Question

four hundred consumers were surveyed about a new brand of snack food, crunchicles. their age groups and preferences are given in the table.

	18 - 24	25 - 34	35 - 55	55 and over	total
disliked crunchicles	57	25	54	43	179
no preference	66	31	3	111	211
total	126	61	59	154	400

one consumer from the survey is selected at random. use reduced fractions for your responses to each of the following questions.
what is the probability that the consumer is 18 - 24 years of age, given that he/she dislikes crunchicles?
what is the probability that the selected consumer dislikes crunchicles?
what is the probability that the selected consumer is 35 - 55 years old or likes crunchicles?
if the selected consumer is 70 years old, what is the probability that he/she likes crunchicles?

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:

$\frac{57}{179}$
$\frac{179}{400}$
$\frac{67}{400}$
$0$