The following question is in my text book (forgive the Latex formatting):
If $A, B,$ and $C$ are three equally likely events, what is the smallest value for $P(A)$ such that $P(A \cap B \cap C)$ always exceeds 0.95?
My scratch work to solve this question has largely used the Multiplicative Law of Probability and the Bonferroni inequality, and is attached below, but is also incorrect. The text book provides the answer of P(A) ≥ .9833. This is equal to .95^(1/3), which would imply that P(A \cap B \cap C)=P(A)^(3)≥.95. This solution defeats me... the question in the text book, featured above, in no way implies that the events are independent, a necessity to invoke the idea that P(A \cap B \cap C)=P(A)^3. How can this be? What subtle (or obvious) rule of Probability is eluding me?
Much obliged for your time,