Lesson: Probability of Event A OR Event B

Comment on Probability of Event A OR Event B

If the GMAT assumes inclusivity in OR probability questions, then why do we subject P(A & B)'s probability? Shouldn't P(A&B) increase the overall P(A OR B) if inclusivity is assumed?
gmat-admin's picture

We need to subtract P(A and B) in order to remove the possibility that we are considering some events twice.

Here's an example that shows what I mean.

Let set S = {3, 5, 7, 11, 15, 16}

Let event A = selecting an odd number

Let event B = selecting a prime number

If we select one value from set S, what is the probability of selecting an odd number OR a prime number?

In other words, we want P(A or B)

We know that P(A) = 5/6

And P(B) = 4/6

So, P(A) + P(B) = 5/6 + 4/6 = 10/6 (hmmmmmmmm, the probability is greater than 1!!)
We get 10/6 because considering some events twice. For example, if we select the 3 from the set, we have selected an odd number AND a prime number.

In order to account for this duplication, we must consider outcomes that meet BOTH conditions. So, in this example, we must consider outcomes that are odd AND prime.

So, P(A or B) = P(A) + P(B) - P(A and B)
= 5/6 + 4/6 - 4/6
= 5/6

ASIDE: P(A and B) = 4/6 because, out of the 6 numbers in set S, 4 of them are BOTH odd AND prime.

Does that help?

Cheers,
Brent

gmat-admin's picture

We need to subtract P(A and B) in order to remove the possibility that we are considering some events twice.

Here's an example that shows what I mean.

Let set S = {3, 5, 7, 11, 15, 16}

Let event A = selecting an odd number

Let event B = selecting a prime number

If we select one value from set S, what is the probability of selecting an odd number OR a prime number?

In other words, we want P(A or B)

We know that P(A) = 5/6

And P(B) = 4/6

So, P(A) + P(B) = 5/6 + 4/6 = 10/6 (hmmmmmmmm, the probability is greater than 1!!)
We get 10/6 because considering some events twice. For example, if we select the 3 from the set, we have selected an odd number AND a prime number.

In order to account for this duplication, we must consider outcomes that meet BOTH conditions. So, in this example, we must consider outcomes that are odd AND prime.

So, P(A or B) = P(A) + P(B) - P(A and B)
= 5/6 + 4/6 - 4/6
= 5/6

ASIDE: P(A and B) = 4/6 because, out of the 6 numbers in set S, 4 of them are BOTH odd AND prime.

Does that help?

Cheers,
Brent

Hi Brent,

In a later video in this module titled "Probability of Event A AND Event B" you showcase a new formula such that:

P(A and B)= P(A) x P(B|A)
**P(B|A) represents the probability of event B given that event A has occurred

As a result, why does this formula not work for the above example in this video? For instance, in this example you state that P(A and B)=2/7.

However, given the formula of P(A and B) would this not become:

P(A and B)= P(A) x P(B|A)= (4/7) x (3/7) =12/7

Could you kindly explain why we do not use this "AND formula" in this scenario? I'm not making much sense of it given that "P(A and B)" appears in the OR formula.

Such that P(A OR B)= P(A) + P(B) - P(A AND B).

Thank you!

gmat-admin's picture

The formula, P(A and B) = P(A) x P(B|A), works here.
However, you must remember what P(B|A) means.

Event A = selected number is odd
Event B = selected number is prime

So, P(B|A) = the probability that the selected number is prime GIVEN THAT the selected number is odd.

Let's say I select a number from {2, 3, 6, 7, 8, 9, 15}, and I tell you that I just selected an odd number.
What is the probability that the selected number is prime?

Well, GIVEN THAT I selected an odd number, we know that I must have selected the 3, 7, 9 or 15 (4 possibilities).
Among those 4 possibilities, there are 2 prime numbers.
So, given that the selected number is odd, the probability is 2/4 that the number is also prime.

In other words, P(B|A) = 2/4 = 1/2

So, P(A and B) = P(A) x P(B|A)
= 4/7 x 1/2
= 2/7
Voila!!

Of course, this just proves that the AND probability formula works. However, it doesn't help us answer the question at hand.
This brings us to...

YOUR QUESTION: Could you kindly explain why we do not use this "AND formula" in this scenario?
The question asks us to find the probability that the selected number is either odd OR prime.
So, it's fitting to use the OR formula.

Does that help?

Cheers,
Brent

Wow, that was a perfect explanation. Thanks a million!

How come you don't have any videos on set? in the OG book they talk about Unions and other topics related to sets.
gmat-admin's picture

Good question!
The Official Guide uses set notation to teach concepts of AND and OR when it comes to probability and counting. However, these concepts can be taught without using that notation.

Cheers,
Brent

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