# 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?

• It's not always true. If $A\cap B\neq \emptyset$, then the probability to have $A$ or $B$ is not $\mathbb P(A)+\mathbb P(B)$, but $\mathbb P(A)+\mathbb P(B)-\mathbb P(A\cap B)$. – Surb Mar 26 at 7:50
• I understand, but why did we add the probabilities at first, why didn't we multiply them before subtraction? – Positron12 Mar 26 at 7:52

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}

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

$$\text{#(black or ball)} = \text{#black} + \text{#balls}.$$

In the end it just boils down to the fact that if $$A$$ and $$B$$ are disjoint finite sets, then $$|A\cup B|=|A| + |B|$$

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.