Bayesian Approach to Fault Analysis in Manufacturing

In manufacturing industries, it is essential to detect and analyze faults at an early stage to ensure the quality and reliability of the products. One common scenario is determining the probability that a defective product was made by a specific machine. In this article, we will discuss the application of Bayes' Theorem to solve such a problem. We will utilize the provided data to calculate the probability that a defective product was made by machine B3.

Introduction to Manufacturing and Defects

Manufacturing processes are inherently prone to faults. These faults can lead to the production of defective products. In the context of our scenario, three machines (B1, B2, and B3) produce products in different proportions. The machines have different defect rates as well. Our goal is to find out the probability that a defective product was produced by machine B3.

Given Information

The proportion of products made by each machine: P(B1) 0.30 (30%) P(B2) 0.45 (45%) P(B3) 0.25 (25%) The probability of a product being defective given the machine: P(D|B1) 0.02 (2%) P(D|B2) 0.03 (3%) P(D|B3) 0.02 (2%)

Step 1: Calculate the Total Probability of a Defective Product (PD)

Using the law of total probability, we can find the total probability of a product being defective:

PD P(D|B1) * P(B1) P(D|B2) * P(B2) P(D|B3) * P(B3)

Substituting the values:

PD 0.02 * 0.30 0.03 * 0.45 0.02 * 0.25

Calculating each term:

0.02 * 0.30 0.006

0.03 * 0.45 0.0135

0.02 * 0.25 0.005

Adding them together:

PD 0.006 0.0135 0.005 0.0245

Step 2: Calculate the Posterior Probability (PB3|D) using Bayes' Theorem

Bayes' Theorem is given by:

P(B_3|D) P(D|B_3) * P(B_3) / PD

Substituting the values we have:

P(B_3|D) (0.02 * 0.25) / 0.0245

Calculating the numerator:

0.02 * 0.25 0.005

Now substituting back into Bayes' Theorem:

P(B_3|D) 0.005 / 0.0245 ≈ 0.2041

Final Result

The probability that a defective product was made by machine B3 is approximately 0.2041 or 20.41%.

So, we can say that there is a 20.41% chance that a defective product was produced by machine B3, given the provided data.

Conclusion

Bayes' Theorem is a powerful tool in solving real-world problems involving conditional probabilities. By utilizing the given proportions and defect rates, we can accurately determine the likelihood of a faulty product originating from a specific machine. This method has significant applications in manufacturing, where early detection of potential issues can prevent costly problems and ensure product quality.