1.4 Conditional Probability and Independence

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Definition 1.4.1: Conditional probability

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 $P({\cup }_{n=1}^{\infty }{B}_n|A)={\sum }_{n=1}^{\infty }P(B_n|A)$

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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

$$\begin{array}{rl}P(A_j|B)& =\frac{P(A_j)P(B|A_j)}{{\sum }_{i=1}^{k}P(A_i)P(B|A_i)}\\ & \\ P(A|B)& =\frac{P(B|A)P(A)}{P(B)}\end{array}$$Link to original

Definition 1.4.2: Independency

Let $A,B$ : Events

If $P(A\cap B)=P(A)P(B)$

Then we say that $A$ and $B$ are independent

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See also

• https://www.youtube.com/watch?v=XQoLVl31ZfQ
• https://www.youtube.com/watch?v=ibINrxJLvlM
• https://www.youtube.com/watch?v=OYT0AcuLXu8