To solve for the probability that a person is from Texas given that they prefer brand B, we use the concept of conditional probability. The conditional probability \( P(\text{Texas} \mid \text{Brand B}) \) is given by:
[tex]\[ P(\text{Texas} \mid \text{Brand B}) = \frac{P(\text{Texas and Brand B})}{P(\text{Brand B})} \][/tex]
From the table provided:
- The number of people from Texas who prefer brand B, \( P(\text{Texas and Brand B}) \), is 45.
- The total number of people who prefer brand B, \( P(\text{Brand B}) \), is 105.
Substituting these values into the formula provides:
[tex]\[ P(\text{Texas} \mid \text{Brand B}) = \frac{45}{105} \][/tex]
Next, we perform the division:
[tex]\[ P(\text{Texas} \mid \text{Brand B}) = 0.42857142857142855 \][/tex]
Finally, we round this result to two decimal places:
[tex]\[ P(\text{Texas} \mid \text{Brand B}) \approx 0.43 \][/tex]
So, the probability that a randomly selected person from those tested is from Texas, given that they prefer brand B, is approximately [tex]\( 0.43 \)[/tex] or 43%.
