Are all disjoint events dependent?

Question

This is a basic question about probability theory.

My reasoning goes as follows:

1. If $A$ and $B$ are independent events, the probability they both happen is their multiplication: $$\Pr(A \text{ and } B) = \Pr(A) \times \Pr(B)$$

2. If their marginal probability is not impossible, also their product is non-zero: $$\Pr(A) > 0,\, \Pr(B) > 0 \implies \Pr(A \text{ and } B) > 0$$

3. Hence, independent events cannot be disjoint

4. Hence, only dependent events can be disjoint

5. Hence, all disjoint events are dependent.

Can you help me point out the error in my argument?

Answer 1

Yes, as long as A and B are mutually exclusive and sum up to 1 or 100%, P(A) + P(B) = 1. Example : You are either dead or alive. The chance of being alive or dead always sums up tp 1 or 100%, no matter what condition you live in.