to reason about the outcome of an experiment, based on partial information
    - Suppose that the outcome is within some given event $B$, we wish to quantify the ==likelihood== that that the outcome also belongs some other given event $A$ 
    - Using a new probability law, we have the **conditional problbility of $A$ given $B$**, denoted by $P(A|B)$, which is defined as $$P(A|B) = \frac{P(A \cap B)}{P(B)}$$
        - If $P(B)$ has zero probability, $P(A|B)$ is undefined.
        - We can think of $P(A|B)$ as out of the total problbility of the elements of $B$ , the fraction that is assigned to possible outcomes that also belong to $A$.

### Conditional Probability Example
$$
P(A_1 \cap A_2 \cap A_3) =P(A_1)P(A_2|A_1)P(A_3|A_1\cap A_2)
$$
<!-- 打兩個$$的記得要單行 不然後面會爛掉 -->
## Conditional Probabilities Satisfy the Three Axioms 

## Conditional Probabilities Satisfy General Probability Laws

### Simple Examples using Conditional Probabilities

### Using Conditional Probability for Modeling

## Multiplication (Chain) Rule

### Multiplication (Chain) Rule: Examples

## Total Probability Theorem 
### Some Examples Using Total Probability Theorem

## Bayes’ Rule

## Bayes’ Rule and Inference

### Inference Using Bayes’ Rule 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