Problem
Five hundred consumers are surveyed about a new brand of snack food, Crunchicles. Their age groups and preferences are given in the table. \begin{tabular}{|c|r|r|r|r|r|r|} \hline & $18-24$ & $25-34$ & $35-55$ & 55 and over & Total \\ \hline \hline Liked Crunchicles & 29 & 14 & 24 & 44 & 111 \\ \hline \hline Disliked Crunchicles & 40 & 6 & 11 & 64 & 121 \\ \hline No Preference & 40 & 50 & 72 & 106 & 268 \\ \hline \hline Total & 109 & 70 & 107 & 214 & 500 \\ \hline \end{tabular} One consumer from the survey is selected at random. Leave all answers in a reduced fraction. a. What is the probability that the consumer is $18-24$ years of age, given that he/she dislikes Crunchicles? b. What is the probability that the selected consumer dislikes Crunchicles? c. What is the probability that the selected consumer is $35-55$ years old or likes Crunchicles? d. If the selected consumer is 70 years old, what is the probability that he/she likes Crunchicles?
Solution
Step 1 ：Find the probabilities P(A and B) and P(B) from the given table:
Step 2 ：\(P(A \text{ and } B) = \frac{40}{500} = 0.08\)
Step 3 ：\(P(B) = \frac{121}{500} = 0.242\)
Step 4 ：Calculate the conditional probability P(A|B) using the formula P(A|B) = P(A and B) / P(B):
Step 5 ：\(P(A|B) = \frac{0.08}{0.242} = \frac{40}{121}\)
Step 6 ：\(\boxed{\frac{40}{121}}\)
