At Least One. In Exercises 5–12, find the probability.

Wi-Fi Based on a poll conducted through the e-edition of USA Today, 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 of selecting a user who is more careful about personal information when using public Wi-Fi hotspot is 0.67.

Four users are selected at random.

The probability of an event occurring "at least once" is the likelihood of a situation occurring once or more than once.

It is also equal to one minus the chance that the event will not occur at all.

$P\left(\text{A}\text{happening}\text{at}\text{least}\text{once}\right)=1-P\left(\text{not}\text{A}\right)$

Let A be the event of selecting a user who is careful about personal information when using public Wi-Fi hotspot.

Then, the complementary event would be selecting a user who is not very careful about personal information when using public Wi-Fi hotspot.

The probability of selecting such a user is given by:

$$\begin{gathered} P\left(A\right)=\frac{67}{100} \\ =0.67 \end{gathered}$$

Thus, P(A) is 0.67.

The probability of selecting a user who is not very careful about personal information when using public Wi-Fi hotspot is equal to:

$$\begin{gathered} P\left(\overline{A}\right)=1-P\left(A\right) \\ =1-0.67 \\ =0.33 \end{gathered}$$

Thus,$P\left(\overline{A}\right)$ is equal to 0.33.

Four users are selected at random.

The probability of getting all four users who are not careful about personal information when using public Wi-Fi hotspot is given by:

$$\begin{gathered} P\left(\text{all}\text{good}\right)=\left(0.33\right)\times \left(0.33\right)\times \left(0.33\right)\times \left(0.33\right) \\ =0.0119 \end{gathered}$$

The probability of getting at least one careful user is one minus the probability of getting no careful user:

$$\begin{gathered} P\left(\text{at}\text{least}\text{1}\text{careful}\text{user}\right)=1-P\left(\text{no}\text{careful}\text{user}\right) \\ =1-0.0119 \\ =0.988 \end{gathered}$$

Therefore, the probability of getting at least one who is not careful about personal information when using public Wi-Fi hotspot is equal to 0.988.

It is known that the data is based on the information collected through the e-edition of USA Today. The users voluntarily responded.

As the users responded voluntarily and were not chosen randomly, the results will be considerably influenced and not 100% correct.