Let's carefully analyze each of the given probabilities and relationships using the provided survey data:
1. Given data:
- [tex]\( P(F) = 0.48 \)[/tex]: Probability that a person is a fan of professional football.
- [tex]\( P(C) = 0.12 \)[/tex]: Probability that a person is a fan of car racing.
- [tex]\( P(F \cap C) = 0.09 \)[/tex]: Probability that a person is a fan of both professional football and car racing.
2. Calculate [tex]\( P(F \mid C) \)[/tex]:
- This represents the conditional probability that a person is a football fan given that they are a car racing fan.
- [tex]\( P(F \mid C) = \frac{P(F \cap C)}{P(C)} = \frac{0.09}{0.12} = 0.75 \)[/tex].
3. Calculate [tex]\( P(C \mid F) \)[/tex]:
- This represents the conditional probability that a person is a car racing fan given that they are a football fan.
- [tex]\( P(C \mid F) = \frac{P(F \cap C)}{P(F)} = \frac{0.09}{0.48} = 0.1875 \)[/tex].
4. Check if [tex]\( P(C \cap F) = P(F \cap C) \)[/tex]:
- By the definition of intersection in probability, [tex]\( P(C \cap F) \)[/tex] should always be equal to [tex]\( P(F \cap C) \)[/tex].
- [tex]\( P(C \cap F) = 0.09 \)[/tex] and [tex]\( P(F \cap C) = 0.09 \)[/tex].
- Therefore, [tex]\( P(C \cap F) = P(F \cap C) \)[/tex].
5. Check if [tex]\( P(C \mid F) = P(F \mid C) \)[/tex]:
- [tex]\( P(C \mid F) = 0.1875 \)[/tex] and [tex]\( P(F \mid C) = 0.75 \)[/tex].
- These two conditional probabilities are not equal.
Given the calculated values and the analysis, the true statements from the options 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 true options are:
- [tex]\( P(F \mid C) = 0.75 \)[/tex]
- [tex]\( P(C \cap F) = 0.09 \)[/tex]
- [tex]\( P(C \cap F) = P(F \cap C) \)[/tex]
