Question

a survey finds that 48% of people identify themselves as fans of professional football, 12% as fans of car racing, and 9% as fans of both professional football and car racing. let event f be choosing a person who is a fan of professional football and let event c be choosing a person who is a fan of car racing. which statements are true? select three options. \\(\square\\) \\(p(f|c) = 0.75\\) \\(\square\\) \\(p(c|f) = 0.25\\) \\(\square\\) \\(p(c\cap f) = 0.09\\) \\(\square\\) \\(p(c\cap f) = p(f\cap c)\\) \\(\square\\) \\(p(c|f) = p(f|c)\\)

Explanation:

Step1: Recall conditional probability formula

The conditional probability formula is $P(A|B)=\frac{P(A\cap B)}{P(B)}$

Step2: Calculate $P(F|C)$

Substitute $P(F\cap C)=0.09$, $P(C)=0.12$
$P(F|C)=\frac{0.09}{0.12}=0.75$

Step3: Calculate $P(C|F)$

Substitute $P(C\cap F)=0.09$, $P(F)=0.48$
$P(C|F)=\frac{0.09}{0.48}=0.1875$

Step4: Verify intersection properties

By definition, $P(C\cap F)=P(F\cap C)=0.09$

Step5: Compare conditional probabilities

$P(C|F)=0.1875
eq0.75=P(F|C)$

Answer:

• $P(F|C) = 0.75$
• $P(C\cap F) = 0.09$
• $P(C\cap F) = P(F\cap C)$