Lesson: Probability of Event A OR Event B

Comment on Probability of Event A OR Event B

total numbers=7
numbers which are odd=4 (3,7,9,15)
numbers which are prime=3 (2,3,7)
numbers which are both odd and prime =2 (3,7)
so no. which are either odd or prime=3 (9,15,2)
so,shouldn't the probability be 3/7 instead of 5/7?
as we need to exclude numbers which satisfy both the events.
greenlight-admin's picture

"so no. which are either odd or prime=3 (9,15,2)"
The part above is not correct. You're missing 3 and 7.
The complete list of numbers that are either odd or prime = {2, 3, 7, 9, 15}
So, P(odd or prime) = 5/7

Sir, I didn't include 3 and 7 as Question is asking about either odd or prime and 3 and 7 are both odd and prime.
greenlight-admin's picture

Ah, I see the issue now. You are using "or" in the exclusive sense, but on the GMAT, we need to use the inclusive definition of "or" 3 and 7 still qualify as being either odd or prime.

We have the numbers {2, 3, 6, 7, 8, 9, 15}. The opposite of a number being either odd or prime is a number that is neither odd nor prime. Those are the only two possibilities. The numbers that are neither odd nor prime are: 6 and 8. So, the remaining numbers (2, 3, 7, 9, and 15) are either odd or prime.

Thank you sir. It is clear to me now. :)

If "or" is considered to be inclusive, why are we subtracting P(A and B)...? Having trouble wrapping my head around this concept.

Is it related to the fact that since the two events are not mutually exclusive we must remove the overlap... but doing this still somehow keeps "or" inclusive (in the sense that A AND B is still included)?
greenlight-admin's picture

On the GRE, OR is considered inclusive (unless stated otherwise).

So, for example, if we're told that "Joe can have pie or cake for dessert," it's possible that Joe will have both pie and cake.

Subtracting P(A and B) reduces duplication. Consider this question.

Set Q: {2, 3, 4, 5, 6, 7}

If one number is randomly selected from Set Q, what is the probability that the selected number is PRIME or ODD

Let event A = selecting a PRIME number
Let event B = selecting an ODD number

There are 6 numbers and 4 are PRIME. So, P(A) = 4/6
There are 6 numbers and 3 are ODD. So, P(B) = 3/6

At this point, if we add the two probabilities, we get:
P(A or B) = 4/6 + 3/6 = 7/6
This should make no sense, since we can't have a probability greater than 1.

The problem here is that there's a lot of overlap between ODD numbers and PRIME numbers.
In fact, we've included 3, 5 and 7 in BOTH calculations.
So, we can account for this DUPLICATION by subtracting P(A and B)
P(A and B) = 3/6 (since 3, 5 and 7 are both prime and odd)

Now apply the complete OR probability:
P(prime OR odd) = P(A or B)
= P(A) + P(B) - P(A and B)
= 4/6 + 3/6 - 3/6
= 4/6

Does that help?


Yes! Thank you so much.

Hey Brent,

When determining P(A and B) why do we not multiply (4/7)*(3/7) as (4/7)= P(A) an (3/7)= P(B). While I intuitively understand that there are two numbers in that set that share both characteristics, I wanted to know what was wrong with my math.
greenlight-admin's picture

Good question.
Since A and B are not independent, we can't say that P(A and B) = P(A) x P(B)

This is covered in the following video: https://www.greenlighttestprep.com/module/gre-probability/video/755

I hope that helps.

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