Question

Based on a​ poll, 66​% 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?

Answer 1

The probability that at least one internet user is more careful is 0.973

Explanation:

Answer 2

98.66% probability that among four randomly selected Internet​ users, at least one is more careful about personal information when using a public​ Wi-Fi hotspot

Explanation:

For each internet user, there are only two possible outcomes. EIther they are more careful about personal information when using a public​ Wi-Fi hotspot, or they are not. The probability of an internet user being more careful about personal information when using a public​ Wi-Fi hotspot is independent of other users. So we use the binomial probability distribution to solve this question.

Binomial probability distribution

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

`P(X = x) = C_(n,x).p^(x).(1-p)^(n-x)`

In which
`C_(n,x)` is the number of different combinations of x objects from a set of n elements, given by the following formula.

`C_(n,x) = (n!)/(x!(n-x)!)`

And p is the probability of X happening.

66​% of Internet users are more careful about personal information when using a public​ Wi-Fi hotspot.

This means that
`p = 0.66`

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?

This is
`P(X \geq 1)` when n = 4.

We know that either none of them are more careful, or at least one is. The sum of the probabilities of these events is decimal 1. So

`P(X = 0) + P(X \geq 1) = 1`

`P(X \geq 1) = 1 - P(X = 0)`

In which

`P(X = x) = C_(n,x).p^(x).(1-p)^(n-x)`

`P(X = 0) = C_(4,0).(0.66)^(0).(0.34)^(4) = 0.0134`

`P(X \geq 1) = 1 - P(X = 0) = 1 - 0.0134 = 0.9866`

98.66% probability that among four randomly selected Internet​ users, at least one is more careful about personal information when using a public​ Wi-Fi hotspot