Problem 39 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

PACKAGE DELIVERY Suppose that the number of overnight packages a business receives during a business day follows a Poisson distribution and that, on average, the company receives four overnight packages per day.| a. Find the probability that the company receives exactly four overnight packages on a randomly selected business day. b. Find the probability that the company does not receive any overnight packages on a randomly selected business day. c. Find the probability that the company receives fewer than four overnight packages on a randomly selected business day.

Step-by-Step Solution

Answer

a) \(0.1954\) b) \(0.0183\) c) \(0.4333\)

Step 1: Understand the Poisson Distribution

The Poisson distribution is used to model the number of events occurring within a given time period. In this case, the arrival of overnight packages can be modeled using a Poisson distribution with an average rate, or parameter, \(\lambda\), of 4 packages per day.

Step 2: Use the Poisson Probability Formula

The probability mass function for a Poisson distribution is given by the formula: \[ P(X=k) = \frac{\lambda^k e^{-\lambda}}{k!} \] where \(\lambda = 4\) is the average rate, and \(k\) is the number of events (packages).

Step 3: Find the Probability for Part (a)

For part (a), we need to find the probability that the company receives exactly four packages. Here, \(k = 4\). Thus, substituting into the formula, we have: \[ P(X=4) = \frac{4^4 e^{-4}}{4!} \] Calculate this value to get the result.

Step 4: Simplify the Expression for Part (a)

First, find the factorial: \(4! = 24\). Now calculate \(4^4 = 256\) and \(e^{-4} \approx 0.0183\) (using a calculator). So, \[ P(X=4) = \frac{256 \times 0.0183}{24} \approx 0.1954 \]

Step 5: Find the Probability for Part (b)

For part (b), we need to find the probability that the company receives zero packages. Here, \(k = 0\). Thus, \[ P(X=0) = \frac{4^0 e^{-4}}{0!} = e^{-4} \approx 0.0183 \]

Step 6: Find the Probability for Part (c)

For part (c), we need to find the probability that the company receives fewer than four packages, which means calculating the cumulative probability for \(k = 0, 1, 2, 3\). Therefore, \[ P(X < 4) = P(X=0) + P(X=1) + P(X=2) + P(X=3) \]

Step 7: Calculate Individual Probabilities for Part (c)

We already have \(P(X=0)\). For \(P(X=1)\), \[ P(X=1) = \frac{4^1 e^{-4}}{1!} = \frac{4 \times 0.0183}{1} = 0.0732 \] For \(P(X=2)\), \[ P(X=2) = \frac{4^2 e^{-4}}{2!} = \frac{16 \times 0.0183}{2} = 0.1464 \] For \(P(X=3)\), \[ P(X=3) = \frac{4^3 e^{-4}}{3!} = \frac{64 \times 0.0183}{6} = 0.1954 \]

Step 8: Sum to Find Total Probability for Part (c)

Add the probabilities from the previous step: \[ P(X < 4) = 0.0183 + 0.0732 + 0.1464 + 0.1954 = 0.4333 \]

Key Concepts

probability theory

Probability theory is a branch of mathematics that deals with the likelihood or probability of different outcomes. It helps in making predictions about uncertain events. For example, in business, it can predict the number of customers arriving in a store or the demand for a product.
A key part of probability theory is understanding different types of distributions, such as the Poisson distribution. These distributions model how events occur over time or space.
By understanding probability theory, businesses can make more informed decisions based on the likelihood of different events happening.

Poisson probability formula

The Poisson distribution is instrumental in probability theory, especially when modeling the number of events in a fixed interval of time or space. It's defined by its average rate or parameter, \( \lambda\).
The Poisson probability formula is given by:
\[ P(X=k) = \frac{\lambda^k e^{-\lambda}}{k!} \]Here, \( P(X=k) \) is the probability of observing \k\ events, \( \lambda \) is the average number of events per interval, and \( k! \) represents the factorial of \( k\).
For example, if a business receives an average of 4 packages per day, \( \lambda = 4 \), and to find the probability of getting exactly 4 packages (k=4), you would substitute these values into the formula.

cumulative probability calculation

Cumulative probability calculation involves finding the probability of an event happening up to a certain number of times. In Poisson distribution, this means adding up the probabilities for different occurrences.
For instance, to calculate the probability that a business receives fewer than 4 packages in a day, you'd sum the probabilities for receiving 0, 1, 2, and 3 packages:
\[ P(X < 4) = P(X=0) + P(X=1) + P(X=2) + P(X=3) \]
This involves using the Poisson probability formula for each value of k (0 to 3) and then adding those probabilities together.

factorial in probability

Factorials are a fundamental concept in probability calculations, especially in the context of the Poisson distribution. The factorial of a non-negative integer \( k \), denoted as \( k! \), is the product of all positive integers less than or equal to \( k \).
For example, \( 4! = 4 \times 3 \times 2 \times 1 = 24 \).
Factorials are used in the denominator of the Poisson probability formula:
\[ P(X=k) = \frac{\lambda^k e^{-\lambda}}{k!} \]Understanding how to calculate factorials is essential for correctly applying the Poisson distribution.