Do I use the rule of multiplication here? Why not here? A die is rolled.  What is the probability that the number is even and
  less than 4?
  
  Event $A$:  Numbers on a die that are even:  2, 4, 6
  Event $B$:  Numbers on a die that are less than 4:  1, 2, 3
  There is only one number (2) that is in both events A and B.
  Total outcomes $S$:  Numbers on a die: 1, 2, 3, 4, 5, 6  (total = 6)


Ok, so obviously the one possible outcome in this example is $1/6$. But if I use the rule of multiplication which states that if the sample space is the same, which I think it is, then $P(a)P(b)$ should give me the right answer, which it doesn't: $1/4$. 

(My logic here is that less than 4 is 50% chance and the even numbers are 50% chance). 

So the big question is, why doesn't the multiplication rule work here?
There is a way of recovering a  version of the multiplication rule.

Let $A$ and $B$ be as in your post. We cannot write $\Pr(A\cap B)=\Pr(A)\Pr(B)$, since $A$ and $B$ are not independent.

However, we have 
$$\Pr(A\cap B)=\Pr(A)\Pr(B|A),$$
 where $\Pr(B|A)$ means the (conditional) probability $B$ happens given that $A$ happens. 

Given that $A$ happened, meaning the number is even, the probability it is between $1$ and $3$ is $1/3$, since only one-third of evens are between $1$ and $3$. So $\Pr(A\cap B)=(1/2)(1/3)$.
To apply multiplication rule, the intersection of values of the individual sets should be null matrix. 
i.e A \cap B = null matrix