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Ch. 4 - Discrete Probability Distributions
Larson - Elementary Statistics: Picturing the World 8th Edition
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook
All textbooksLarson 8th EditionCh. 4 - Discrete Probability DistributionsProblem 4.T.1a
Chapter 4, Problem 4.T.1a

In Exercises 1–3, find the indicated probabilities using the geometric distribution, the Poisson distribution, or the binomial distribution. Then determine whether the events are unusual. If convenient, use a table or technology to find the probabilities.
One out of every 42 tax returns for incomes over \$1 million requires an audit. An auditor is examining tax returns for over \$1 million. Find the probability that (a) the first return requiring an audit is the 25th return the tax auditor examines, (b) the first return requiring an audit is the first or second return the tax auditor examines, and (c) none of the first five returns the tax auditor examines require an audit. (Source: Kiplinger)

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Step 1: Identify the type of probability distribution to use. Since the problem involves finding the probability of the first success (audit) occurring on a specific trial or within a certain number of trials, the geometric distribution is appropriate. The geometric distribution models the number of trials until the first success in a sequence of independent Bernoulli trials.
Step 2: Define the parameters of the geometric distribution. The probability of success (p) is given as 1/42 (since one out of every 42 tax returns requires an audit). The probability of failure (q) is therefore 1 - p = 41/42.
Step 3: Solve part (a). To find the probability that the first return requiring an audit is the 25th return, use the probability mass function (PMF) of the geometric distribution: P(X = k) = q^(k-1) * p, where k is the trial number of the first success. Substitute k = 25, p = 1/42, and q = 41/42 into the formula.
Step 4: Solve part (b). To find the probability that the first return requiring an audit is the first or second return, calculate P(X = 1) + P(X = 2). Use the PMF formula for each value of k (k = 1 and k = 2), and then sum the results.
Step 5: Solve part (c). To find the probability that none of the first five returns require an audit, calculate the probability of five consecutive failures. This is given by q^5, where q = 41/42. Substitute the value of q into the formula and compute q^5.

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Key Concepts

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

Geometric Distribution

The geometric distribution models the number of trials needed to achieve the first success in a series of independent Bernoulli trials. In this context, it is used to find the probability that the first tax return requiring an audit occurs on a specific trial, such as the 25th return examined. The probability of success (an audit) is constant, making this distribution suitable for the problem.
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Poisson Distribution

The Poisson distribution is used to model the number of events occurring within a fixed interval of time or space, given a known average rate of occurrence. While not directly applicable to the specific questions posed, understanding this distribution is essential for scenarios where events happen independently and at a constant average rate, such as audits in tax returns over time.
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Binomial Distribution

The binomial distribution describes the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. In this case, it can be used to determine the probability of a certain number of audits occurring within a set of tax returns examined, providing a framework for evaluating the likelihood of unusual events based on the defined parameters.
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Related Practice
Textbook Question

In Exercises 1–7, consider a grocery store that can process a total of four customers at its checkout counters each minute.

The mean increases to five arrivals per minute, but the store can still process only four per minute. Generate a list of 20 random numbers with a Poisson distribution for mu = 5 . Then create a table that shows the number of customers waiting at the end of 20 minutes.

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Textbook Question

In Exercises 1–7, consider a grocery store that can process a total of four customers at its checkout counters each minute.

The mean number of arrivals per minute is four. Find the probability that

a. three, four, or five customers will arrive during the third minute.

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Textbook Question

In Exercises 1–7, consider a grocery store that can process a total of four customers at its checkout counters each minute.

The mean number of arrivals per minute is four. Find the probability that

c. one customer is waiting in line after one minute and no customers are waiting in line after the second minute..

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Textbook Question

In Exercises 1–3, find the indicated probabilities using the geometric distribution, the Poisson distribution, or the binomial distribution. Then determine whether the events are unusual. If convenient, use a table or technology to find the probabilities.

One out of every 42 tax returns for incomes over \$1 million requires an audit. An auditor is examining tax returns for over \$1 million. Find the probability that (c) none of the first five returns the tax auditor examines require an audit.

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Textbook Question

The table shows the ages of students in a freshman orientation course.

b. Graph the probability distribution using a histogram and describe its shape.

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Textbook Question

The table shows the ages of students in a freshman orientation course.

a. Construct a probability distribution.

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