Let A and B be events such that $P(A)=P(B)=0.95$. I know that $$P(A\cup B) = P(A)+P(B)-P(A\cap B)$$ and hence $P(A \cap B) \geq 0.9$
How can I use the information above to disprove the misconception that a 95% confidence interval for a quantity $\theta$ implies then there is a 95% chance that $\theta$ lies in the interval?