Silicon chips are manufactured by four machines, A, B, C and D. Machines A, B, C and D manufacture \(20 \%, 25 \%, 35 \%\) and \(20 \%\) of the components respectively. Of those silicon chips manufactured by machine A, \(2.1 \%\) are faulty. The respective figures for machines \(\mathrm{B}, \mathrm{C}\) and \(\mathrm{D}\) are \(3 \%, 1.6 \%\) and \(2.5 \%\). A silicon chip is selected at random. Calculate the probability that it is (a) made by machine \(\mathrm{C}\) and is faulty (b) made by machine \(\mathrm{A}\) and is not faulty (c) faulty.

Understanding conditional probability is essential for solving many real-world problems, including quality control in manufacturing processes. Conditional probability essentially answers the question ‘What is the probability of event A occurring given that event B has already occurred?’. Mathematically, this is represented as P(A|B) and is calculated using the formula P(A and B) / P(B), provided P(B) is not zero.

In the context of our silicon chip manufacturing problem, the probability of a chip being faulty given it was made by machine C, which is denoted as P(Faulty|C), is a classic example of conditional probability. Having these conditional probabilities for each machine allows us to address complex questions about the manufacturing process, such as the overall rate of defective products, which is critical for ensuring quality and efficiency.

The application of probability in engineering fields, such as manufacturing, is crucial for making informed decisions and predictions about the performance and reliability of systems and components. In engineering, probability helps to model uncertainty and manage risk. For example, determining the probability of a silicon chip being faulty involves understanding both the production rate of each machine and the likelihood of producing a defective item.

By utilizing probabilities, engineers can assess the effectiveness of different machines, plan for maintenance, manage inventory, and set quality control standards. This probabilistic approach enables engineers to minimize defects, optimize production schedules, and save costs, thereby improving the overall performance of the manufacturing system.

Probability of independent events is another fundamental concept, particularly when the events in question do not influence one another. When two events, A and B, are independent, the probability of both events occurring is simply the product of their individual probabilities: P(A and B) = P(A) * P(B).

However, it's essential to distinguish between independent and non-independent (dependent) events to correctly calculate probabilities. In our textbook exercise, the event of selecting a chip made by a specific machine (A, B, C, or D) and the event of that chip being faulty are treated as independent. But in reality, if a certain machine starts producing more faulty chips, this might imply a dependence that would require a revised approach to probability calculations. Recognizing such dependencies is critical for accurate predictions and decision-making in various applications involving probability.