Based on a poll, \( 70 \% \) of Internet users are more careful about personal information when using a public Wi-Fi hotspot. What is the pro more careful about personal information when using a public Wi-Fi hotspot? How is the result affected by the additional information the The probability that at least one of them is careful about personal information is (Round to three decimal places as needed.)

It looks like you're working on a probability problem related to Internet users' caution with personal information on public Wi-Fi hotspots. Let's break down the problem and solve it step by step. ### **Problem Statement:** *Based on a poll, **70%** of Internet users are more careful about personal information when using a public Wi-Fi hotspot. What is the probability that **at least one** of **two** users is careful about personal information when using a public Wi-Fi hotspot? **Round your answer to three decimal places.*** ### **Understanding the Problem:** - **Probability that one user is careful (P):** 70% or 0.7 - **Number of users (n):** 2 - **Desired Probability:** At least one user is careful. ### **Approach:** To find the probability that **at least one** of the two users is careful, it's easier to calculate the probability that **none** of the users are careful and then subtract that from 1. 1. **Probability that one user is **not** careful (Q):** \( Q = 1 - P = 1 - 0.7 = 0.3 \) 2. **Probability that both users are **not** careful:** Since the users are independent, \( Q_{\text{both}} = Q \times Q = 0.3 \times 0.3 = 0.09 \) 3. **Probability that **at least one** user is careful:** \( P_{\text{at least one}} = 1 - Q_{\text{both}} = 1 - 0.09 = 0.91 \) ### **Final Answer:** The probability that at least one of the two users is careful about personal information when using a public Wi-Fi hotspot is **0.910** when rounded to three decimal places. ### **Additional Information Impact:** If there is **additional information** about the users being dependent or if there are more users involved, the calculation would differ. For instance: - **Dependence:** If the users' behaviors are dependent, the calculation of \( Q_{\text{both}} \) would need to account for that dependency. - **More Users:** For more than two users, the formula generalizes to: \( P_{\text{at least one}} = 1 - (1 - P)^n \) where \( n \) is the number of users. However, based on the given problem with two independent users, the probability remains **0.910**. **Answer:** The probability that at least one of the two users is careful is **0.910**.