I am trying to prove the multiplication rule for mutually independent events for all n.
I know the base step for $A$ and $B$ if they are independent events, but is it enough to say that :
- if $A, B, C$ are mutually independent,
- then
- $P(A \cap B) = P(A) \times P(B)$,
- and $P((A \cap B) \cap C) = (P(A) \times P(B)) \times P(C)$?
If so, then I can generalize this to any event. If not, it's looking like proof by induction.
Best Answer
The events $A$, $B$, $C$ are pairwise independent if $$ P(A\cap B)=P(A)P(B), \ P(B\cap C)=P(B)P(C), \ \mathrm{and} \ P(C\cap A)=P(C)P(A). $$
The events $A$, $B$, $C$ are (mutually) independent if
$A$, $B$, $C$ are pairwise independent, and
$P(A\cap B\cap C)=P(A)P(B)P(C)$.
We have some examples showing that 1. does not imply 2., also there are examples showing that 2. does not imply 1.