Why do we add probabilities? I know that there is the addition rule of probability, but I want to understand the intuition behind it. Specifically, why does OR signifies addition in probability theory?
 A: I suppose that you question is this:

If two events $A$ or $B$ cannot occur simultaneously, then why is the probability that $A$ occurs or $B$ occurs equal to the probability that $A$ occurs plus the probability that $B$ occurs?

Suppose that there are, say, $100$ possibilities, that $A$ takes place in $50$ of them and that $B$ takes place in $20$ of them. Then the probability that $A$ occurs is $\frac12\left(=\frac{50}{100}\right)$ and the probability that $B$ occurs is $\frac15\left(=\frac{20}{100}\right)$. What is the probability that $A$ occurs or $B$ occurs? Well, out of those $100$ possibilities, $A$ occurs or $B$ occurs exactly in $70$ of them (this is where I use the fact that $A$ or $B$ cannot occur simultaneously). So, the probability that $A$ occurs or $B$ occurs is\begin{align}\frac{70}{100}&=\frac{50}{100}+\frac{20}{100}\\&=\text{probability that $A$ occurs}+\text{probability that $B$ occurs.}\end{align}
A: Because the probability is the number of favorable draws over the total number of draws.
And the number of favorable draws are additive: the number of [red or green] balls is the number of red plus the number of green.

Caution:
The additive rule is only valid for disjoint categories. For example, if you have black/white balls and dice, it is not necessarily true that
$$\text{#(black or ball)} = \text{#black} + \text{#balls}.$$
A: In the end it just boils down to the fact that if $A$ and $B$ are disjoint finite sets, then $$|A\cup B|=|A| + |B|$$
A: Think about it like this
Lets say there is a $6$ sided dice. 
Now, when asked to find $P(3 \mathbf {or} 6)$, consider what this statement actually means. The probability that either $3$ occurs or $6$ occurs. This is equivalent to adding the chances of $3$ and $6$. Therefore $\mathbf{or}$ represents addition. 
