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In probability, independent events refer to two or more events where the outcome of one event does not affect the outcomes of the others. This concept is essential in probability calculations because it affects how probabilities are combined.

For instance, let's consider the scenario from the exercise, where two items are sold, and we need to determine the likelihood of both being returned. Whether the first item is returned does not influence the probability of the second item being returned. Thus, these events are independent. The probability of both items being returned is calculated by multiplying the individual probabilities of return for each item. This multiplication is valid only because the events are independent.

The probability of an item being returned is 0.05, so the probability that both items are returned (an independent scenario) would be:

• \( P(R \cap R) = P(R) \times P(R) = 0.05 \times 0.05 = 0.0025 \).

Understanding the independence of these events enables accurate calculations when determining the chances that certain outcomes will occur.

A probability tree diagram is a visual representation that helps illustrate all potential outcomes of a series of events and their associated probabilities. This tool is especially helpful to organize and compute probabilities when dealing with multiple stages in an event.

In our exercise, the tree diagram starts with the first item and its two potential outcomes: "Returned" with probability 0.05 and "Not Returned" with probability 0.95.
From each of these outcomes, the next item's outcomes are drawn, splitting further down the branches:

• Under the "Returned" branch, the second item can be "Returned" (0.05) or "Not Returned" (0.95).
• Similarly, under the "Not Returned" branch, you branch out to "Returned" (0.05) or "Not Returned" (0.95).

At the end of these branches, you can find the probabilities of all possible combinations. These branches help in visualizing complex problems and systematically understanding possible outcomes which can be matched to the probability calculations you perform.

Calculating probability involves identifying the likelihood of specific outcomes occurring among all possible outcomes. In this scenario, the key steps involve determining the probabilities of all individual events initially, and then using the rules of probability to find the required outcomes.

From the given probability of returning an item, denoted as 0.05, and not returning an item, as 0.95, we can determine:

• The probability that both items are returned: \(P(R \cap R)\) which calculates to 0.0025.
• The probability that neither item is returned: \(P(NR \cap NR)\) which calculates to 0.9025.

These calculations are achieved by multiplying the related probabilities of each independent event. This approach helps avoid counting overlap or missing combinations, ensuring that your calculations reflect the true complexity of real-world scenarios.