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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.6b
Chapter 4, Problem 4.T.6b

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.

Verified step by step guidance
1
Step 1: Recognize that this problem involves a Poisson distribution because it deals with the number of events (customer arrivals) occurring in a fixed interval of time (one minute). The mean number of arrivals per minute (λ) is given as 4.
Step 2: Recall the formula for the Poisson probability mass function (PMF): P(X = k) = (λ^k * e^(-λ)) / k!, where X is the random variable representing the number of arrivals, λ is the mean, k is the number of arrivals, and e is the base of the natural logarithm (approximately 2.718).
Step 3: To find the probability of 'more than four customers arriving,' calculate the complement of the cumulative probability for X ≤ 4. This means you need to compute P(X > 4) = 1 - P(X ≤ 4).
Step 4: Compute P(X ≤ 4) by summing the probabilities for X = 0, 1, 2, 3, and 4 using the Poisson PMF formula. Specifically, calculate P(X = 0), P(X = 1), P(X = 2), P(X = 3), and P(X = 4), then add these probabilities together.
Step 5: Subtract the cumulative probability P(X ≤ 4) from 1 to find P(X > 4). This result represents the probability that more than four customers will arrive during the first minute.

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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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Mean (λ) in Poisson Distribution

In the context of the Poisson distribution, the mean (denoted as λ) represents the average number of occurrences in a specified interval. For this question, λ is equal to four, indicating that, on average, four customers arrive at the checkout counters each minute. This parameter is crucial for calculating probabilities related to the number of arrivals.
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Calculating Probability

To find the probability of more than four customers arriving in the first minute, we can use the cumulative distribution function (CDF) of the Poisson distribution. Specifically, we calculate the probability of zero to four customers arriving and subtract this from one. This approach allows us to determine the likelihood of observing more than the average number of arrivals in a given time frame.
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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.

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 1–7, 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 1–3, 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

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.