Problem 28 - Basic Statistics for Business and Economics with Student

Question

Automobiles arrive at the Elkhart exit of the Indiana Toll Road at the rate of two per minute. The distribution of arrivals approximates a Poisson distribution. a. What is the probability that no automobiles arrive in a particular minute? b. What is the probability that at least one automobile arrives during a particular minute?

Step-by-Step Solution

Answer

a. Probability is 0.1353; b. Probability is 0.8647.

Step 1: Understand the Problem

Given a Poisson distribution with an average rate of arrival \( \lambda = 2 \) automobiles per minute. We need to find probabilities based on this distribution.

Step 2: Use Poisson Formula for No Arrivals

To find the probability of no automobiles arriving in a particular minute, apply the Poisson probability formula: \( P(X=k) = \frac{e^{-\lambda} \lambda^{k}}{k!} \) where \( k = 0 \) and \( \lambda = 2 \).

Step 3: Calculate Probability for No Arrivals

Substitute the values into the Poisson formula: \( P(X=0) = \frac{e^{-2} \times 2^{0}}{0!} = \frac{e^{-2}}{1} \). Use a calculator to find \( e^{-2} \approx 0.1353 \).

Step 4: Determine Probability for At Least One Arrival

The probability of at least one arrival can be found by subtracting the probability of no arrivals from 1: \( P(X \geq 1) = 1 - P(X=0) \).

Step 5: Calculate Probability for At Least One Arrival

Using the calculated value for no arrivals, \( P(X \geq 1) = 1 - 0.1353 = 0.8647 \).

Key Concepts

Probability Calculations

The Poisson distribution is a helpful tool for computing the probability of a given number of events, like automobile arrivals, in a specific period of time. This distribution is particularly suited for situations where the events occur independently and at a constant average rate. Probability calculations can be accomplished using the Poisson probability formula.
The formula is \( P(X=k) = \frac{e^{-\lambda} \lambda^{k}}{k!} \), where \( \lambda \) represents the average rate of occurrences (here, 2 automobiles per minute) and \( k \) is the number of occurrences you are interested in. When \( k = 0 \), you are looking at the probability of no automobile arrivals.

• Plug in \( \lambda = 2 \) and \( k = 0 \) to find the probability of no arrivals.

This straightforward calculation is important, especially in scenarios involving predictions over time, such as traffic flow or entry rates in a queue.

Arrival Rate

The arrival rate, denoted as \( \lambda \), is a fundamental part of the Poisson distribution. It defines the expected number of events, such as cars arriving, in a given period. In our scenario, the arrival rate is 2 cars per minute.
This constant rate allows us to apply the Poisson distribution effectively. When using a Poisson model, the assumption is that the number of cars arriving in different intervals of one minute follows the same distribution pattern.

• The arrival rate helps determine the probability of a range of outcomes, such as zero cars arriving, or more than one.
• It is a crucial parameter that models the randomness of events in time-bound processes.

Understanding \( \lambda \) helps in planning and managing resources effectively; whether it be optimizing toll booth staffing or managing flow at busy exits.

Mathematical Formulae

Mathematical formulae are central to calculating and understanding probabilities in the Poisson distribution model. They not only allow us to compute exact probabilities but also to express them in a language that can be universally understood.
The primary formula for the Poisson distribution is \( P(X=k) = \frac{e^{-\lambda} \lambda^{k}}{k!} \).

• \( e \) is the base of the natural logarithm, approximately equal to 2.71828.
• \( k! \) (k factorial) means multiplying all positive integers up to \( k \).

For instance, when calculating the probability of no arrivals: substitute \( k = 0 \), which simplifies the multiplication due to \( 0! = 1 \). The formula highlights the elegance of mathematics in making complex predictions, like estimating traffic, simple and manageable.