Independence

Statistical Independence

If knowing $B$ doesn’t give you any information to the probability of $A$ occurring, then $A$ and $B$ are independent.

Thus:

$\text{if}P(A|B)=P(A)$ The events $A$ and $B$ are independent.

$A$ and $B$ are independent $\text{iff}P(A\cap B)=P(A)P(B)$

If the events $A$ and $B$ are independent, $P(A|B^{\complement })=P(A)$

Checking for Independence

Method 1

Check $P(A|B)=P(A)$ or $P(B|A)=P(B)$

Method 2

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

Statistical Dependence

If knowing $B$ provides some information on the probability of $A$ occurring, then $A$ and $B$ are dependent.

Thus:

$\text{if}P(A|B)\ne P(A)$ The events $A$ and $B$ are dependent.

Independence of more than two events (Mutual Independence)

The events $A_1,A_2,\dots ,A_k$ are mutually independent if, for any subset of the events, the joint probability is equal to the product of the individual probabilities.

Similarly, the conditional probability for any event $A_i$ given any subset of the other events is equal to the unconditional probability of $A_i$.

Ex: 50 events

$$\begin{array}{r}P(A_1\cap A_2)=P(A_1)P(A_2)\text{and similarly for any 2 events in}\{A_1,\dots ,A_{50}\}\\ P(A_1\cap A_2\cap A_3)=P(A_1)P(A_2)P(A_3)\text{and similarly for any 3 events in}\{A_1,\dots ,A_{50}\}\\ \dots \\ \dots \\ \dots \\ P(A_1\cap A_2\cap \dots \cap A_3)=P(A_1)P(A_2)\dots P(A_{50})\end{array}$$