Math Problem Statement

Four 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 Liked Crunchicles 14 17 22 48 101 Disliked Crunchicles 26 43 14 70 153 No Preference 7 10 8 121 146 Total 47 70 44 239 400

One consumer from the survey is selected at random. Leave all answers in a reduced fraction. 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?

Solution

Let's break down each question one by one using the data provided in the table. The total number of consumers surveyed is 400, and we have different categories based on age groups and preferences.

1. What is the probability that the consumer is 18–24 years of age, given that he/she dislikes Crunchicles?

We use conditional probability: P(18-24Dislikes Crunchicles)=P(18-24 and Dislikes Crunchicles)P(Dislikes Crunchicles)P(\text{18-24} \mid \text{Dislikes Crunchicles}) = \frac{P(\text{18-24 and Dislikes Crunchicles})}{P(\text{Dislikes Crunchicles})}

From the table, the number of consumers who are 18-24 and dislike Crunchicles is 26, and the total number of consumers who dislike Crunchicles is 153. Therefore:

P(18-24Dislikes Crunchicles)=26153P(\text{18-24} \mid \text{Dislikes Crunchicles}) = \frac{26}{153}

2. What is the probability that the selected consumer dislikes Crunchicles?

This is simply the ratio of consumers who dislike Crunchicles to the total number of consumers surveyed:

P(Dislikes Crunchicles)=153400P(\text{Dislikes Crunchicles}) = \frac{153}{400}

3. What is the probability that the selected consumer is 35–55 years old or likes Crunchicles?

This question asks for the probability that the consumer belongs to either of two mutually inclusive events (being 35-55 years old or liking Crunchicles). We use the formula for the union of two events:

P(35-55 or Likes Crunchicles)=P(35-55)+P(Likes Crunchicles)P(35-55 and Likes Crunchicles)P(\text{35-55 or Likes Crunchicles}) = P(\text{35-55}) + P(\text{Likes Crunchicles}) - P(\text{35-55 and Likes Crunchicles})

From the table:

  • The number of 35-55-year-olds is 44.
  • The number of people who like Crunchicles is 101.
  • The number of people who are 35-55 and like Crunchicles is 22.

Now calculate:

P(35-55 or Likes Crunchicles)=44400+10140022400=123400P(\text{35-55 or Likes Crunchicles}) = \frac{44}{400} + \frac{101}{400} - \frac{22}{400} = \frac{123}{400}

4. If the selected consumer is 70 years old, what is the probability that he/she likes Crunchicles?

Since age group 55 and over contains consumers who are 70 years old, we need to calculate the conditional probability:

P(Likes Crunchicles55 and over)=P(Likes Crunchicles and 55 and over)P(55 and over)P(\text{Likes Crunchicles} \mid \text{55 and over}) = \frac{P(\text{Likes Crunchicles and 55 and over})}{P(\text{55 and over})}

From the table:

  • The number of consumers 55 and over who like Crunchicles is 48.
  • The total number of consumers 55 and over is 239.

Thus:

P(Likes Crunchicles55 and over)=48239P(\text{Likes Crunchicles} \mid \text{55 and over}) = \frac{48}{239}


Recap of Answers:

  1. 26153\frac{26}{153}
  2. 153400\frac{153}{400}
  3. 123400\frac{123}{400}
  4. 48239\frac{48}{239}

Would you like more details on any of these calculations?

Related Questions:

  1. What is the probability that the consumer is 25-34, given they like Crunchicles?
  2. What is the probability that a randomly selected consumer is 18–24 years of age?
  3. What is the probability that a consumer has no preference?
  4. What is the probability that a consumer dislikes Crunchicles and is over 55 years of age?
  5. What is the probability that the consumer is 35-55 years old, given that they have no preference?

Tip:

When calculating probabilities involving "or" (union) of events, remember to subtract the overlap to avoid double-counting!

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Math Problem Analysis

Mathematical Concepts

Conditional Probability
Probability Rules
Union of Events

Formulas

P(A | B) = P(A and B) / P(B)
P(A or B) = P(A) + P(B) - P(A and B)

Theorems

Conditional Probability Theorem
Addition Rule for Probability

Suitable Grade Level

Grades 9-12