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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.6c
Chapter 4, Problem 4.T.6c

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. more than four customers will arrive during each of the first four minutes.

Verified step by step guidance
1
Step 1: Recognize that this problem involves a Poisson distribution because we are dealing with the number of arrivals in a fixed interval of time (per minute) and the mean number of arrivals is given as 4. The Poisson probability mass function is given by P(X = k) = (λ^k * e^(-λ)) / k!, where λ is the mean number of arrivals, k is the number of arrivals, and e is the base of the natural logarithm.
Step 2: To find the probability of 'more than four customers arriving,' calculate the complement of the probability of 'four or fewer customers arriving.' This means we need to compute P(X > 4) = 1 - P(X ≤ 4).
Step 3: Compute P(X ≤ 4) by summing the probabilities for k = 0, 1, 2, 3, and 4 using the Poisson formula. Specifically, calculate P(X = 0), P(X = 1), P(X = 2), P(X = 3), and P(X = 4), and then sum them: P(X ≤ 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4).
Step 4: Once P(X ≤ 4) is calculated, find P(X > 4) for one minute using the complement rule: P(X > 4) = 1 - P(X ≤ 4).
Step 5: Since the problem asks for the probability that more than four customers will arrive during each of the first four minutes, raise the probability P(X > 4) to the power of 4 (because the events across the four minutes are independent). The final expression is [P(X > 4)]^4.

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Key Concepts

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

Poisson Distribution

The Poisson distribution is a probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space, given a known average rate of occurrence. It is particularly useful for modeling the number of arrivals in a fixed period, such as customers arriving at a grocery store. In this scenario, the mean arrival rate is four customers per minute.
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Intro to Frequency Distributions

Mean and Variance

In the context of the Poisson distribution, the mean (λ) represents the average number of events (customer arrivals) in a given time frame, while the variance is also equal to the mean. This property simplifies calculations, as both the mean and variance are four in this case. Understanding these concepts is crucial for calculating probabilities related to customer arrivals.
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Difference in Means: Hypothesis Tests

Cumulative Probability

Cumulative probability refers to the probability that a random variable takes on a value less than or equal to a certain number. In this question, to find the probability of more than four customers arriving in the first four minutes, one would first calculate the cumulative probability of four or fewer arrivals and then subtract this from one. This approach is essential for determining the likelihood of exceeding a specific threshold.
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Introduction to Probability
Related Practice
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 (b) the first return requiring an audit is the first or second return the tax auditor examines, 

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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

b. more than four customers will arrive during the first minute.

Textbook Question

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

Minitab was used to generate 20 random numbers with a Poisson distribution for . Let the random number represent the number of arrivals at the checkout counter each minute for 20 minutes. 3 3 3 3 5 5 6 7 3 6 3 5 6 3 4 6 2 2 4 1During each of the first four minutes, only three customers arrived. These customers could all be processed, so there were no customers waiting after four minutes.

b. Create a table that shows the number of customers waiting at the end of 1 through 20 minutes.

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.

Textbook Question

The Centers for Disease Control and Prevention (CDC) is required by law to publish a report on assisted reproductive technology (ART). ART includes all fertility treatments in which both the egg and the sperm are used. These procedures generally involve removing eggs from a patient’s ovaries, combining them with sperm in the laboratory, and returning them to the patient’s body or giving them to another patient.

You are helping to prepare a CDC report on young ART patients and select at random 6 ART cycles of patients under 35 years of age for a special review. None of the cycles resulted in a live birth. Your manager feels it is impossible to select at random 10 ART cycles that do not result in a live birth. Use the pie chart at the right and your knowledge of statistics to determine whether your manager is correct.

b. What probability distribution do you think best describes the situation? Do you think the distribution of the number of live births is discrete or continuous? Explain your reasoning.