Problem 24 - Algebra and Trigonometry

Question

Quality Control An assembly line that manufactures fuses for automotive use is checked every hour to ensure the quality of the finished product. Ten fuses are selected randomly, and if any one of the ten is found to be defective, the process is halted and the machines are recalibrated. Suppose that at a certain time 5\(\%\) of the fuses being produced are actually defective. What is the probability the assembly line is halted at that hour's quality check?

Step-by-Step Solution

Answer

The probability the assembly line is halted is 40.13%.

Step 1: Define the Problem

We need to find the probability of finding at least one defective fuse from a sample of ten, considering 5% of the fuses are defective. This results in halting the assembly line for recalibration.

Step 2: Recognize the Concept

This is a binomial probability problem because we have a fixed number of trials (10 fuses), two possible outcomes for each fuse (defective or not defective), and a constant probability of defectiveness (5%). We need to find the probability of at least one success (finding a defective fuse).

Step 3: Binomial Probability Formula

The binomial probability formula is given by \( P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \), where \( n \) is the number of trials, \( k \) is the number of successes, and \( p \) is the probability of success. Here, \( n = 10 \), \( p = 0.05 \).

Step 4: Calculate Probabilities for Zero Defects

We first calculate the probability of finding no defective fuses (0 defects). This is \( P(X = 0) \): \[ P(X = 0) = \binom{10}{0} (0.05)^0 (0.95)^{10} = 1 \times 0.95^{10} \approx 0.5987. \]

Step 5: Compute Complement Probability

The probability that the line is halted (at least 1 defective fuse) is the complement of finding no defective fuses. Therefore, \[ P(X \geq 1) = 1 - P(X = 0) = 1 - 0.5987 = 0.4013. \]

Step 6: Conclusion

Based on the calculations, there is a 40.13% probability that at least one fuse is defective in a sample of ten, leading to the assembly line being halted.

Key Concepts

Quality Control and Its Importance

Quality control is a crucial aspect of modern manufacturing, especially in industries like automotive production. Ensuring the quality of products not only maintains a company’s reputation but also ensures the safety and satisfaction of consumers. In the context of the problem, quality control is achieved by regularly checking products, such as fuses, for defects.
It involves assessing samples from the production line to confirm product standards are met. If a defect is detected, corrective action is taken immediately.

• Regular checks maintain production consistency.
• It helps identify when a process needs adjustment, reducing waste and costs in the long run.
• Protects customers from faulty products.

Maintain a rigorous quality control system, and you will minimize the risk of distributing defective items.

Understanding Defective Items

Defective items are those that fail to meet the set quality standards. In manufacturing, identifying defects is a critical step to ensure that products perform their intended function effectively. The exercise focuses on detecting defective fuses out of a batch, with a defect rate of 5%.
Defects can occur due to various reasons, such as:

• Imprecise machinery settings.
• Faulty materials used in production.
• Human errors during assembly.

Identifying and understanding the causes of defects allows for targeted troubleshooting and prevention of future errors. By analyzing batch samples, companies can determine defect rates, helping gauge how often defects occur and under what conditions.

Probability Calculation Explained

Probability calculations can seem complex, but they are invaluable for predicting outcomes in quality control. In this exercise, we're calculating the probability that at least one defective fuse will be found amongst a sample of ten fuses.
This exercise uses the binomial probability formula, which is ideal for scenarios with fixed numbers of trials and binary outcomes (defective or not). The probability of finding no defective fuses in this scenario was calculated first, leading to the complementary probability of halting the assembly line.

• The formula used is: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \] where \( n = 10 \), \( p = 0.05 \).
• The probability of finding zero defective items is about 59.87%.
• The complementary probability (line halted) is 40.13%.

Understanding these calculations empowers decision-makers to take proactive steps based on data-driven insights.

Assembly Line Production Insights

Assembly line production is a method of manufacturing where products are assembled in a sequence of steps. Each step adds specific parts or performs specific functions until the final product is complete.
Key advantages of assembly line production include:

• Increased production speed due to repetitive tasks.
• Reduced production costs as a result of automation and efficiency.
• Consistent product quality due to standardized procedures.

In the exercise, the assembly line produces fuses, which need to be consistently high-quality to ensure the safety of automotive systems. Frequent checks mitigate the risk of passing defective units further down the line, which could result in costly customer returns or safety issues. Understanding how assembly lines operate helps in understanding the importance of quality checks and defect prevention.