70 likes both soccer and basketball so 70/200 or 0.35

What is the probability of a randomly selected person likes basketball given that they like soccer?

The answer choices are

7/20

3/7

7/10

1/2

When I work out this problem, the answer is not one of the choices. I think I have correctly solved the problem using conditional probability. Please advise. Thank you.

The only people who can possibly like basketball and soccer is the 3rd bullet (70 out of 200).
This is known because it is given that the other two populations only like one sport.
Reduce the fraction and that’s 7/20.

Very much appreciate your response TMosh. I am still a tad confused.

The question asks, “What is the probability of a randomly selected person likes basketball given that they like soccer?” From your response, if the answer is 7/20 or 70 out of 200 people which represent people who likes both soccer and basketball, wouldn’t that me P(A intersection B) and not P(A | B) where A is people who likes baseketball and B is people who likes soccer ? The question asks for P(A | B) and not P(A intersection B) if I am not mistaken?

From the Product Rule we have:

(1) P(A intersection B) = P(A) * P(B | A)

and let’s substitute

P(A intersection B) = 70/200 from the problem statement for
people who likes both basketball and soccer.
P(A) = people who likes soccer or 30/200
P(B | A) = people who likes basketball given that they like soccer since A represents people who likes soccer and B represents people who likes basketball.

Therefore, we have, using (1) from above w/substitution:

70/200 = 30/200 * P(B | A) or

70/200 * 200/30 = P(B | A)

70/30 = P(B | A) but 70/30 is not any of the answer choices?

Thank you for your response. If the question is asking for P(A intersection B) then I am really confused why the question is asking " a randomly selected person likes basketball given that they like soccer?" The “given” is specifically stated in the question. Wouldn’t the question be asking what’s the probability of the people liking BOTH soccer and basketball as oppose to liking basketball GIVEN that they like soccer?

I am still very confused unfortunately. If the correct answer is 7/20 then may I suggest to change the wording on the question please.

Note that from the facts given about the statistics, you can say there is no intersection between the two groups that only like one sport. Because, it says they like only one sport. So no intersection is possible.

Thank you for the prompt response. I am sorry I am not following.

It doesn’t intersect with the other two one-sport-only populations.

But how do we extract that information from the problem statement? The problem statement simply states X # of people likes soccer and Y # of people like basketball and Z # of people likes both soccer and basketball. From that 3 facts alone, how do we extract the information that Z is not an intersectoin of X and Y ? I guess this is where I am still confused that there is no intersection. Thank you for taking the time to respond to my questions. Very much appreciate it.

I was traveling and didn’t get a chance to continue our discussion. I tried the solution you suggested and this is the result I got. Please advise. Thank you.

The population being considered in the question isn’t the whole 200 people. The “given that they like soccer” means we’re only considering those who only (30) like soccer, and those who like soccer and basketball (70). So the denominator is 30 + 70 = 100.

The numerator is then 70, because we know the other 30 only like soccer.

So I’m going to go out on a limb and say the correct answer is 7/10.