Consider two events $A$ and $B$, with $P(A) = 0.4$ and $Pr(B) = 0.7$. Determine the maximum and the minimum possible values for $P(A \& B)$ and the conditions under which each of these values is attained.
To solve, I considered the event with the lowest probability $A$ to be a subset of the other, so maximum value is attained under that circumstance giving a probability of $0.4$. But the book states that the minimum is $0.1$, if $P(A \cup B) = 1$.
I don't understand why! Because I thought that the minimum value is get when the two events are disjoint... So the minimum value must be $0$...