Question

five 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	22	63	25	122	232
no preference	23	28	19	28	98
total	91	98	71	240	500

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: Recall conditional - probability formula

The formula for conditional probability is $P(A|B)=\frac{P(A\cap B)}{P(B)}$. For the first question, let $A$ be the event that the consumer is 18 - 24 years old and $B$ be the event that the consumer dislikes Crunchicles.
$P(A\cap B)=\frac{22}{500}$ and $P(B)=\frac{232}{500}$, so $P(A|B)=\frac{22/500}{232/500}=\frac{22}{232}=\frac{11}{116}$.

Step2: Find probability of disliking Crunchicles

The number of consumers who dislike Crunchicles is 232 out of 500. So the probability $P(\text{dislike})=\frac{232}{500}=\frac{58}{125}$.

Step3: Use the addition - rule for probability

The addition - rule for probability is $P(A\cup B)=P(A)+P(B)-P(A\cap B)$. Let $A$ be the event that the consumer is 35 - 55 years old and $B$ be the event that the consumer likes Crunchicles. $P(A)=\frac{71}{500}$, $P(B)=\frac{170}{500}$, and $P(A\cap B)=\frac{27}{500}$. Then $P(A\cup B)=\frac{71 + 170-27}{500}=\frac{214}{500}=\frac{107}{250}$.

Step4: Since 70 years old is in the 55 and over age group

Let $A$ be the event that the consumer likes Crunchicles and $B$ be the event that the consumer is 55 and over. $P(A\cap B)=\frac{90}{500}$ and $P(B)=\frac{240}{500}$, so $P(A|B)=\frac{90/500}{240/500}=\frac{90}{240}=\frac{3}{8}$.

Answer:

$\frac{11}{116}$
$\frac{58}{125}$
$\frac{107}{250}$
$\frac{3}{8}$