EX 2.33 mutually exclusive vs independent

• Let $A$ and $B$ be any two mutually exclusive events with $P\left(A\right)>0$ .
• For example, for a randomly chosen automobile, let
  • $A=\{$ the car has a four cylinder engine $\}$
  • $B=\{$ the car has a six cylinder engine $\}$ .
• Since the events are mutually exclusive,
  • if $B$ occurs,
  • then $A$ cannot possibly have occurred,
  • so $P\left(A\mid B\right)=0\ne P\left(A\right)$ .
• The message here is that
  • if two events are mutually exclusive,
  • they cannot be independent.
• When $A$ and $B$ are mutually exclusive,
  • the information that $A$ occurred says something about $B$ (it cannot have occurred),
  • so independence is precluded.