Let's use the binomial distribution to solve these questions. We are given a group size of 15 and the rate of users in America is 0.15.
(a) At least one of them uses the website.
To find this probability, we will use the complement rule which subtracts the possibility that none of the people in the group use the website from 1. With a rate of 0.15, the probability that none of the 15 people in the group use the website is (0.85)^15. To find the probability that at least one person uses the website, we subtract this from 1. Hence, the probability that at least one person uses the website is 0.913.
(b) More than two of them use the website.
To calculate this, we use the cumulative density function (CDF) of the binomial distribution. The CDF calculates the probability of at most a certain number of successes. However, we want the probability of *more* than 2 users, so we subtract the CDF of 2 or fewer users from 1 to give us the probability of more than 2 users. Hence, the probability that more than two of them use the website is 0.396.
(c) None of them use the website.
To find the probability that none use the website, we calculate the same number for zero successes in the binomial distribution with a group size of 15 and a success rate of 0.15. Hence, the probability that none of them use the website is 0.087.
(d) At least 13 of them do not use the website.
We can calculate this by using the binomial distribution with a success rate of 'non-users' which is 1-0.15 = 0.85. So, we can calculate the probability that out of 15, at most 2 are users i.e., at least 13 are non-users. Hence, the probability that at least 13 of them do not use the website is 0.0.