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Ch. 5 - Discrete Probability Distributions
Triola - Elementary Statistics 14th Edition
Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook
All textbooksTriola 14th EditionCh. 5 - Discrete Probability DistributionsProblem 5.1.20a
Chapter 5, Problem 5.1.20a

Using Probabilities for Significant Events


a. Find the probability of getting exactly 1 match.

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1
Step 1: Understand the problem. We are tasked with finding the probability of getting exactly 1 match in a given scenario. This typically involves a probability distribution, such as the binomial distribution, where we calculate the likelihood of a specific number of successes (matches) in a series of trials.
Step 2: Identify the parameters of the problem. Determine the total number of trials (n), the probability of success in a single trial (p), and the number of successes (k) we are interested in. For this problem, k = 1 since we are looking for exactly 1 match.
Step 3: Use the binomial probability formula to calculate the probability of exactly 1 match. The formula is: P(X = k) = C(n, k) * pk * (1 - p)n-k, where C(n, k) is the number of combinations and can be calculated as C(n, k) = n! / (k! * (n - k)!).
Step 4: Substitute the values of n, k, and p into the formula. Compute the combination term C(n, k), the probability term pk, and the complement term (1 - p)n-k. Multiply these together to find the probability.
Step 5: Interpret the result. The calculated probability represents the likelihood of getting exactly 1 match in the given scenario. Ensure the result makes sense in the context of the problem and check for any errors in the calculations.

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Key Concepts

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

Probability

Probability is a measure of the likelihood that a particular event will occur, expressed as a number between 0 and 1. A probability of 0 indicates that the event cannot happen, while a probability of 1 indicates certainty. In the context of finding the probability of getting exactly one match, it involves calculating the chances based on the total number of trials and the desired outcome.
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Binomial Distribution

The binomial distribution is a statistical distribution that models the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. It is characterized by two parameters: the number of trials (n) and the probability of success (p). This concept is essential for calculating the probability of getting exactly one match in scenarios where there are multiple attempts.
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Combinatorics

Combinatorics is a branch of mathematics dealing with combinations and permutations of objects. It is crucial for determining the number of ways to choose a specific number of successes from a larger set. In the context of finding the probability of exactly one match, combinatorial calculations help in identifying how many different ways one match can occur among the total trials.
Related Practice
Textbook Question

In Exercises 31 and 32, assume that hybridization experiments are conducted with peas having the property that for offspring, there is a 0.75 probability that a pea has green pods (as in one of Mendel’s famous experiments).


Hybrids Assume that offspring peas are randomly selected in groups of 16.


a. Find the mean and standard deviation for the numbers of peas with green pods in the groups of 16.

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Textbook Question

Lottery. In Exercises 15–20, refer to the accompanying table, which describes probabilities for the California Daily 4 lottery. The player selects four digits with repetition allowed, and the random variable x is the number of digits that match those in the same order that they are drawn (for a “straight” bet).


Using Probabilities for Significant Events


a. Find the probability of getting exactly 2 matches.

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Textbook Question

 In Exercises 5–8, assume that the Poisson distribution applies; assume that the mean number of Atlantic hurricanes in the United States is 5.5 per year, as in Example 1; and proceed to find the indicated probability.


Hurricanes


a. Find the probability that in a year, there will be 7 hurricanes.

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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

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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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?


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