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The image you uploaded contains two probability-based questions related to customer surveys and service calls. Let's break down the tasks for each problem:

### Question 1:
A company is launching a new product, and a survey indicates that 70% of the respondents are likely to purchase it. The company makes an initial batch of 20 products and wants to find the probabilities for the following scenarios:
1. Exactly 10 customers will purchase the new product.
2. Fewer than 5 customers will purchase the new product.
3. More than 12 customers will purchase the new product.

This question involves **binomial probability** because each customer either buys or doesn't buy the product (a binomial outcome). The probability of success (a customer purchasing the product) is 70% or 0.7.

#### Approach:
- Use the **binomial probability formula**:  
  \[
  P(X = k) = \binom{n}{k} \cdot p^k \cdot (1-p)^{n-k}
  \]
  Where:
  - \(n = 20\) (number of trials),
  - \(k\) is the number of successes (purchases),
  - \(p = 0.7\) (probability of success).

### Question 2:
A call center receives an average of 10 calls per hour. The company wants to find the probabilities for the following:
1. Exactly 4 calls will be received in a given hour.
2. Fewer than 8 calls will be received in a given hour.
3. More than 12 calls will be received in a given hour.

This question involves the **Poisson distribution** because it deals with the number of events (calls) happening in a fixed time period, with an average rate (10 calls per hour).

#### Approach:
- Use the **Poisson probability formula**:  
  \[
  P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}
  \]
  Where:
  - \(\lambda = 10\) (average rate of calls),
  - \(k\) is the number of calls.

Would you like a detailed calculation for each part of these questions?

Here are five related questions to expand your understanding:
1. What are the key differences between the binomial and Poisson distributions?
2. How would the probabilities change if the acceptance rate for the new product was 50% instead of 70%?
3. How can we calculate cumulative probabilities, such as "less than 5 customers"?
4. How is the Poisson distribution related to the exponential distribution in call center scenarios?
5. Can the binomial distribution be approximated using the normal distribution when the sample size is large?

**Tip:** Always check the assumptions behind the distribution models you are using (binomial for fixed trials, Poisson for rate-based events).

A company is launching a new product, and they have conducted a survey to estimate the product's acceptance rate among potential customers. From the survey, they found that 70% of the respondents indicated that they would be likely to purchase the new product. The company decides to use this information to forecast sales and makes an initial batch of 20 products to test the market. Find the probability that: a) exactly 10 customers will purchase the new product, b) less than 5 customers will purchase the new product, c) more than 12 customers will purchase the new product. Additionally, a call center receives an average of 10 customer service calls per hour. Find the probability that: a) exactly 4 calls will be received in a given hour, b) less than 8 calls will be received in a given hour, c) more than 12 calls will be received in a given hour.

Learn how to calculate probabilities using the binomial distribution for product acceptance rates and the Poisson distribution for call center events. Detailed step-by-step solutions for undergraduate-level students are provided.

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