# Addition law for non mutually exclusive events

The addition law for non mutually exclusive events is given in my textbook as

$$P(A \space \text{or} \space B)=P(A \space\cup \space B)= P(A)+P(B)-P(A \space \cap \space B)$$

I understand the logic behind this and would be fine if it were written as $$P(A \space \text{or} \space B)=P(A)+P(B)-P(A \space \cap \space B)$$ however I am fairly certain that in this case it is incorrect to say that $$P(A \space \text{or} \space B)=P(A \space\cup \space B)$$

Here is a diagram to further elaborate

From the diagram would it not be true that $$P(A \space \cup \space B)=P(A \space \text{or} \space B) + P(A \space \text{and} \space B)$$?

Is there a mistake in my textbook or am I missing something crucial?

In this context “A or B” means “A, or B, or both”. So it is correct to say that $$P(A \text{ or } B) = P(A \cup B)$$. There’s no mistake in your textbook.

• If that is the case then why have they subtracted the $P(A \space \cap B)$ term , would $P(A \space \cap B)$ not qualify as both A,B? Commented Apr 26, 2021 at 10:59
• Because if you just add P(A) and P(B), you’ll be counting the intersection area twice. You have to subtract one of the two copies. Commented Apr 26, 2021 at 11:52
• Forget about probabilities for a minute — just think about areas. Commented Apr 26, 2021 at 11:53
• Area of overall figure-eight = area of A + area of B - area of overlap. Commented Apr 26, 2021 at 11:54

To be more precise...

$$P(x \in A ~\text{or}~ x \in B) = P(x \in (A \cup B)).$$

Bonus:

$$P(x \in A ~\text{and}~ x \in B) = P(x \in (A \cap B)).$$

It seems by the definition that when they talk about $$P(A\text{ or }B)$$ they actually mean $$P(\text{at least }A\text{ or }B)$$, that is $$P(A,B\text{ or }A\text{ and }B$$)

• Yes, "or" is used in the inclusive sense. Exclusive or can be written "xor". So A xor B means A or B but not both. Commented Apr 26, 2021 at 10:52