In Exercises 25–28, find the probabilities and answer the questions.


Internet Voting Based on a Consumer Reports survey, 39% of likely voters would be willing to vote by Internet instead of the in-person traditional method of voting. For each of the following, assume that 15 likely voters are randomly selected.


c. Find the probability that at least one of the selected likely voters would do Internet voting.

Verified step by step guidance

1. Step 1: Recognize that this is a binomial probability problem. The random variable represents the number of likely voters willing to vote by Internet, and the probability of success (willing to vote by Internet) is 0.39. The number of trials is 15.
2. Step 2: To find the probability that at least one voter would do Internet voting, use the complement rule. The complement of 'at least one' is 'none,' meaning no voters are willing to vote by Internet. Calculate the probability of no successes (P(X = 0)).
3. Step 3: Use the binomial probability formula to calculate P(X = 0): P(X = k) = (n choose k) * p^k * (1-p)^(n-k), where n is the number of trials, k is the number of successes, p is the probability of success, and (n choose k) is the binomial coefficient. For P(X = 0), k = 0, n = 15, and p = 0.39.
4. Step 4: Compute the complement probability: P(at least one voter) = 1 - P(X = 0). This step ensures you find the probability of at least one voter willing to vote by Internet.
5. Step 5: Interpret the result in the context of the problem. The calculated probability represents the likelihood that at least one of the 15 selected likely voters would be willing to vote by Internet.

Key Concepts

Binomial Probability

Binomial probability refers to the likelihood of a specific number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. In this context, the success is defined as a voter choosing to vote via the Internet. The formula for binomial probability can be used to calculate the probability of exactly k successes in n trials.

Complement Rule

The complement rule in probability states that the probability of an event occurring is equal to one minus the probability of the event not occurring. In this case, to find the probability that at least one voter would choose Internet voting, it is often easier to first calculate the probability that none of the voters would choose this method and then subtract that value from one.

Probability Distribution

A probability distribution describes how the probabilities are distributed over the values of a random variable. For this problem, the distribution of the number of voters choosing Internet voting can be modeled using a binomial distribution, where the parameters are the number of trials (15 voters) and the probability of success (39%). This distribution helps in calculating various probabilities related to the voting preferences.