Problem 28 - Algebra Form and Function

Question

A high-tech company makes silicon wafers for computer chips, and tests them for defects. The test identifies \(90 \%\) of all defective wafers, and misses the remaining \(10 \% .\) In addition, it misidentifies \(20 \%\) of all non- defective wafers as being defective. (a) Suppose 5000 wafers are made. Of the \(5 \%\) of these wafers that contain defects, how many are correctly identified by the test as being defective? (b) How many of the non-defective wafers are incorrectly identified by the test as being defective? (c) What is the probability, given as a percentage, that a wafer identified as defective is actually defective?

Step-by-Step Solution

Answer

Answer: 225 wafers are correctly identified as defective. 2. How many of the non-defective wafers are incorrectly identified as defective by the test? Answer: 950 non-defective wafers are incorrectly identified as defective. 3. What is the probability, in percentage, that a wafer identified as defective is actually defective? Answer: The probability is 19.15%.

Step 1: ( Step 1: Calculate Defective and Non-Defective Wafers )

As the problem states that \(5 \%\) of the wafers are defective, we first need to identify how many wafers are defective and how many are non-defective among the total number of 5000 wafers. Defective wafers: \(5000\times (5/100)=250\) Non-defective wafers: \(5000-250=4750\)

Step 2: ( Step 2: Calculate Correctly Identified Defective Wafers )

The test identifies \(90 \%\) of all defective wafers correctly. Therefore, of the 250 defective wafers, we need to find how many are correctly identified as defective: Correctly identified defective wafers: \(250\times (90/100)=225\)

Step 3: ( Step 3: Calculate Incorrectly Identified Non-Defective Wafers )

The test misidentifies \(20 \%\) of all non-defective wafers as defective. Therefore, of the 4750 non-defective wafers, we need to find how many are incorrectly identified as defective: Mistakenly identified non-defective wafers: \(4750\times (20/100)=950\)

Step 4: ( Step 4: Calculate Probability of Wafer Identified as Defective is Actually Defective )

To calculate the probability that a wafer identified as defective is actually defective, we first need to find the total number of wafers identified as defective (true positives and false positives): Total identified defective wafers: \(225+950=1175\) Now, we calculate the probability that a wafer identified as defective is actually defective: Probability (percentage) = \((225/1175) \times 100 = 19.15 \%\)

Step 5: ( Answer )

(a) Among the 5000 wafers, with a \(5 \%\) defect rate, 225 wafers are correctly identified by the test as being defective. (b) 950 non-defective wafers are incorrectly identified as defective by the test. (c) The probability that a wafer identified as defective is actually defective is \(19.15 \%\).

Key Concepts

Defective Wafers

In the production of silicon wafers for computer chips, defects can occur due to various reasons including material imperfections or processing errors. The identification of these defective wafers is crucial as they can affect the performance and reliability of the final product. In this exercise, we are given that 5% of the wafers produced, from a total of 5000, are defective. This percentage represents the initial condition before any testing is done to confirm which are indeed defective.
To assess the defective wafers, a testing system is used. The system is set up to correctly identify 90% of the defective wafers, known as the true positive rate.
Here is a quick breakdown of the numbers involved:

• Total Wafers Produced: 5000
• Defective Wafers: 250 (because 5% of 5000 is 250)
• Correctly Identified Defective Wafers: 225 (90% of 250)

Thus, the test successfully identifies the majority of defective wafers, but a small fraction (10%) may go unnoticed. This undetected portion represents the false negatives. Understanding and analyzing defective wafer statistics is key to improving testing methods and enhancing production quality.

False Positives

False positives are an essential concept in conditional probability, particularly when testing for defects in products. In this scenario, a false positive refers to the non-defective wafers that have been incorrectly identified as defective by the testing process. This type of error can lead to unnecessary wastage and increased costs, as good wafers might be mistakenly discarded.
For the 4750 non-defective wafers, the test incorrectly labels 20% of them as defective:

• Non-Defective Wafers: 4750
• Falsely Identified as Defective: 950 (20% of 4750)

False positives can significantly impact the yield of production lines as valuable time and resources are wasted on addressing unfounded defects. Mitigating false positives is vital for enhancing testing accuracy and maintaining cost efficiency.

True Positives

The term "true positives" in defect detection refers to the number of defective wafers that are correctly identified by the testing setup. True positives are crucial because they measure the effectiveness of a testing procedure, indicating how well the test can correctly confirm defects. In probabilities, this is frequently represented by the sensitivity rate of a test or its true positive rate.
From a total of 250 defective wafers, the system correctly identifies 90%, which translates into:

• True Positives (Correctly Identified Defective Wafers): 225

Achieving a high true positive rate ensures that the majority of defective items are caught before they proceed further down the production line or reach the consumer. This capability is fundamental to quality assurance because missing a defect (which occurs in the complementary false negative scenario) can have reputational and financial repercussions.
Understanding how true positives interact with false positives helps in making more informed decisions about improving the reliability and accuracy of testing systems.