Independent Events

Definition 12.15 -- independent events. Events are said to be independent if the occurrence or non-occurrence of one of them does not affect the probability of the occurrence or non-occurrence of the other. Formally, two events $A$ and $B$ are independent if and only if

Note 12.6. (1) This is exactly equivalent to $P(A/B)=P(A)$ (when $P(B)>0$) and $P(B/A)=P(B)$ (when $P(A)>0$) -- independence means the conditional probability collapses back to the plain, unconditional probability; knowing one event happened genuinely tells you nothing new about the other. (2) The events $A_1,A_2,\ldots,A_n$ are mutually independent if EVERY sub-collection satisfies the product rule: $P(A_1\cap A_2\cap\cdots\cap A_n)=P(A_1)\cdot P(A_2)\cdots P(A_n)$ (and every smaller sub-collection too).

Theorem 12.8. If $A$ and $B$ are independent, then: (i) $\bar A$ and $B$ are independent; (ii) $A$ and $\bar B$ are independent; (iii) $\bar A$ and $\bar B$ are also independent. Proof of (i): since $A,B$ are independent, $P(A\cap B)=P(A)P(B)$. By De Morgan's law, $P(\bar A\cap\bar B)=P(\overline{A\cup B})=1-P(A\cup B)=1-[P(A)+P(B)-P(A\cap B)]=1-P(A)-P(B)+P(A)P(B)=[1-P(A)][1-P(B)]=P(\bar A)\cdot P(\bar B)$, which proves $\bar A,\bar B$ are independent (part iii); parts (i) and (ii) follow by the same style of argument.

Note 12.7. Independence is a property of PROBABILITY, but mutual exclusiveness is a SET-THEORETIC property. Independent events are identified from their probability VALUES ($P(A\cap B)$ compared with $P(A)P(B)$); mutually exclusive events are identified from the EVENTS themselves ($A\cap B=\varnothing$ or not).

Theorem 12.9. Suppose $A,B$ are two events with $P(A)\ne0, P(B)\ne0$. (1) If $A,B$ are mutually exclusive, they CANNOT be independent. (2) If $A,B$ are independent, they CANNOT be mutually exclusive (without proof). The reason for (1): independence would require $P(A\cap B)=P(A)P(B)>0$ (a strictly positive value, since both factors are positive), but mutual exclusivity requires $P(A\cap B)=0$ -- these two requirements directly contradict each other, so genuinely non-trivial mutually exclusive events can never also be independent. …