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The addition rule for probabilities is a fundamental concept in probability theory. It helps us calculate the likelihood of either one of two events occurring. If you have two events, say Event A and Event B, the rule is expressed as follows:\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]This equation states that the probability of either Event A or Event B occurring is equal to the sum of their individual probabilities, subtracting the probability that both events occur simultaneously. It accounts for the double-counting of the overlap where both events happen at the same time. Using this rule, we can find probabilities of combined events in situations like the pump system scenario where you want to know about any failures.

Conditional probability is used to determine the likelihood of an event occurring, given that another event has already occurred. In symbols, the conditional probability of Event B given Event A is represented as:\[ P(B \mid A) = \frac{P(A \cap B)}{P(A)} \]This formula shows us that the probability of Event B, given that Event A has already happened, is equal to the probability of both events happening together divided by the probability of Event A. It's a way to update probabilities based on new information. In our exercise, once Pump #1 fails, the probability that Pump #2 will also fail becomes higher. This is an application of conditional probability because the failure of Pump #2 is dependent on the failure of Pump #1.

System reliability refers to the ability of a system to function under specified conditions for a planned duration. In systems with components like pumps, reliability often means that the system continues to work even if one component fails, as long as others remain functional. Redundancies are designed to ensure continued operation. However, increased strain on remaining components, like a single operational pump having to do the work of two, may lead to higher chances of failure as seen in our problem. Understanding system reliability involves figuring out these interactions and making sure that the failure of one component does not lead to a total system breakdown. Such calculations ensure systems are designed to last, even when individual parts start failing.

Event union and intersection are basic operations in set theory applied to probabilities.- **Event Union (\(A \cup B\))**: Represents the occurrence of either Event A or Event B or both. Calculated using the addition rule, it signifies any conditions that lead to at least one of the events happening. - **Event Intersection (\(A \cap B\))**: Represents the occurrence of both Event A and Event B simultaneously. It's the shared space where both probabilities overlap.Understanding these concepts helps in complex probability scenarios, such as determining the occurrence of pump failures in our exercise. Calculating these intersections and unions is critical to solving problems where overlapping probabilities exist and can help predict the likelihood of multiple events occurring together.