Many wholesale stores offer free samples for customers to taste things in order to increase sales. Let’s assume that Costco provides free samples of cheese for everyone who walks by the cheese aisle. Historically, 45% of the customers who sampled a product purchased it. Round to four decimal places.

A) What is the probability exactly 12 of the next 30 customers who taste the cheese will purchase?

This problem is a classic example of a **binomial probability distribution**. We are given:

- The probability that a customer who samples the cheese purchases it (\( p \)) is 45%, or \( p = 0.45 \).
- The number of customers who sample the cheese (\( n \)) is 30.
- We are asked to find the probability that exactly 12 customers purchase the cheese (\( x = 12 \)).

The formula for the binomial probability is:

\[
P(X = x) = \binom{n}{x} p^x (1 - p)^{n - x}
\]

Where:
- \( \binom{n}{x} = \frac{n!}{x!(n - x)!} \) is the binomial coefficient.
- \( p^x \) is the probability of \( x \) successes.
- \( (1 - p)^{n - x} \) is the probability of \( n - x \) failures.

### Step-by-step solution:

1. **Set the parameters**:
   - \( n = 30 \)
   - \( x = 12 \)
   - \( p = 0.45 \)
   - \( 1 - p = 0.55 \)

2. **Calculate the binomial coefficient**:
   \[
   \binom{30}{12} = \frac{30!}{12!(30 - 12)!} = \frac{30!}{12!18!}
   \]

3. **Plug everything into the binomial probability formula**:
   \[
   P(X = 12) = \binom{30}{12} (0.45)^{12} (0.55)^{18}
   \]

Let me calculate that for you.The probability that exactly 12 of the next 30 customers who taste the cheese will purchase it is approximately **0.1265** or **12.65%**.

Would you like further details or explanations?

Here are 5 related questions to expand your understanding:
1. How would the probability change if only 10 customers purchased instead of 12?
2. What is the probability that fewer than 12 customers will purchase the cheese?
3. How does the probability change if the purchase probability increases to 50%?
4. What is the expected number of customers to purchase the cheese out of 30?
5. How does the binomial distribution differ from other probability distributions?

**Tip:** When solving binomial problems, the expected value is simply \( n \times p \), making it a useful quick check.

What is the probability exactly 12 of the next 30 customers who taste the cheese will purchase?

Learn how to calculate the probability that exactly 12 out of 30 customers who sample cheese at Costco will make a purchase, given a 45% purchase probability. This problem demonstrates the use of the binomial probability distribution formula, suitable for high school students in Grades 10-12.

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