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In probability, events are termed **independent** if the occurrence of one event doesn't affect the probability of the other event occurring. For instance, we have two pumps in a system, where each can fail on a given day. The pumps failing independently means that if one pump fails, it doesn’t change the likelihood of the other pump failing. This characteristic is crucial when calculating probabilities in systems with multiple components or scenarios.

Here’s a friendly tip to remember: if two events are independent, the probability of both occurring is the product of their individual probabilities. Mathematically, if events A and B are independent, then:

• \( P(A \cap B) = P(A) \times P(B) \)

This can simplify calculations significantly when dealing with complex systems like parallel pumps.

**Joint Probability** refers to the probability of two or more events occurring at the same time. In the context of our pumps, it represents the probability that both pumps fail simultaneously on a given day.

To calculate joint probabilities, especially for independent events, we consider the independent nature of events. If event O is the older pump failing and N is the newer pump failing, the joint probability of both events together, \( P(O \cap N) \), is of key interest. Notice, for independent events, the formula simplifies to the multiplication of individual probabilities:

• \( P(O \cap N) = P(O) \times P(N) \)

However, if these achievements stretch beyond independent events to dependent scenarios, other tools are needed to evaluate the joint probability.

In any system with multiple components, **Failure Analysis** is a critical concept. It involves understanding and calculating the chances of a system or its components ceasing to function correctly. In our problem, the goal was to determine the likelihood of complete system failure — meaning both pumps fail.

Several steps help in analyzing failures:

• Identifying critical components (in our case, the two pumps)
• Understanding the interplay between components (independent events mean each pump's performance is unaffected by the other)
• Using probability to predict system failure outcomes

When both components must fail for a system to fail, understanding the failure interdependencies and applying the right probability rules is the way to go.

The **Total Probability Rule** is a fundamental rule used to relate marginal probabilities to conditional probabilities. It's especially useful when analyzing all possible scenarios that can result in an event.

In our scenario, the Total Probability Rule helps us examine the probability of the pump system failing. The system can fail under different conditions such as one pump failing or both pumps failing. Thus, the overall probability for system failure involves combining the probabilities of all these events.

This rule uses the concept of partitioning the probability space and accounting for all possible disjoint outcomes:

• If \( E_1, E_2, \ldots, E_n \) are mutually exclusive and exhaustive events, for any event A: \( P(A) = \sum P(A \cap E_i) \)

Applying these, we can wrap together various individual scenario probabilities to find the complete probability of an event happening, like our pump system's total failure probability.