To find the probability that a fan prefers pizza given that the fan prefers football, we need to use the concept of conditional probability. The conditional probability of an event [tex]\(A\)[/tex] given an event [tex]\(B\)[/tex] is calculated using the formula:
[tex]\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \][/tex]
Here, event [tex]\(A\)[/tex] is "the fan prefers pizza" and event [tex]\(B\)[/tex] is "the fan prefers football."
From the given table, we have the following data regarding fans who prefer football:
- Fans who prefer wings: 14
- Fans who prefer pizza: 20
- Fans who prefer hot dogs: 6
First, let's find the total number of football fans. Summing up the numbers for all food preferences for football fans, we get:
[tex]\[ \text{Total football fans} = 14 + 20 + 6 = 40 \][/tex]
Next, we know that the number of football fans who prefer pizza is 20.
To find the conditional probability that a football fan prefers pizza, we divide the number of football fans who prefer pizza by the total number of football fans:
[tex]\[ P(\text{pizza}|\text{football}) = \frac{\text{Number of football fans who prefer pizza}}{\text{Total number of football fans}} = \frac{20}{40} = \frac{1}{2} \][/tex]
Thus, the probability that a fan prefers pizza given that the fan prefers football is:
[tex]\[ \boxed{\frac{1}{2}} \][/tex]
