To determine the probability that a person is from Texas given that they prefer brand A, we need to use the concept of conditional probability. Specifically, we are looking for [tex]\( P(\text{Texas} \mid \text{Brand A}) \)[/tex].
The formula for conditional probability [tex]\( P(A \mid B) \)[/tex] is given by:
[tex]\[ P(A \mid B) = \frac{P(A \cap B)}{P(B)} \][/tex]
Where:
- [tex]\( P(A \cap B) \)[/tex] is the probability of both events A and B occurring.
- [tex]\( P(B) \)[/tex] is the probability of event B occurring.
In this context:
- Event A is "the person is from Texas."
- Event B is "the person prefers brand A."
First, find [tex]\( P(A \cap B) \)[/tex], which is the probability that a person is from Texas and prefers brand A. From the table, the number of people from Texas who prefer brand A is [tex]\( 80 \)[/tex].
Next, find [tex]\( P(B) \)[/tex], which is the probability that a person prefers brand A regardless of their origin. The total number of people who prefer brand A is [tex]\( 170 \)[/tex].
Now, calculate the conditional probability:
[tex]\[ P(\text{Texas} \mid \text{Brand A}) = \frac{\text{number of people from Texas who prefer Brand A}}{\text{total number of people who prefer Brand A}} = \frac{80}{170} \][/tex]
Next, perform the division:
[tex]\[ \frac{80}{170} \approx 0.4706 \][/tex]
When rounded to two decimal places, this value is:
[tex]\[ 0.47 \][/tex]
Therefore, the probability that a randomly selected person is from Texas given that they prefer brand A is [tex]\( 0.47 \)[/tex].
So, the correct answer is:
A. 0.47
