Components are manufactured by three machines, \(\mathrm{A}, \mathrm{B}\) and \(\mathrm{C}\). Machine A makes \(30 \%\) of the components, machine B makes \(25 \%\) of the components and machine \(\mathrm{C}\) makes the rest. Of those components made by machine A, \(6 \%\) are substandard; when made by machine B, \(3 \%\) are substandard; and when made by machine C, \(5 \%\) are substandard. A component is picked at random. Calculate the probability that it is (a) substandard (b) made by machine B given it is substandard (c) made by either machine \(\mathrm{A}\) or machine \(\mathrm{B}\) (d) substandard and made by machine B (e) substandard, given it is made by machine \(\mathrm{A}\) (f) made by machine A or is substandard.

Conditional probability is a fundamental concept in probability theory, which expresses how the probability of an event changes when another related event is known to occur. In simpler terms, it tells us the probability of an event happening, given that another event has already taken place.

For example, when trying to determine the likelihood of a component being substandard from machine A (\(P(S|A)\)), we are dealing with conditional probability. It allows engineers to consider the specific context of one machine's output and gives them precise information about the quality of components from that machine. The formula for conditional probability is outlined as follows: \[\begin{equation} P(A|B) = \frac{P(A \cap B)}{P(B)} \end{equation}\]where \(P(A|B)\) is the probability of event A given event B has occurred, and \(P(A \cap B)\) represents the probability of both events A and B occurring.

• Understand that the sum of probabilities for all possible events equals 1, and conditional probabilities adhere to this rule within their specific contexts.
• Consider conditional probability as a means of updating our beliefs about the likelihood of an event based on new information.

Applying conditional probability is not only key in textbook exercises but also critical in real-world applications where outcomes depend on given conditions, such as quality control in manufacturing processes.

Bayes' theorem, named after the Reverend Thomas Bayes, is a powerful formula used to update probabilities when new evidence is introduced. It is an application of conditional probability and is written as follows: \[\begin{equation} P(A|B) = \frac{P(B|A)P(A)}{P(B)} \end{equation}\]Bayes' theorem allows us to reverse the conditionality; if we're seeking the probability that a component is made by machine B given it is substandard \(P(B|S)\), we can calculate it using the likelihood of a component being substandard given it's made by machine B \(P(S|B)\) and the overall probability of a component being substandard \(P(S)\).

In the provided exercise, utilizing Bayes' theorem helps students not to just understand the abstract formula but to see its practical calculation in action. This real-world connection solidifies the theorem's fundamental role in decision-making processes within engineering fields, where gauging the impact of new information on existing beliefs is crucial.

Union of events probability refers to the likelihood of at least one of multiple events happening. This concept is essential when outcomes are not mutually exclusive and can occur simultaneously. The probability of the union of events A and B is represented by \(P(A \cup B)\) and can be calculated using the formula: \[\begin{equation} P(A \cup B) = P(A) + P(B) - P(A \cap B) \end{equation}\]It ensures we do not double-count the chances of overlapping events, such as a component being made by machine A or B, or a component being substandard. This calculation instructs us to add the individual probabilities and subtract the probability of both events occurring together, which has been included in both individual probabilities.

• To avoid confusion, visualize Venn diagrams to see how probabilities overlap between events.
• Remember that the union probability cannot exceed \(1\) as it represents the complete set of outcomes.

Understanding the union of events probability is invaluable, especially when analyzing complex systems with multiple sources or pathways that might lead to a particular outcome.