Conditional probability based on tabulated data

Question

I have a table of containing data that shows the "likes" and "dislikes of football and basketball for a population where:

                              Doesn't like basketball     Likes basketball
Doesn't like football                  0.7                   0.1
Likes football                         0.05                  0.15

I'm trying to find the probability of a person liking football given that the person does not like basketball.

Here's where i get confused. Conditional probability is given by:

P(X = x | Y = y) = P(x  ∩  y) / P(y)

So I'm torn between two choices, my first attempt was to obtain the probability straight from the table above where

P(likes football | does not like basketball) = 0.05 #since they intersect

and when i follow the equation above,

P(likes football | does not like basketball) = 0.05/(0.7 + 0.05)
                                             = 0.0666 or 1/15

I was wondering which of the attempts are the correct solution to the problem. Would appreciate some help on this.

Answer 1

It's often helpful to look at simpler examples:

                              Doesn't like basketball     Likes basketball
Doesn't like football                  0                     0.8
Likes football                         0.05                  0.15

In this example, what's the probability of the person liking football, given that the person doesn't like basketball? The first approach says $0.05$, the second approach says $1$. Which one is right?

Answer 2

Some people find thinking about integers easier

Suppose there were $100$ people in your population. Then the numbers involved would be

                              Doesn't like basketball     Likes basketball
Doesn't like football                  70                   10
Likes football                          5                   15

• How many do not like basketball?
• Of those who do not like basketball, how many do like football?
• What proportion of those who do not like basketball, do like football?