Question

not all visitors to a certain companys 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 2 actually are looking for the website.

round your response to at least three decimal places. (if necessary, consult a list of formulas.)

Explanation:

Identify the distribution type

The problem describes a scenario with a fixed number of independent trials, \(n = 5\) visitors. Each visitor has two mutually exclusive outcomes: either they are looking for other websites (success) or they are not (failure). The probability of success is constant at \(p = 10\% = 0.1\). Therefore, we model this using a Discrete Random Variable following a Binomial Distribution.

Define parameters and target

Using the Probability of Events concept, we define:

• Number of trials: \(n = 5\)
• Probability of success (looking for other websites): \(p = 0.1\)
• Probability of failure: \(q = 1 - p = 0.9\)
• Number of successful trials desired: \(k = 2\)

Apply the binomial formula

The binomial probability formula is:

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

Using Combinations to find the binomial coefficient:

$$\binom{5}{2} = \frac{5!}{2!(5-2)!} = 10$$

Calculate the probability

Substitute the values into the formula:

$$P(X = 2) = 10 \times (0.1)^2 \times (0.9)^3$$

$$P(X = 2) = 10 \times 0.01 \times 0.729$$

$$P(X = 2) = 0.0729$$

Round to three decimal places

Rounding \(0.0729\) to three decimal places yields \(0.073\).

Answer:

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 2 actually are looking for the website.

Round your response to at least three decimal places. (If necessary, consult a list of formulas.)

<blank>0.073</blank>