Are all disjoint events dependent? 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:
$$\Pr(A) &gt; 0,\, \Pr(B) &gt; 0 \implies \Pr(A \text{ and } B) &gt; 0$$
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?
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.
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.