A researcher wishes to conduct a study of the color preferences of new car buyers. Suppose that 40%
 of this population prefers the color red. If 18
 buyers are randomly selected, what is the probability that exactly 8
 buyers would prefer red? Round your answer to four decimal places.



This is a binomial probability problem since we're dealing with two possible outcomes for each buyer (either they prefer red or they don't).

The formula for the binomial probability is given by:

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

Where:
- \( n = 18 \) (the number of trials, i.e., the number of buyers),
- \( k = 8 \) (the number of successes, i.e., buyers who prefer red),
- \( p = 0.40 \) (the probability of success, i.e., a buyer prefers red),
- \( \binom{n}{k} \) is the binomial coefficient, which calculates as \( \frac{n!}{k!(n - k)!} \).

Let's calculate this probability.

\[
P(X = 8) = \binom{18}{8} (0.40)^8 (1 - 0.40)^{18 - 8}
\]

Now, I will compute the value.The probability that exactly 8 buyers out of 18 would prefer the color red is approximately \( 0.1734 \).

Do you have any questions or would you like more details?

Here are 5 related questions for further exploration:
1. What is the probability that fewer than 8 buyers prefer red?
2. What is the probability that more than 8 buyers prefer red?
3. How does the probability change if the number of selected buyers increases to 20?
4. What is the expected number of buyers who prefer red in this sample of 18?
5. How would the probability change if the preference rate for red is 50% instead of 40%?

**Tip:** Binomial distributions are ideal for scenarios with fixed trials and two outcomes, like success/failure.

A researcher wishes to conduct a study of the color preferences of new car buyers. Suppose that 40% of this population prefers the color red. If 18 buyers are randomly selected, what is the probability that exactly 8 buyers would prefer red? Round your answer to four decimal places.

Calculate the probability that exactly 8 out of 18 buyers prefer the color red when 40% of new car buyers favor red. This problem uses the binomial probability formula and involves key probability concepts like success/failure outcomes.

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