/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 59 Use the following information to... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 59

Use the following information to answer the next six exercises: On average, a clothing store gets 120 customers per day. What is the probability of getting 35 customers in the first four hours? Assume the store is open 12 hours each day.

Short Answer

Expert verified
The probability of getting exactly 35 customers in the first four hours is calculated using the Poisson distribution with \( \lambda = 40 \), resulting in a very small value.

Step by step solution

01

Interpret the problem as a Poisson distribution

We are dealing with a Poisson process since we want to find the probability of a given number of events (customers) in a fixed interval of time. The problem provides the average number of customers as 120 per 12-hour day, from which we can determine the average number per hour.
02

Determine the average rate for four hours

If the store gets 120 customers in 12 hours, then the rate per hour is \( \lambda = \frac{120}{12} = 10 \) customers per hour. Therefore, for four hours the rate \( \lambda = 10 \times 4 = 40 \) customers.
03

Apply the Poisson probability formula

The Poisson probability mass function is given by:\[ P(X = k) = \frac{e^{-\lambda} \cdot \lambda^k}{k!} \]Where \( \lambda = 40 \) is the average rate we found and \( k = 35 \) is the number of customers we want to find the probability for.
04

Substitute values into the Poisson formula

Substitute \( \lambda = 40 \) and \( k = 35 \) into the Poisson formula:\[ P(X = 35) = \frac{e^{-40} \cdot 40^{35}}{35!} \]
05

Calculate the probability

Using a calculator or software, compute \( e^{-40} \), \( 40^{35} \), and \( 35! \), then use these to find \( P(X = 35) \). This calculation approximately yields a very small probability, indicating it is unlikely for exactly 35 customers to arrive in the first four hours.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability Theory
Probability Theory is a branch of mathematics focused on analyzing random events and outcomes. In simple terms, it helps us measure the likelihood of different events happening.
One of the essential aspects of Probability Theory is interpreting real-world situations mathematically, like determining the likelihood of specific occurrences.
The Poisson distribution, as in the original exercise, is a powerful tool within Probability Theory.
  • It is used for predicting the probability of a number of events happening within a fixed interval of time.
  • The distribution is particularly useful for modeling events that occur independently and sporadically.
In the context of the clothing store, Probability Theory helps in gauging the probability of having a certain number of customer arrivals in a specified time. By converting the average daily customer count into an hourly rate and applying it to a shorter time interval, students can calculate specific probabilities using this distribution.
Statistical Methods
Statistical Methods involve organizing, summarizing, and analyzing data. These methods are crucial for interpreting data and deriving meaningful conclusions.
When applying Statistical Methods to the exercise regarding the clothing store, several steps are undertaken to arrive at a solution:
  • First, the problem is understood statistically by identifying it as a suitable case for the Poisson distribution.
  • Then, we calculate the average rate of customer arrival per hour to suit the statistical model.
  • Finally, with the necessary data, the Poisson formula is applied to determine specific probabilities.
Using these methods provides a systematic approach to solve the problem, enabling students to understand and compute probabilities effectively. It's a bridge between pure calculations and meaningful real-world applications.
Customer Arrival Rate
Customer Arrival Rate is a concept used to quantify the number of customers visiting a location over a specific period.
This metric can vary from store to store and even within different times of the day.
In the textbook exercise, the average Customer Arrival Rate is crucial for calculating desired probabilities using the Poisson distribution model.
  • Firstly, the problem sets the customer base as 120 customers every 12 hours, averaging 10 customers per hour.
  • Secondly, by extending this rate to the four-hour interval in question, it helps determine the suitable Poisson parameter (\( \lambda \) becomes 40).
Understanding the Customer Arrival Rate allows students to make justified assays on specific probabilities. It's a vital concept in statistical models predicting the likelihood of customer patterns and behaviors, ensuring businesses can prepare and plan operations efficiently.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Use the following information to answer the next two exercises: The probability that the San Jose Sharks will win any given game is 0.3694 based on a 13-year win history of 382 wins out of 1,034 games played (as of a certain date). An upcoming monthly schedule contains 12 games. It has been estimated that only about 30% of California residents have adequate earthquake supplies. Suppose you randomly survey 11 California residents. We are interested in the number who have adequate earthquake supplies. a. In words, define the random variable \(X.\) b. List the values that \(X\) may take on. c. Give the distribution of \(X . X \sim\) ___ (___,____) d. What is the probability that at least eight have adequate earthquake supplies? e. Is it more likely that none or that all of the residents surveyed will have adequate earthquake supplies? Why? f. How many residents do you expect will have adequate earthquake supplies?

Use the following information to answer the next six exercises: The Higher Education Research Institute at UCLA collected data from 203,967 incoming first-time, full-time freshmen from 270 four-year colleges and universities in the U.S. 71.3% of those students replied that, yes, they believe that same-sex couples should have the right to legal marital status. Suppose that you randomly select freshman from the study until you find one who replies 鈥測es.鈥� You are interested in the number of freshmen you must ask. What values does the random variable \(X\) take on?

Use the following information to answer the next seven exercises: A ballet instructor is interested in knowing what percent of each year's class will continue on to the next, so that she can plan what classes to offer. Over the years, she has established the following probability distribution. \(\bullet\) Let \(X=\) the number of years a student will study ballet with the teacher. \(\bullet\) Let \(P(x)=\) the probability that a student will study ballet \(x\) years. What does the column \(" P(x)\) "sum to and why?

Use the following information to answer the next eight exercises: The Higher Education Research Institute at UCLA collected data from 203,967 incoming first-time, full-time freshmen from 270 four year colleges and universities in the U.S. 71.3% of those students replied that, yes, they believe that same-sex couples should have the right to legal marital status. Suppose that you randomly pick eight first-time, full-time freshmen from the survey. You are interested in the number that believes that same sex-couples should have the right to legal marital status. What is the probability that at most five of the freshmen reply 鈥測es鈥�?

Suppose that a technology task force is being formed to study technology awareness among instructors. Assume that ten people will be randomly chosen to be on the committee from a group of 28 volunteers, 20 who are technically proficient and eight who are not. We are interested in the number on the committee who are not technically proficient. a. In words, define the random variable X. b. List the values that X may take on. C. Give the distribution of \(X . X \sim\) ____ (___,____) d. How many instructors do you expect on the committee who are not technically proficient? e. Find the probability that at least five on the committee are not technically proficient. f. Find the probability that at most three on the committee are not technically proficient.
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Calculus

Read Explanation

Logic and Functions

Read Explanation

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Applied Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.