Four hundred consumers are surveyed about a new brand of snack food, *Crunchicles.* Their age groups and preferences are given in the table.

 18–2425–3435–5555 and overTotalLiked *Crunchicles*

19

17

7

33

76

Disliked *Crunchicles*

36

27

7

99

169

No Preference

34

8

5

108

155

Total

89

52

19

240

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*?

  


Let's solve each part step by step:

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

This is a conditional probability problem. The formula for conditional probability is:

\[
P(A \mid B) = \frac{P(A \cap B)}{P(B)}
\]

Where:
- \( A \) = the consumer is 18–24 years of age.
- \( B \) = the consumer dislikes *Crunchicles*.

From the table:
- The number of consumers who are 18–24 years old and dislike *Crunchicles* is 36.
- The total number of consumers who dislike *Crunchicles* is 169.

Thus, the probability is:

\[
P(\text{18–24} \mid \text{Dislikes}) = \frac{36}{169}
\]

---

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

This is the simple probability that a consumer dislikes *Crunchicles*. The formula is:

\[
P(\text{Dislikes}) = \frac{\text{Number of people who dislike *Crunchicles*}}{\text{Total number of people surveyed}}
\]

From the table:
- The number of people who dislike *Crunchicles* is 169.
- The total number of people surveyed is 400.

Thus, the probability is:

\[
P(\text{Dislikes}) = \frac{169}{400}
\]

---

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

This is the probability of the union of two events:
- The consumer is 35–55 years old (\(A\)).
- The consumer likes *Crunchicles* (\(B\)).

The formula for the union of two events is:

\[
P(A \cup B) = P(A) + P(B) - P(A \cap B)
\]

Where:
- \( P(A) \) is the probability the consumer is 35–55 years old.
- \( P(B) \) is the probability the consumer likes *Crunchicles*.
- \( P(A \cap B) \) is the probability the consumer is 35–55 years old and likes *Crunchicles*.

From the table:
- The number of people aged 35–55 is 19.
- The number of people who like *Crunchicles* is 76.
- The number of people aged 35–55 who like *Crunchicles* is 7.

Thus:

\[
P(\text{35–55 or Likes}) = \frac{19}{400} + \frac{76}{400} - \frac{7}{400} = \frac{19 + 76 - 7}{400} = \frac{88}{400} = \frac{22}{100}
\]

---

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

Here we assume the consumer is part of the "55 and over" age group, which includes people aged 70.

This is another conditional probability:

\[
P(\text{Likes} \mid \text{55 and over}) = \frac{P(\text{Likes and 55 and over})}{P(\text{55 and over})}
\]

From the table:
- The number of people in the 55 and over group who like *Crunchicles* is 33.
- The total number of people in the 55 and over group is 240.

Thus, the probability is:

\[
P(\text{Likes} \mid \text{55 and over}) = \frac{33}{240} = \frac{11}{80}
\]

---

### Summary of Answers:
1. \( P(\text{18–24} \mid \text{Dislikes}) = \frac{36}{169} \)
2. \( P(\text{Dislikes}) = \frac{169}{400} \)
3. \( P(\text{35–55 or Likes}) = \frac{22}{100} \)
4. \( P(\text{Likes} \mid \text{55 and over}) = \frac{11}{80} \)

Would you like to go over any details or have any further questions?

---

Here are some related questions for further practice:
1. What is the probability that a randomly selected consumer is aged 25–34 and dislikes *Crunchicles*?
2. What is the probability that the consumer is aged 18–24 and has no preference?
3. What is the probability that the selected consumer likes *Crunchicles* or has no preference?
4. What is the probability that a consumer aged 35–55 dislikes *Crunchicles*?
5. If a consumer has no preference, what is the probability that they are 25–34 years old?

**Tip:** Conditional probabilities focus on a subset of the total population, so always make sure to adjust the total when calculating them.

Four hundred consumers are surveyed about a new brand of snack food, *Crunchicles.* Their age groups and preferences are given in the table. One consumer from the survey is selected at random. 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*?

Explore the probability questions from a consumer survey on *Crunchicles*, a new snack food. This problem covers conditional probability, event unions, and uses key formulas such as P(A | B) and P(A ∪ B). Detailed solutions and step-by-step calculations are provided. Suitable for students in Grades 9-12.

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