Questions: Based on a poll, 63% of Internet users are more careful about personal information when using a public Wi-Fi hotspot. What is the probability that among three randomly selected Internet users, at least one is more careful about personal information when using a public Wi-Fi hotspot? How is the result The probability that at least one of them is careful about personal information is (Round to three decimal places as needed.).

Solution Steps

We are tasked with finding the probability that at least one of three randomly selected Internet users is more careful about personal information when using a public Wi-Fi hotspot, given that \(63\%\) of Internet users exhibit this behavior.

Let:

• \(p = 0.63\) (the probability that a user is careful)
• \(q = 1 - p = 0.37\) (the probability that a user is not careful)
• \(n = 3\) (the number of users selected)

To find the probability that at least one user is careful, we first calculate the probability that none of the users are careful, denoted as \(P(X = 0)\). This can be calculated using the binomial probability formula:

\[ P(X = x) = \binom{n}{x} \cdot p^x \cdot q^{n-x} \]

For \(x = 0\):

\[ P(X = 0) = \binom{3}{0} \cdot (0.63)^0 \cdot (0.37)^3 = 1 \cdot 1 \cdot (0.37)^3 = 0.051 \]

The probability that at least one user is careful is given by:

\[ P(X \geq 1) = 1 - P(X = 0) \]

Substituting the value we calculated:

\[ P(X \geq 1) = 1 - 0.051 = 0.949 \]

Final Answer