Addition Rule of Probability

CFA Level I Quant
#1

I have two question concerning the addition rule of probability:

1.) What is the probability that event A or B occurs?

2.) What is the probability that event A or B or both of them occur?

Here is what’s given:

P(A)=40%

P(B)=60%

P(AB)=40%

#2

P(A) + P(B) - P(AB) = (0.4 + 0.6) - 0.4 = 0.6

I think the answer to number 2 is the same as 1, I could be wrong though.

#3

That’s exactly what I am confused about. But how can the answer be the same to these two questions? Questions 1 has less possibilities of success. The answer to A should be lower than the answer to B in my opinion.

#4

That’s a weird question. Where did you get that? I think the answer to #2 is 1.

Why? P(A or B) can be shown as a venn diagram. That is think of A as one circle, b as another, and AB as the intersection.

P(A or B) = P(A) + P(B) - P(AB).

So if we say what is the probabillity that all can occur, that is 1. Just my thoughts.

#5

My reasoning for my answer is as follows (and to preface, I’m not 100% sure, so it’d be nice to hear what the actual answer is from OP).

Probability of A or B or AB, should be alike to the P(A or B) OR P(AB)…

P(A or B) + P(AB) - [Joint P(A or B*AB)]

=0.6 + 0.4 - 0.4

=0.6

-logically, that is the answer that makes sense to me, but my head hurts now lol. Hopefully someone else can come in and clarify.

ps. another reason I think it cannot be 1, is these events are not independent. To add these probabilities, as we saw, in #1, you need to subtract the joint probability…so you cannot have 100% probability in any combination here IMO.

#6

Yeah…it’s a weird question. I would argue that P(A), P(B), and P(AB) make up the entire possible probabllity range. But yeah, anyone else wants to chime in that would be welcome.

#7

I’m not going to lie, I’m about 97% sure of my answer lol.

Here’s something to consider as well…for a more visual way of looking at it (w/ your diagram).

My question to you would be what is the Probability of A or B?? (that is, the question #1). If you look at that diagram, the probability of A or B is the entire thing (including AB in the middle). That’s why for non-independent probabilities, you need to SUBTRACT (AB) in the middle. You would agree that the probability of A or B is equal to the P(A) + (B) subtracting that portion in the middle (AB). When we subtract that portion we are not DISCOUNTING it in our probability, we are simply not counting it TWICE. When you add that entire pie, if you added just P(A) + P(B) you are in essence, counting that intersection TWICE. So you subtract it, so it’s only counted ONCE in the A or B calculation. That entire probability is = 0.6

Now when you ask OR probability of AB in 2nd question, you’ve already accounted for that space in the 1st question, like I said above. You counted it, but once (as you should…).

I don’t know if that makes anymore sense lol, but yeah, if OP could share answer, or someone else who either is better at explaining, or has a different answer, go for it (ie. "magic"man2000… lol).

#8

Conventionally, “A or B” means “A or B or both”. If they meant A or B but not both , they’d have to specify “but not both”.

By the way, the probability of A or B but not both is:

P(A) + P(B) – 2P(AB)

#9

I saw this question on a mock. Made no sense. I doubt questions will be this ambiguous on the test.