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When we say two events are independent, it means that the occurrence or non-occurrence of one event does not affect the occurrence of the other event. This is a fundamental concept in probability, providing a basis to calculate the joint probability of such events easily.

Let's say we have two independent events, A and B. The probability of both events happening together, denoted as \( P(A \cap B) \), is simply the product of their individual probabilities. Therefore, if A and B are independent, we have:

• \( P(A \cap B) = P(A) \cdot P(B) \)

This rule makes calculations straightforward as we don't need to worry about any interaction between events. Each event remains unaffected by the other, simplifying our approach to solving probability problems.

By knowing this, students can tackle problems involving independent events more efficiently, applying the straightforward multiplication of probabilities to find combinations.

The probability of the union of two events, A and B, is the probability that either event A occurs, event B occurs, or both occur. This is represented mathematically as \( P(A \text{ or } B) \).

The formula to find this probability for any two events is:

• \[ P(A \text{ or } B) = P(A) + P(B) - P(A \cap B) \]

This equation ensures that we do not double-count the scenario where both events happen simultaneously, represented by \( P(A \cap B) \).

If A and B are independent, this formula becomes even simpler. Since \( P(A \cap B) \) for independent events equals \( P(A) \cdot P(B) \), we substitute this directly into the formula to get:

• \[ P(A \text{ or } B) = P(A) + P(B) - P(A) \cdot P(B) \]

This modified formula helps in comparing and understanding the combinations of independent events further.

Probability inequalities help us to understand relationships between different probabilities in a more general way. In this context, we focus on the inequality \( P(A \text{ or } B) \geq P(A) \).

This inequality implies that when considering the occurrence of either of two events, A or B, it is at least as likely as one of the individual events (in this case, A). The reason is simple - the possibility of both events increases the likelihood compared to just one of them. After all, you're adding another chance of occurrence when including B in the scenario.

By examining the inequality mathematical simplification, we see:

• \[ P(B)(1 - P(A)) \geq 0 \]

Since probabilities \( P(A) \) and \( P(B) \) are both between 0 and 1, the term \( 1 - P(A) \) is always positive, and thus \( P(B)(1 - P(A)) \geq 0 \) is verified. The product is non-negative, affirming that \( P(A \text{ or } B) \) is indeed greater than or equal to \( P(A) \), aligning with our understanding of probability properties.

Understanding this inequality helps in making sense of how probabilities can be reliably expected to behave in more complex scenarios.