Question

based on a poll, 67% of internet users are more careful about personal information when using a public wi - fi hotspot. what is the probability that among four 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? the probability that at least one of them is careful about personal information is 0.988 (round to three decimal places as needed.) how is the result affected by the additional information that the survey subjects volunteered to respond? a. it is very possible that the result is not valid because not everyone uses public wi - fi b. it is very possible that the result is not valid because the sample may not be representative of the people who use public wi - fi c. the result should not be impacted by this because volunteers are likely to have the most relevant responses. d. the result should not be impacted by this because there is no reason to lie when responding

Explanation:

Step1: Find the probability that none are careful

The probability that an internet - user is careful is $p = 0.67$. So the probability that an internet - user is not careful is $q=1 - p=1 - 0.67 = 0.33$. We have $n = 4$ randomly selected internet - users. The probability that none of them is careful is given by the binomial probability formula $P(X = 0)=C(n,0)\times p^{0}\times q^{n}$, where $C(n,k)=\frac{n!}{k!(n - k)!}$. Here, $C(4,0)=\frac{4!}{0!(4 - 0)!}=1$, $p = 0.67$, $q = 0.33$ and $n = 4$. So $P(X = 0)=1\times1\times(0.33)^{4}=0.01185921$.

Step2: Find the probability that at least one is careful

The probability that at least one is careful is the complement of the event that none are careful. Let $P(X\geq1)$ be the probability that at least one is careful. Then $P(X\geq1)=1 - P(X = 0)$. Substituting the value of $P(X = 0)$ we found in Step 1, we get $P(X\geq1)=1-(0.33)^{4}=1 - 0.01185921 = 0.98814079\approx0.988$.

For the second part:
When the survey subjects volunteered to respond, there is a possibility of non - response bias. Volunteers may have different characteristics from non - volunteers. In the context of this survey about public Wi - Fi and personal information, the sample of volunteers may not be representative of all people who use public Wi - Fi.

Answer:

The probability that at least one of them is careful about personal information is $0.988$.
B. It is very possible that the result is not valid because the sample may not be representative of the people who use public Wi - Fi.