1.4 Conditional Probability and Independence

<< 1.3 The Probability Set Function | 1.5 Random Variables >>

Definition 1.4.1: Conditional probability

Definition

Let

• $B,A$ : Events
• $P(A)>0$

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

Then we say $P(B|A)$ is the conditional probability of $B$ given $A$

Moreover,

1. $P(B|A)\ge 0$
2. $P(A|A)=1$
3. If $B_1,\dots ,B_n$ are mutually exclusive events, then $P(\cup_{n=1}^\infty B_{n}|A)=\sum_{n=1}^\infty P(B_{n}|A)$$

Theorem 1.4.1: Bayes theorem

Definition

Let

• $A_1,\dots ,A_k$ : Events
• $P(A_i)>0,i=1,\dots ,k$
• $B$ : Event

Suppose $A_1,\dots ,A_k$ form a partition of $\mathcal{C}$

Then $ \begin{align} P(A_{j}|B) & = \frac{P(A_{j})P(B|A_{j})}{\sum_{i=1}^kP(A_{i})P(B|A_{i})} \ \

P(A|B) & = \frac{P(B|A);P(A)}{P(B)} \end{align}

## Definition 1.4.2: Independency ## Definition Let $A,B$ : [[3 Reference/Def-events\|Events]] If $ P(A\cap B) = P(A)P(B) $$ Then we say that $A$ and $B$ are **independent** ## See also - https://www.youtube.com/watch?v=XQoLVl31ZfQ - https://www.youtube.com/watch?v=ibINrxJLvlM - https://www.youtube.com/watch?v=OYT0AcuLXu8