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?
 A: 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 $\Pr(A\cap B)=(1/2)(1/3)$.
A: Observe that Event A and Event B are statistically independent iff $P(A \cap B) = P(A)P(B)$, which is decidedly not the case here ($P(A\cap B)=P(2)=1/6, P(A)P(B)=1/4$).
As a result you can not use the multiplication rule for independent events.
https://en.wikipedia.org/wiki/Independence_%28probability_theory%29#More_than_two_events
A: Consider these two events: 


*

*$A$: the result of the die is not odd: $\{2, 4, 6\}, P(A)=1/2$

*$B$: the result of the die is divisible by 2: $\{6, 4, 2\}, P(B)=1/2$


Now if we were able to use the multiplication rule we would say $\mathsf P(A\cap B) \mathop{=}^{?} \mathsf P(A)\,\mathsf P(B) = 1/4$.  
This is, of course, nonsense.   The questioned equality does not hold.
The events are the same, so the probability of their intersection must be the probability of the events themselves: $\mathsf P(A\cap B) = 1/2$.
So clearly the multiplication rule is not useable when the events are not independent.
A: To apply multiplication rule, the intersection of values of the individual sets should be null matrix. 
i.e A \cap B = null matrix
