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Probability offers a way to understand how likely an event is to occur. But sometimes, we are more interested in when it does not. This is where the **complement of an event** comes into play. For any event, the complement refers to scenarios where the event does not happen. Mathematically, the probability of the complement of an event is calculated by subtracting the probability of the event from 1.
\( P(\text{Not A}) = 1 - P(A) \).
This formula works because probabilities for all possible outcomes sum to 1. For example, if the probability of rainfall (Event R) is 0.30, the probability of no rainfall is \( 1 - 0.30 = 0.70 \). This has practical uses like planning for scenarios where the preferred event does not occur. In the exercise, we found the complement of \( P(J \text{ AND } K) \) as 0.90, which means there is a 90% chance that either event J or event K, or both, DO NOT occur simultaneously.

The **intersection of events** is a probability concept that identifies when two or more events occur at the same time. Think of this as finding a common area where events overlap. This is represented symbolically by 鈥淎ND鈥� (e.g., \( J \text{ AND } K \)) which means both events J and K happen simultaneously.
Mathematically, it's often denoted as \( P(J \cap K) \) where \( \cap \) is the intersection symbol.
To find the probability of this intersection, we used the formula:

• \( P(J \text{ OR } K) = P(J) + P(K) - P(J \text{ AND } K) \)

This formula ensures we don't double count the intersection when adding probabilities of multiple events. From the exercise, using given values, this provided \( P(J \text{ AND } K) = 0.10 \). So, there is a 10% chance that both events J and K happen at the same time.

The **union of events** relates to any scenario where one or more of the events occur. This is visualized as combining event outcomes to create a broader possibility space where at least one event takes place. In probability terms, the union of events is represented with 鈥淥R鈥� (e.g., \( J \text{ OR } K \)), meaning either event J occurs, event K occurs, or both.
Representing this mathematically, we use the formula:

• \( P(J \cup K) = P(J) + P(K) - P(J \text{ AND } K) \)

Here, \( \cup \) indicates union, and the formula ensures that any overlapping probability (\( P(J \text{ AND } K) \)) isn't counted twice.
In our exercise, \( P(J \text{ OR } K) = 0.45 \) was given, meaning there's a 45% probability that either one or both of the events J and K will happen. This concept is used widely in scenarios where outcomes are large, such as in risk management or predictive modeling.