Problem 124 - Applied Statistics and Probability for Engineers

Question

Suppose \(2 \%\) of cotton fabric rolls and \(3 \%\) of nylon fabric rolls contain flaws. Of the rolls uscd by a manufacturcr. \(70 \%\) arc cotton and \(30 \%\) are nylon. What is the probability that a randomly selected roll used by the manufacturer contains flaws?

Step-by-Step Solution

Answer

The probability that a randomly selected roll contains flaws is 2.3%.

Step 1: Define the Probabilities

Let the probability that a roll is made of cotton be \(P(C) = 0.7\) and the probability that a roll is made of nylon be \(P(N) = 0.3\). The probability that a cotton roll contains flaws is \(P(F|C) = 0.02\), and the probability that a nylon roll contains flaws is \(P(F|N) = 0.03\).

Step 2: Apply Total Probability Theorem

We need to find the probability that a roll contains flaws, \(P(F)\). Using the law of total probability, we calculate \(P(F)\) as follows: \[ P(F) = P(F|C)P(C) + P(F|N)P(N) \].

Step 3: Calculate Each Component

Substitute the given probabilities into the equation: \[ P(F) = (0.02)(0.7) + (0.03)(0.3) \].

Step 4: Perform the Calculations

Calculate \(0.02 \times 0.7 = 0.014\) and \(0.03 \times 0.3 = 0.009\). Add these results: \[ P(F) = 0.014 + 0.009 = 0.023 \].

Step 5: Interpret the Result

The probability that a randomly selected roll contains flaws is \(2.3\%\).

Key Concepts

Total Probability Theorem

The Total Probability Theorem is a crucial concept in probability theory. It helps us find the probability of an event by considering all possible ways that event can happen. In the context of our exercise, we want to find out the probability that any roll of fabric has flaws. We do this by accounting for both types of fabric: cotton and nylon.

Here's how it works:

• First, we know that each roll could either be cotton or nylon, and each type has a different probability of having flaws.
• Next, we multiply the probability of choosing a cotton roll with the probability that a cotton roll has flaws, and do the same for nylon rolls.
• Finally, we sum these probabilities to find the overall probability of a roll having flaws.

This method ensures that we're considering all possible sources of flaws accurately.

Conditional Probability

Conditional probability is used to find the probability of an event, given that another event has already occurred. In our exercise, we are using conditional probabilities to determine the flaw probabilities for each type of fabric roll.

Let's break it down:

• For a cotton roll, the probability that it has flaws given that it is cotton is written as \(P(F|C) = 0.02\). This reads as "the probability of flaws given cotton is 2%".
• Similarly, for a nylon roll, the conditional probability \(P(F|N) = 0.03\) means "the probability of flaws given nylon is 3%".

These conditional probabilities are essential components of our calculations and allow us to distinguish between the different flaw probabilities for cotton and nylon.

Bayes' Theorem

Bayes' Theorem provides a way to update our knowledge about the probability of an event based on new evidence. Although this exercise does not directly compute probabilities using Bayes' Theorem, understanding it helps us see how prior knowledge can be combined with new data.

In scenarios similar to our exercise, Bayes' Theorem could be used if we discovered new information about additional rolls or other qualities about the fabric. It assists in recalibrating probabilities when given new context or data.

• This theorem is especially useful in complex statistical models and decision-making processes.
• It tweaks and updates the probability of hypotheses considering new insights.

Bayes' overarching principle is that it's a way to learn from data and update our understanding as more information becomes available.

Applied Statistics

Applied statistics involves using statistical methods to solve real-world problems and make informed decisions. In our exercise, we apply the principles of probability to determine outcomes relevant to a manufacturer.

Some common uses of applied statistics include:

• Quality control in manufacturing, as shown in our example where mistaking fabric flaws could impact product quality.
• Making predictions or identifying trends based on data analysis, such as estimating the frequency of flawed rolls.
• Conducting surveys and experimental designs in various industries.

By understanding these principles, manufacturers can anticipate issues such as flaws and make adjustments in their production processes to improve overall quality and efficiency. Applied statistics provides the practical tools to use theoretical knowledge effectively.