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When we say that events are independent, it means that the occurrence or non-occurrence of one event does not affect the likelihood of the other. In the context of probability, two events, say event A and event B, are independent if the probability of event A happening is the same regardless of whether event B has occurred or not. Mathematically, if A and B are independent, then:\[ P(A \cap B) = P(A) \times P(B) \]This concept is crucial when calculating probabilities in systems like series or parallel systems, where different components or events work together. For our example of two components in series, this independence assumption allows us to multiply the individual probabilities to find the system's total probability of functioning.

System reliability is a measure of a system's ability to function without failure over a specific period. In a series system, each component must perform its function for the system to operate successfully. If any component fails, the entire system fails. This makes reliability calculation critical for system designs where uninterrupted service is essential.
When calculating the reliability of systems configured in series, the overall system reliability is lower than that of the individual components. This is because the failure of any single component results in system failure. Therefore, engineers and designers often work to ensure that each component’s reliability is as high as possible to maximize overall system reliability.

Probability calculation in the context of systems is about determining the likelihood of a system performing its intended function. It's a straightforward process when the events are independent, as seen in our exercise. To calculate the probability of a system working with two independent components in series:

• Identify the probability of each component working. For example, probability of component A, \( P(A) = 0.90 \), and probability of component B, \( P(B) = 0.90 \).
• Since the components are independent, multiply their probabilities: \[ P(\text{System Works}) = P(A) \times P(B) = 0.90 \times 0.90 \]
• Perform the multiplication to find the system probability: \[ 0.90 \times 0.90 = 0.81 \]
• The resultant probability, 0.81 or 81%, is the likelihood that the system will work.

This step-by-step calculation demonstrates the simplicity yet importance of understanding the independence of events when working with probabilities in series systems.