Problem 208 - Applied Statistics and Probability for Engineers

Question

Asbestos fibers in a dust sample are identified by an electron microscope after sample preparation. Suppose that the number of fibers is a Poisson random variable and the mean number of fibers per square centimeter of surface dust is \(100 .\) A sample of 800 square centimeters of dust is analyzed. Assume that a particular grid cell under the microscope represents \(1 / 160,000\) of the sample. (a) What is the probability that at least one fiber is visible in the grid cell? (b) What is the mean of the number of grid cells that need to be viewed to observe 10 that contain fibers? (c) What is the standard deviation of the number of grid cells that need to be viewed to observe 10 that contain fibers?

Step-by-Step Solution

Answer

(a) 39.35% probability for at least one fiber, (b) mean is 25.43 cells, (c) standard deviation is 7.5 cells.

Step 1: Finding the Grid Cell Area

Each grid cell under the microscope represents \(\frac{1}{160,000}\) of the sample. The total area of the sample is 800 square centimeters. Therefore, the area of one grid cell is \(\frac{800}{160,000}\) square centimeters.

Step 2: Calculate Mean Number of Fibers in a Grid Cell

The mean number of fibers per square centimeter is 100, so for the grid cell, the mean is \(100 \times \frac{800}{160,000}\). Simplify it to find \(\lambda = \frac{1}{2}\).

Step 3: Calculate Probability Using Poisson Distribution

Use the Poisson distribution formula, where \(P(X = 0) = \frac{e^{-\lambda} \cdot \lambda^0}{0!}\). Substitute \(\lambda = \frac{1}{2}\) to find \(P(X=0)\).

Step 4: Determine Probability of At Least One Fiber

The probability of at least one fiber is \(P(X \geq 1) = 1 - P(X = 0)\). Calculate it using the probability found in the previous step.

Step 5: Determine Distribution for Observing Grid Cells with Fibers

For parts (b) and (c), use a geometric distribution. The probability of a fiber in a grid cell is the complement of no fiber, where \(p = 1 - P(X=0)\) calculated before.

Step 6: Mean of Grid Cells to Observe 10 with Fibers (Part b)

The mean for a geometric distribution of observing 10 successes is given by \(\frac{10}{p}\), where \(p\) is the probability calculated in step 4.

Step 7: Standard Deviation of Grid Cells to Observe 10 with Fibers (Part c)

The standard deviation for the geometric distribution of observing 10 cells is given by \(\sqrt{\frac{10 \cdot (1-p)}{p^2}}\), using the \(p\) found earlier.

Key Concepts

Probability Theory

Probability theory helps us understand the chances of events happening. It's the foundation of statistics and is used to analyze random events. In this exercise, we look at calculating probabilities using different distribution models.
Like the Poisson distribution which is used to model the number of times an event happens in a fixed interval of time or space. Probability theory involves:

• Random Variables: These are variables that consist of possible outcomes from a random phenomenon.
• Distribution of Variables: This refers to how probabilities are assigned over different outcomes.

Understanding how these variables behave allows us to calculate the likelihood of certain events and make informed decisions based on those probabilities.

Geometric Distribution

The geometric distribution is a probability distribution that models the number of trials until the first success in a series of Bernoulli trials, where each trial has the same probability of success. It's useful when we're dealing with situations like flipping a coin multiple times until we see heads. In this exercise, it helps us understand how many grid cells we need to observe, until we see a certain number of grid cells containing asbestos fibers. Here’s why it’s important:

• It models scenarios where you repeat an experiment until a defined number of successes are achieved.
• Each trial is independent, which means the result of one does not affect the others.

For example, part (b) and (c) of the exercise uses geometric distributions to calculate the mean and standard deviation for observing 10 cells containing fibers.

Random Variables

Random variables are a fundamental concept in probability theory. They act as functions that assign numerical values to the outcomes of a random process. In simpler terms, they help quantify uncertainty. There are mainly two types:

• Discrete Random Variables: These take on a countable number of values, like rolling a dice.
• Continuous Random Variables: These can take on any value within a given continuum or interval, like measuring rainfall.

In our particular problem, the number of asbestos fibers is treated as a discrete random variable. We can calculate the probability of observing a certain number of fibers using distributions, like Poisson and geometric distributions.

Asbestos Fibers Analysis

Asbestos fibers analysis is a process of determining the concentration and distribution of asbestos particles in a sample. These fibers are microscopic, often requiring an electron microscope to identify them accurately. In the given exercise, we have an analysis process which involves:

• Sample Preparation: Dust samples are prepared for microscopic examination.
• Microscopic Analysis: Electron microscopes are used to identify asbestos fibers.
• Statistical Modeling: Using Poisson distribution to calculate the likelihood of fibers being present in specific areas.

Understanding the distribution of asbestos fibers can help in assessing exposure risks and ensuring safety in environments where asbestos is present.