Problem 125 - Applied Statistics and Probability for Engineers

Question

The edge roughness of slit paper products increases as knife blades wour, Only \(1 \%\) of products slit with ncw bladcs have rough edges, \(3 \%\) of products slit with blades of average sharpness exhibit roughness, and \(5 \%\) of products slit with worn bladcs cxhibit roughncss. If \(25 \%\) of the bladcs in manufacturing are ncw, \(60 \%\) are of average sharpncss, and \(15 \%\) are worn, what is the proportion of products that exhibit edge roughness?

Step-by-Step Solution

Answer

The proportion of products with edge roughness is 2.8%.

Step 1: Identify the Given Probabilities

We are given that:- The probability of a product having rough edges with new blades is 1%, denoted as \( P(R|N) = 0.01 \).- The probability with average sharpness blades is 3%, denoted as \( P(R|A) = 0.03 \).- The probability with worn blades is 5%, denoted as \( P(R|W) = 0.05 \).- The probability of a blade being new is 25%, \( P(N) = 0.25 \).- The probability of a blade being of average sharpness is 60%, \( P(A) = 0.60 \).- The probability of a blade being worn is 15%, \( P(W) = 0.15 \).

Step 2: Apply the Law of Total Probability

To find the overall probability that a product exhibits edge roughness, you must use the Law of Total Probability:\[ P(R) = P(R|N)P(N) + P(R|A)P(A) + P(R|W)P(W) \]

Step 3: Substitute the Given Values into the Formula

Insert the values into the Total Probability formula: \[ P(R) = (0.01 \times 0.25) + (0.03 \times 0.60) + (0.05 \times 0.15) \]

Step 4: Calculate Each Term Separately

Compute the products for each term:1. For new blades: \( 0.01 \times 0.25 = 0.0025 \).2. For average sharpness blades: \( 0.03 \times 0.60 = 0.018 \).3. For worn blades: \( 0.05 \times 0.15 = 0.0075 \).

Step 5: Sum the Results

Add the results of each term to find \( P(R) \):\[ P(R) = 0.0025 + 0.018 + 0.0075 = 0.028 \]

Step 6: Interpret the Result

The total probability that a product exhibits rough edges is 0.028, or 2.8%.

Key Concepts

Law of Total Probability

The Law of Total Probability is essential when determining the overall probability of an event that can occur in multiple distinct ways. Imagine you have several paths leading to the same destination; the law helps you calculate the probability of reaching that destination through any of those paths.
In our exercise, the destination is the product having rough edges, denoted as event \( R \). The paths leading to this are the states of the knife blades: new, average sharpness, or worn.
The formula for the Law of Total Probability sums up the probabilities of the event \( R \) occurring through each path:

• \( P(R|N) \times P(N) \): Rough edges with new blades
• \( P(R|A) \times P(A) \): Rough edges with average sharpness blades
• \( P(R|W) \times P(W) \): Rough edges with worn blades

To find \( P(R) \), each path's probability is multiplied by the chance of that path (blade state) and then all products are summed. This offers a comprehensive view of the overall probability as we've done in the original solution.

Conditional Probability

Conditional Probability helps us assess the likelihood of an event occurring given another event has already occurred. In simple terms, it's about understanding the probability of an event under certain conditions or circumstances.When we look at the problem, we assess the probability of a product having rough edges when given specific conditions of blade sharpness. For instance, \( P(R|N) \) indicates the probability that a product will have rough edges (\( R \)) given that the blades are new (\( N \)).This form of reasoning is fantastic for practical applications like manufacturing, where each condition (in this case, the sharpness of the blade) affects the outcome (edge roughness). By using conditional probabilities, we can predict and improve the processes based on different scenarios or conditions.

Probability in Manufacturing

In manufacturing, probability plays a vital role in quality control and operational efficiency. Imagine a factory producing items with different machinery states, like new, average, and worn-out equipment. Each machine's condition influences the quality of the final product. For blade sharpness:

• New blades result in a 1% chance of rough edges
• Average blades increase this to 3%
• Worn blades escalate it to 5%

This information helps manufacturers anticipate and rectify potential issues. By calculating the total probability of producing rough-edged products, the factory can strategize maintenance schedules or quality checks, minimizing defects and improving product standards. Using probability in manufacturing enables informed decisions, ensuring efficiency while maintaining high-quality outputs.