Questions: Based on a poll, 65% 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 affected by the additional information that the survey subjects volunteered to respond?

Solution Steps

We are given that \(65\%\) of Internet users are more careful about personal information when using a public Wi-Fi hotspot. We need to find the probability that among three randomly selected Internet users, at least one is more careful.

Let \( p = 0.65 \) be the probability that an Internet user is more careful, and \( q = 1 - p = 0.35 \) be the probability that an Internet user is not more careful.

We use the binomial probability formula to find the probability that none of the three users is more careful: \[ P(X = 0) = \binom{3}{0} \cdot p^0 \cdot q^3 = 1 \cdot 1 \cdot 0.35^3 = 0.042875 \] Rounding to three decimal places, we have: \[ P(X = 0) = 0.043 \]

The probability that at least one user is more careful is the complement of the probability that none of them is careful: \[ P(\text{at least one careful}) = 1 - P(X = 0) = 1 - 0.043 = 0.957 \]

The result might be affected by the fact that the survey subjects volunteered to respond. This introduces a potential bias, as those who are more concerned about privacy might be more likely to respond. Therefore, the actual probability in the general population might differ.

Final Answer