four hundred consumers were surveyed about a new brand of snack food, crunchicles. their age groups and…
four hundred consumers were surveyed about a new brand of snack food, crunchicles. their age groups and preferences are given in the table.\n| |18 - 24|25 - 34|35 - 55|55 and over|total|\n|--|--|--|--|--|--|\n|liked crunchicles|3|5|2|0|10|\n|disliked crunchicles|57|25|54|43|179|\n|no preference|66|31|3|111|211|\n|total|126|61|59|154|400|\none consumer from the survey is selected at random. use reduced fractions for your responses to each of the following questions.\nwhat is the probability that the consumer is 18 - 24 years of age, given that he/she dislikes crunchicles?\nwhat is the probability that the selected consumer dislikes crunchicles?\nwhat is the probability that the selected consumer is 35 - 55 years old or likes crunchicles?\nif the selected consumer is 70 years old, what is the probability that he/she likes crunchicles?
Answer
Explanation:
Step1: Identify conditional probability components
A: 18-24, B: Disliked. n(A∩B)=57, n(B)=179
Step2: Calculate P(A|B)
$P(A|B)=\frac{57}{179}$
Step1: Total disliked over total surveyed
n(disliked)=179, total=400
Step2: Simplify fraction
$\frac{179}{400}$ (179 prime, no common factors)
Step1: Apply addition rule P(A or B)=P(A)+P(B)-P(A∩B)
A:35-55 (n=59), B:Liked (n=10), A∩B=2 (35-55 and liked)
Step2: Compute and simplify
$\frac{59+10-2}{400}=\frac{67}{400}$
Step1: 70 is 55+, check liked in 55+
Liked in 55+: 0, total 55+:154
Step2: Probability is 0
$\frac{0}{154}=0$
Answer:
$\frac{57}{179}$ $\frac{179}{400}$ $\frac{67}{400}$ $0$