Proof that if Two Events are Mutually Exclusive, They are Dependent
Theorem
For two events $A,B$, if $B=A^c$ then $P(A\cap B) \neq P(A)P(B)$
Explanation
Even without a formal proof through equations, it is common sense that if events are mutually exclusive, they cannot be independent. If one event occurring means the other cannot, this already implies an influence. However, knowing or not knowing this makes a big difference when solving problems regarding true or false judgments.
Proof
Let’s say for two events $A,B$, we have $P(A)>0, P(B)>0$. Since the events are mutually exclusive, $P(A\cap B)=0$ holds. However, because of $P(A)>0, P(B)>0$, $P(A)P(B)>0$, and consequently, $P(A\cap B)\neq P(A)P(B)$ must be true.
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