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.

a. What is the probability that exactly 12 of those selected would do Internet voting?

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Step 1: Recognize that this is a binomial probability problem. The binomial distribution is used when there are a fixed number of trials (n), two possible outcomes (success or failure), a constant probability of success (p), and the trials are independent. Here, n = 15 (number of voters), p = 0.39 (probability of a voter choosing Internet voting), and we are asked to find the probability of exactly 12 successes (k = 12).

Step 2: Write the formula for the binomial probability: P(X = k) = (n choose k) * p^k * (1 - p)^(n - k). Here, (n choose k) is the binomial coefficient, which is calculated as (n! / (k! * (n - k)!)).

Step 3: Substitute the given values into the formula. For this problem, n = 15, k = 12, and p = 0.39. The formula becomes: P(X = 12) = (15 choose 12) * (0.39)^12 * (1 - 0.39)^(15 - 12).

Step 4: Calculate the binomial coefficient (15 choose 12). This is computed as 15! / (12! * (15 - 12)!), which simplifies to 15! / (12! * 3!).

Step 5: Substitute the binomial coefficient and probabilities into the formula. Compute (0.39)^12 and (1 - 0.39)^3, then multiply these values by the binomial coefficient to find the probability. This will give you the final result for P(X = 12).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. In this context, the 'success' is a voter choosing to vote via the Internet, with a probability of 0.39. The distribution is defined by two parameters: the number of trials (n) and the probability of success (p).

Probability Mass Function (PMF)

The probability mass function gives the probability of obtaining exactly k successes in n trials for a binomial distribution. It is calculated using the formula P(X = k) = (n choose k) * p^k * (1-p)^(n-k). This function is essential for determining the likelihood of a specific outcome, such as exactly 12 voters choosing Internet voting out of 15.

Combinatorial Coefficient

The combinatorial coefficient, often represented as 'n choose k' or C(n, k), calculates the number of ways to choose k successes from n trials. It is crucial in the binomial probability formula, as it accounts for the different arrangements of successes and failures. For example, in this scenario, it helps determine how many ways 12 voters can be selected from 15.