A Machine Producing Defective Bolts and Probability Calculations

Understanding the production and probability of defective bolts is crucial in quality control for any manufacturing process. The scenario where a machine produces a large batch, such as 250 bolts with 50 being defective, provides insights into the probability of drawing defective items from that batch. This analysis can be useful for predicting and mitigating issues in manufacturing, ensuring product quality, and improving overall efficiency.

Introduction to the Scenario

Consider a scenario where a machine produces 250 bolts, out of which 50 are found to be defective. This implies that 200 bolts are non-defective. If we randomly pick two bolts from this batch with replacement, we can calculate the probability of both picking defective bolts.

Calculating the Probability with Replacement

The probability of the first bolt being defective is given by:

$$frac{50}{250} frac{1}{5}$$

Since the process involves replacement, the probability of picking a defective bolt on the second draw remains the same, which is again (frac{1}{5}).

The probability that both bolts are defective is calculated as:

$$frac{1}{5} times frac{1}{5} frac{1}{25}$$

This result provides us with the probability that both randomly selected bolts will be defective.

Alternative Calculation Without Replacement

Alternatively, if the bolts are not replaced after the first pick, the probability changes as follows:

$$frac{50}{250} times frac{49}{249} frac{1225}{31125} approx 0.039357$$

This calculation shows the adjusted probability of picking two defective bolts without replacement. The reduction in the total number of bolts and defective bolts after the first draw influences the overall probability.

Implications in Quality Control

Without a functioning machine, it is impractical to test the bolts for quality. Therefore, it is essential for the manufacturer to address the issue of defective bolts promptly. Quality control measures such as regular machine maintenance, inspection, and testing can help in identifying and fixing the cause of excessive defectivity.

Monitoring the production process is crucial to ensure that defective rates remain within acceptable limits. If a machine consistently produces a high number of defective items, it may need to be fixed or replaced. This proactive approach helps in maintaining product reliability and customer satisfaction.

Conclusion

Predicting the probability of defective items in a batch is essential for maintaining quality standards. Whether through calculation with or without replacement, understanding these probabilities can help manufacturers identify and address issues in their production process. Regular maintenance and quality control measures can significantly reduce the number of defective items, ensuring a higher quality product and increased customer trust.

Keywords: probability, machine production, defective parts, quality control, reliability