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A taste test asks people from Texas and California which pasta brand they prefer, brand [tex]$A$[/tex] or brand [tex]$B$[/tex]. This table shows the results:

\begin{tabular}{|l|c|c|c|}
\hline & Brand A & Brand B & Total \\
\hline Texas & 80 & 45 & 125 \\
\hline California & 90 & 60 & 150 \\
\hline Total & 170 & 105 & 275 \\
\hline
\end{tabular}

A person is randomly selected from those tested. What is the probability that the person is from Texas, given that the person prefers brand B? Round your answer to two decimal places.



Answer :

GinnyAnswer
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%.

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