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.