Ch. 5 - Discrete Probability Distributions

Triola - Elementary Statistics 14th Edition

Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook

Triola 14th Edition

Ch. 5 - Discrete Probability Distributions

Problem 5.2.27a

Chapter 5, Problem 5.2.27a

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?

Verified step by step guidance

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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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Correlation Coefficient

Related Practice

Textbook Question

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

Too Young to Tat Based on a Harris poll, among adults who regret getting tattoos, 20% say that they were too young when they got their tattoos. Assume that five adults who regret getting tattoos are randomly selected, and find the indicated probability.

a. Find the probability that none of the selected adults say that they were too young to get tattoos.

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Textbook Question

One of Mendel’s famous experiments with peas resulted in 580 offspring, and 152 of them were yellow peas. Mendel claimed that under the same conditions, 25% of offspring peas would be yellow. Assume that Mendel’s claim of 25% is true, and assume that a sample consists of 580 offspring peas.

a. Use the range rule of thumb to identify the limits separating values that are significantly low and those that are significantly high. Based on the results, is the result of 152 yellow peas either significantly low or significantly high?

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Textbook Question

Using Probabilities for Significant Events

a. Find the probability of getting exactly 1 match.

Textbook Question

Politics The County Clerk in Essex, New Jersey, was accused of cheating by not using randomness in assigning the order in which candidates’ names appeared on voting ballots. Among 41 different ballots, Democrats were assigned the desirable first line 40 times. Assume that Democrats and Republicans are assigned the first line using a method of random selection so that they are equally likely to get that first line.

a. Use the range rule of thumb to identify the limits separating values that are significantly low and those that are significantly high. Based on the results, is the result of 40 first lines for Democrats significantly high?

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Textbook Question

Expected Value in North Carolina’s Pick 4 Game In North Carolina’s Pick 4 lottery game, you can pay \(1 to select a four-digit number from 0000 through 9999. If you select the same sequence of four digits that are drawn, you win and collect \)5000.

a. How many different selections are possible?

Textbook Question

In Exercises 29 and 30, assume that different groups of couples use the XSORT method of gender selection and each couple gives birth to one baby. The XSORT method is designed to increase the likelihood that a baby will be a girl, but assume that the method has no effect, so the probability of a girl is 0.5.

Gender Selection Assume that the groups consist of 36 couples.

a. Find the mean and standard deviation for the numbers of girls in groups of 36 births.

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