Difference between P(A)+P(B) and P(A)+P(B) - P(A&B)

Question

COuld you please explain me the difference between

1. P(A) + P(B)
2. P(A) + P(B) - P(A & B)

I thought that:

1. meant "Probability that either A or B or both happen" so basically at least one event happening.

2. meant "Probability that only one of A or B happens but NOT both".

But this seems to be wrong because I am seeing problems that are solved in different ways, like assuming that 2) is the probability of at least one event happening, which is what I thought was 1).

Answer 1

It may be easier to see what is going on if you split $P(A)$ and $P(B)$:

• $P(A)=P(A\cap B)+P(A\cap B^c)$
• $P(B)=P(A\cap B)+P(A^c\cap B)$

So

• Probability of at least one is $$P(A\cap B)+P(A\cap B^c) +P(A^c\cap B) \\= P(A)+P(B)-P(A\cap B)$$ subtracting once to avoid double counting the intersection
• Probability of exactly one is $$P(A\cap B^c) +P(A^c\cap B) \\= P(A)+P(B)-2P(A\cap B)$$ subtracting twice to avoid counting the intersection at all