Question

five hundred consumers are 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	58	53	47	62	220
no preference	36	19	37	40	132
total	137	88	105	170	500

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?

Explanation:

Step1: Recall conditional - probability formula

The formula for conditional probability is $P(A|B)=\frac{P(A\cap B)}{P(B)}$. For part a, let $A$ be the event that the consumer is 18 - 24 years old and $B$ be the event that the consumer dislikes Crunchicles. $P(A\cap B)=\frac{58}{500}$ and $P(B)=\frac{220}{500}$. Then $P(A|B)=\frac{\frac{58}{500}}{\frac{220}{500}}=\frac{58}{220}=\frac{29}{110}$.

Step2: Calculate simple probability for part b

The probability that a selected consumer dislikes Crunchicles is the number of consumers who dislike Crunchicles divided by the total number of consumers. So $P(\text{dislike})=\frac{220}{500}=\frac{11}{25}$.

Step3: Use the addition - rule for probability for part c

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. $P(A)=\frac{105}{500}$, $P(B)=\frac{148}{500}$, and $P(A\cap B)=\frac{21}{500}$. Then $P(A\cup B)=\frac{105 + 148-21}{500}=\frac{232}{500}=\frac{58}{125}$.

Step4: Calculate conditional probability for part d

If the consumer is 70 years old, they are 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. $P(A\cap B)=\frac{68}{500}$ and $P(B)=\frac{170}{500}$. Then $P(A|B)=\frac{\frac{68}{500}}{\frac{170}{500}}=\frac{68}{170}=\frac{2}{5}$.

Answer:

a. $\frac{29}{110}$
b. $\frac{11}{25}$
c. $\frac{58}{125}$
d. $\frac{2}{5}$