This is a basic question about probability theory.
My reasoning goes as follows:
- If $A$ and $B$ are independent events, the probability they both happen is their multiplication: $$\Pr(A \text{ and } B) = \Pr(A) \times \Pr(B)$$
If their marginal probability is not impossible, also their product is non-zero: $$\Pr(A) > 0,\, \Pr(B) > 0 \implies \Pr(A \text{ and } B) > 0$$
Hence, independent events cannot be disjoint
Hence, only dependent events can be disjoint
Hence, all disjoint events are dependent.
Can you help me point out the error in my argument?