Components are manufactured by machines A and B. Machine A makes \(55 \%\) of the components. Of those components made by machine \(\mathrm{A}, 7 \%\) are unacceptable; of those made by machine \(\mathrm{B}, 5 \%\) are unacceptable. A component is picked at random. Calculate the probability that it is (a) made by machine \(\mathrm{B}\) (b) acceptable (c) acceptable and made by machine \(\mathrm{A}\) (d) unacceptable given it is made by machine \(\mathrm{B}\) (e) made by machine A given it is unacceptable.

Understanding conditional probability is crucial when we're dealing with dependent events, where the occurrence of one event affects the likelihood of another. In our example, we want to find the probability of an event, given that another event has occurred. This is written as P(A|B), meaning the probability of A given B.

For instance, to find the probability that a component is unacceptable given it was made by machine B, we apply the formula: P(Unacceptable|B) = 0.05. This represents the proportion of machine B's components that are defective. In conditional probability, the emphasis is on the dependency between two events and how the outcome of one influences the probability of the other.

The total probability rule helps us find the likelihood of a particular outcome based on several distinct possibilities. Essentially, it's a way to account for all the different ways an event can occur. In this problem, we use the total probability rule to calculate the chance that a component is acceptable.

To achieve this, we consider the probability of a component being acceptable from each machine, then multiply by the probability that the component came from that machine, and sum these values up. The formula looks like this: P(Acceptable) = P(Acceptable|A)P(A) + P(Acceptable|B)P(B), giving us \(0.93 \times 0.55 + 0.95 \times 0.45 = 0.9405\). This rule is a powerful tool for managing complex probability scenarios involving multiple sources or processes, like our machines A and B.

When we refer to the probability of acceptance, we're looking for the likelihood that an item meets a certain standard or criteria - in this case, that a component is not defective. This probability is crucial for quality control in manufacturing.

Utilizing the total probability rule, as described above, we determined the probability of a component being acceptable overall. The result, \(P(Acceptable) = 0.9405\), tells us that roughly 94% of all components from both machines are likely to be acceptable. This high percentage indicates a reasonably low defect rate in the production process, something that would be reassuring in a real manufacturing setting. Understanding this probability can help businesses make informed decisions about their manufacturing processes and quality assurance protocols.