What is the probability that the selected consumer is $$ 35-55 $$ years old or likes Crunchicles? 0258 If the selected consumer is 70 years old, what is the probability that he/she likes Crunchicles? 0.267

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To determine the probabilities you've mentioned, we need specific data about the consumer demographics and their preferences regarding "Crunchicles." Since you provided the answers as **0.258** and **0.267**, I'll outline the general approach to solving such problems. If you have additional data (like a contingency table or specific probabilities), please provide them for a more precise calculation.

### 1. Probability that the Selected Consumer is $$ 35-55 $$ Years Old **OR** Likes Crunchicles

**Notation:**
- Let $$ A $$ be the event that a consumer is aged between 35 and 55.
- Let $$ B $$ be the event that a consumer likes Crunchicles.

**Formula:**
$$
P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)
$$

**Explanation:**
- **$$ P(A) $$**: Probability that a consumer is between 35-55 years old.
- **$$ P(B) $$**: Probability that a consumer likes Crunchicles.
- **$$ P(A \text{ and } B) $$**: Probability that a consumer is both between 35-55 years old and likes Crunchicles.

**Given Answer:**
- You've mentioned **0.258**. This implies that:
  $$
  P(A \text{ or } B) = 0.258
  $$
  Without the individual probabilities or intersection, we can't verify this number. However, using the formula above with the correct values will give you the desired probability.

### 2. Probability that a 70-Year-Old Consumer Likes Crunchicles

**Notation:**
- Let $$ C $$ be the event that a consumer is 70 years old.
- We need to find $$ P(B | C) $$, the probability that a consumer likes Crunchicles given they are 70 years old.

**Formula (Conditional Probability):**
$$
P(B | C) = \frac{P(C \text{ and } B)}{P(C)}
$$

**Explanation:**
- **$$ P(C \text{ and } B) $$**: Probability that a consumer is both 70 years old and likes Crunchicles.
- **$$ P(C) $$**: Probability that a consumer is 70 years old.

**Given Answer:**
- You've mentioned **0.267**. This implies that:
  $$
  P(B | C) = 0.267
  $$
  This means there's a 26.7% chance that a 70-year-old consumer likes Crunchicles. To verify this, you'd use the above formula with the appropriate probabilities.

### **Key Points to Consider:**
1. **Data Availability:** Ensure you have all necessary probabilities:
   - Marginal probabilities like $$ P(A) $$, $$ P(B) $$, and $$ P(C) $$.
   - Joint probabilities like $$ P(A \text{ and } B) $$ and $$ P(C \text{ and } B) $$.

2. **Independence:** Determine if the events are independent. If they are, $$ P(A \text{ and } B) = P(A) \times P(B) $$. However, preferences might not always be independent of age groups.

3. **Consistent Data:** Make sure that all probabilities are based on the same population sample to maintain consistency.

### **Example:**

Suppose you have the following data:
- Total consumers: 1000
- Consumers aged 35-55: 300
- Consumers who like Crunchicles: 200
- Consumers who are both aged 35-55 and like Crunchicles: 50

**Calculations:**

1. **$$ P(A \text{ or } B) $$:**
$$
P(A \text{ or } B) = \frac{300}{1000} + \frac{200}{1000} - \frac{50}{1000} = 0.3 + 0.2 - 0.05 = 0.45
$$

2. **$$ P(B | C) $$:**
   - Suppose 40 consumers are 70 years old.
   - Among them, 10 like Crunchicles.
$$
P(B | C) = \frac{10}{40} = 0.25
$$

These example calculations illustrate the process. If you provide specific data, I can offer more accurate computations.
To find the probability that a selected consumer is between 35-55 years old or likes Crunchicles, use the formula: $$ P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B) $$ Given the answer is **0.258**, ensure you have the correct individual probabilities to verify. For a 70-year-old consumer, the probability they like Crunchicles is **0.267**.

What is the probability that the selected consumer is \( 35-55 \) years old or likes Crunchicles? 0258 If the selected consumer is 70 years old, what is the probability that he/she likes Crunchicles? 0.267

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