Let's break down the problem step-by-step to determine which statements are true.
Given:
1. [tex]\(P(F) = 0.48\)[/tex] (probability that a person is a fan of professional football)
2. [tex]\(P(C) = 0.12\)[/tex] (probability that a person is a fan of car racing)
3. [tex]\(P(F \cap C) = 0.09\)[/tex] (probability that a person is a fan of both professional football and car racing)
### Calculate the Conditional Probabilities
First, let's determine the conditional probability [tex]\(P(F \mid C)\)[/tex]:
[tex]\[ P(F \mid C) = \frac{P(F \cap C)}{P(C)} \][/tex]
Substituting the given values:
[tex]\[ P(F \mid C) = \frac{0.09}{0.12} = 0.75 \][/tex]
So, [tex]\(P(F \mid C )=0.75\)[/tex] is true.
Next, let's calculate [tex]\(P(C \mid F)\)[/tex]:
[tex]\[ P(C \mid F) = \frac{P(F \cap C)}{P(F)} \][/tex]
Substituting the given values:
[tex]\[ P(C \mid F) = \frac{0.09}{0.48} = 0.1875 \][/tex]
So, [tex]\(P(C \mid F)=0.25\)[/tex] is false.
### Verification of Probabilities and Equalities
- We are given [tex]\(P(F \cap C)=0.09\)[/tex], so this statement [tex]\(P(C \cap F)=0.09\)[/tex] is true.
- From the definition of intersection, we know [tex]\(P(C \cap F) = P(F \cap C)\)[/tex]. This is always true for any events [tex]\(F\)[/tex] and [tex]\(C\)[/tex], thus the statement [tex]\(P(C \cap F) = P(F \cap C)\)[/tex] is true.
- Finally, considering the symmetry of conditional probabilities, [tex]\(P(C \mid F) = P(F \mid C)\)[/tex] is generally false unless [tex]\(P(F) = P(C)\)[/tex], which is not given here.
### Summary of True Statements
The true statements, therefore, are:
1. [tex]\(P(F \mid C )=0.75\)[/tex]
2. [tex]\(P(C \cap F)=0.09\)[/tex]
3. [tex]\(P( C \cap F )=P(F \cap C )\)[/tex]
Thus, the selected three options should be:
1. [tex]\(P(F \mid C)=0.75\)[/tex]
2. [tex]\(P(C \cap F)=0.09\)[/tex]
3. [tex]\(P(C \cap F) = P(F \cap C)\)[/tex]
These three options are correct based on the given information and calculations.
