Problem 6 - A Modern Introduction to Probability and Statistics: Understanding Why and How

Question

A certain brand of copper wire has flaws about every 40 centimeters. Model the locations of the flaws as a Poisson process. What is the probability of two flaws in 1 meter of wire?

Step-by-Step Solution

Answer

The probability of two flaws in 1 meter is approximately 0.1288.

Step 1: Understand the Problem

In this problem, we're looking to find the probability that there are exactly two flaws in a 1-meter length of copper wire given that flaws occur at an average rate of one flaw every 40 centimeters.

Step 2: Define Parameters of the Poisson Process

Since flaws occur every 40 cm on average, in 1 meter (100 cm), the expected number of flaws is \( \lambda = \frac{100}{40} = 2.5 \). This is the rate (\( \lambda \)) for our Poisson distribution over 1 meter of wire.

Step 3: Apply the Poisson Probability Formula

The probability of observing \( k \) events in a Poisson process is given by \( P(k; \lambda) = \frac{e^{-\lambda} \lambda^k}{k!} \). Here, \( k = 2 \) and \( \lambda = 2.5 \).

Step 4: Calculate the Probability

Substitute the values into the formula: \[P(2; 2.5) = \frac{e^{-2.5} \times 2.5^2}{2!}\]Calculate each of the components:- \( e^{-2.5} \approx 0.0821 \)- \( 2.5^2 = 6.25 \)- \( 2! = 2 \)The probability is:\[P(2; 2.5) = \frac{0.0821 \times 6.25}{2} \approx 0.1288\]

Step 5: Conclude the Calculation

The probability that there are exactly two flaws in a 1-meter segment of copper wire is approximately 0.1288.

Key Concepts

Probability Theory

Probability theory is a branch of mathematics that deals with the analysis of random events. It helps us understand how likely events are to occur. Probabilities range from 0 to 1, where 0 means the event will not happen and 1 means it certainly will. In this context:

• Random Experiment: Any process that can result in different outcomes, like flipping a coin or observing flaws in wire.
• Event: A single result or a set of results of a random experiment.
• Probability: A measure assigned to each event in a space that indicates the likelihood of an event occurring.

In the exercise, we're analyzing the probability that two flaws occur in a 1-meter wire. This involves understanding both the rate at which flaws appear and the specific event (finding two flaws) we're interested in. When dealing with random processes that occur over time or space, like flaws in a wire, probability theory often uses statistical models to describe these phenomena.

Statistical Modelling

Statistical modeling is a powerful tool that helps in summarizing complex real-world data processes. It involves building models or representations of systems to analyze data and make predictions.1. **Understanding the Real World Scenario:**
The problem scenario involves flaws in copper wire. Such random flaws are well-suited to statistical modeling because they can be analyzed as a random process. 2. **Choosing the Right Model:**
A Poisson process is chosen here because it appropriately describes events occurring independently within a fixed space, like flaws per meter of wire.3. **Parameter Estimation:**
The parameter in a Poisson model is typically the average rate (\( \lambda \)) of occurrence. This exercise estimates an average rate of 2.5 flaws per meter based on given flaw intervals.4. **Model Application:**
The model is then used to find probabilities of different outcomes, guiding real-world predictions or decisions based on statistical inference.

Poisson Distribution

The Poisson distribution is a statistical distribution that expresses the probability of a given number of events happening in a fixed interval of time or space. The key characteristics are:

• Random and Independent Events: Events occur independently of one another.
• Constant Average Rate: Events occur at a constant average rate known as \( \lambda \).
• Discrete Outcomes: The number of events can only be whole numbers (0, 1, 2, ...).

In this exercise, the Poisson distribution is utilized to predict the likelihood of finding exactly two flaws in a meter of copper wire:- **Setting the Scene:**
The model's parameter (\( \lambda = 2.5 \)) is crucial, representing the average number of flaws per meter.- **Applying the Formula:**
We use the formula \( P(k; \lambda) = \frac{e^{-\lambda} \lambda^k}{k!} \) to find specific probabilities.
Here \( k \) is the number of events we are interested in, which is 2 flaws.This approach provides the basis for using the Poisson distribution to explore and solve real-world problems involving rare events over a given space or time. By calculating these probabilities, decisions can be made about quality control, risk assessments, and planning in various fields.