Not all visitors to a certain company's website are customers. In fact, the website administrator estimates that about 10% of all visitors to the website are looking for other websites. Assuming that this estimate is correct, find the probability that, in a random sample of 5 visitors to the website, exactly 4 actually are looking for the website.
Round your response to at least three decimal places. (If necessary, consult a list of formulas.)

This problem can be solved using the **binomial probability formula**. The binomial probability formula is used to find the probability of getting exactly \( k \) successes in \( n \) independent Bernoulli trials, each with the same probability of success \( p \).

The binomial probability formula is given by:

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

Where:
- \( n \) is the number of trials (in this case, \( n = 5 \)),
- \( k \) is the number of successes (in this case, \( k = 4 \)),
- \( p \) is the probability of success on a single trial (in this case, \( p = 0.9 \) since 90% are looking for the website),
- \( 1-p \) is the probability of failure on a single trial (in this case, \( 1-p = 0.1 \)).

First, we need to calculate \( \binom{n}{k} \), which represents the number of combinations of \( n \) items taken \( k \) at a time:

\[
\binom{5}{4} = \frac{5!}{4!(5-4)!} = \frac{5 \times 4 \times 3 \times 2 \times 1}{(4 \times 3 \times 2 \times 1)(1)} = 5
\]

Now, substitute the values into the binomial probability formula:

\[
P(X = 4) = 5 \times (0.9)^4 \times (0.1)^1
\]

Let's calculate this step by step:

1. \( (0.9)^4 = 0.9 \times 0.9 \times 0.9 \times 0.9 = 0.6561 \)
2. \( (0.1)^1 = 0.1 \)
3. Multiply them together: \( 0.6561 \times 0.1 = 0.06561 \)
4. Finally, multiply by 5: \( 5 \times 0.06561 = 0.32805 \)

So, the probability that exactly 4 out of 5 visitors are actually looking for the website is approximately **0.328** (rounded to three decimal places).

Would you like more details or have any questions?

Here are 5 related questions to expand your understanding:

1. What is the probability that all 5 visitors are actually looking for the website?
2. How would the probability change if the sample size were increased to 10 visitors?
3. What is the expected number of visitors who are actually looking for the website in a random sample of 5?
4. How would the probability change if the percentage of visitors looking for the website increased to 95%?
5. Can the binomial probability formula be applied if the trials are not independent?

**Tip:** The binomial distribution is useful when you have a fixed number of independent trials, each with the same probability of success.

Learn how to calculate the probability of exactly 4 out of 5 visitors to a website actually looking for the site. Use the binomial probability formula and combinatorial calculations. Suitable for advanced high school students and beyond.

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