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Each participant tastes snack [tex]$A$[/tex] and snack [tex]$B$[/tex] and then chooses their favorite. Some participants have eaten snack [tex]$A$[/tex] before and some have not. The results of the test are shown in the table below. Using the data in the table, the company that makes snack [tex]$A$[/tex] calculates probabilities related to a randomly selected person.
\begin{tabular}{|c|l|l|l|} \hline & Prefers Snack A & Prefers Snack B & Total \\ \hline \begin{tabular}{c} Has Eaten Snack A \\ before \end{tabular} & 144 & 92 & 236 \\ \hline \begin{tabular}{l} Has Not Eaten \\ Snack A before \end{tabular} & 108 & 228 & 336 \\ \hline Total & 252 & 320 & 572 \\ \hline \end{tabular}
Complete the conclusions based on the data in the table.
Given a person who has eaten snack [tex]$A$[/tex] before, the customer will [tex]$\square$[/tex]
Given a person who has not eaten snack [tex]$A$[/tex] before, the customer will want to eat snack [tex]$\square$[/tex]
- The table shows the preferences of participants for snacks A and B, categorized by whether they have eaten snack A before or not.
2. Formulating Probabilities:
- The first part of the question asks about the preference given that a person has eaten snack A before. This translates to the probability of preferring snack A if the person has eaten snack A before. - The second part of the question asks about the preference given that a person has not eaten snack A before. This translates to the probability of preferring snack B if the person has not eaten snack A before.
3. Extracting Relevant Data:
- For those who have eaten snack A before: - Prefers A: 144 - Prefers B: 92 - Total: 236
- For those who have not eaten snack A before: - Prefers A: 108 - Prefers B: 228 - Total: 336
4. Calculating Probabilities:
- Probability that a person who has eaten snack A before prefers snack A: [tex]\[ \text{Probability} = \frac{\text{Number who prefer A}}{\text{Total who have eaten A before}} \][/tex] [tex]\[
\frac{144}{236} \approx 0.610
\][/tex] This means given a person who has eaten snack A before, there is approximately a 61.0% chance they will prefer snack A.
- Probability that a person who has not eaten snack A before prefers snack B: [tex]\[ \text{Probability} = \frac{\text{Number who prefer B}}{\text{Total who have not eaten A before}} \][/tex] [tex]\[
\frac{228}{336} \approx 0.679
\][/tex] This means given a person who has not eaten snack A before, there is approximately a 67.9% chance they will prefer snack B.
Based on this data, we can fill in the blanks:
Given a person who has eaten snack A before, the customer will prefer snack A.
Given a person who has not eaten snack A before, the customer will want to eat snack B.
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