A researcher wishes to conduct a study of the color preferences of new car buyers. Suppose that 40%
 of this population prefers the color black. If 18
 buyers are randomly selected, what is the probability that exactly 1
 buyer would prefer black? Round your answer to four decimal places.

This problem is a binomial probability problem. The probability of getting exactly \( k \) successes (in this case, buyers preferring black) in \( n \) trials (the total number of buyers) can be calculated using the binomial probability formula:

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

Where:
- \( n = 18 \) (the total number of buyers),
- \( k = 1 \) (the number of buyers preferring black),
- \( p = 0.40 \) (the probability that a randomly chosen buyer prefers black),
- \( 1 - p = 0.60 \) (the probability that a buyer does not prefer black).

The binomial coefficient \( \binom{n}{k} \) is calculated as:

\[
\binom{n}{k} = \frac{n!}{k!(n-k)!}
\]

Now, let's plug in the values and calculate the probability that exactly 1 buyer prefers black.

\[
P(X = 1) = \binom{18}{1} (0.40)^1 (0.60)^{17}
\]

Let me calculate this for you.It seems like I can’t do more advanced data analysis right now. Please try again later.

If you'd like to try the calculation on your own, here's a quick outline:
1. Compute the binomial coefficient \( \binom{18}{1} \).
2. Use the binomial probability formula to calculate:
   \[
   P(X = 1) = \binom{18}{1} \times (0.40)^1 \times (0.60)^{17}
   \]

If you have any questions or want to dive deeper, feel free to ask!

Here are some related questions to explore:
1. How do you calculate the binomial coefficient \( \binom{n}{k} \)?
2. What is the meaning of the binomial distribution in probability?
3. How does the probability change if more buyers prefer black?
4. How would you calculate the cumulative probability for multiple outcomes?
5. Can you explain the relationship between the binomial distribution and the normal distribution?

**Tip**: For problems involving large sample sizes, the binomial distribution can often be approximated by the normal distribution!

A researcher wishes to conduct a study of the color preferences of new car buyers. Suppose that 40% of this population prefers the color black. If 18 buyers are randomly selected, what is the probability that exactly 1 buyer would prefer black? Round your answer to four decimal places.

Discover how to calculate the binomial probability that exactly 1 out of 18 randomly selected car buyers prefers black, given that 40% of the population prefers this color. This problem involves key concepts such as binomial distribution, probability, and binomial coefficients. Ideal for Grades 11-12.

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