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Probability theory is a fundamental concept in mathematics that helps us quantify uncertainty. It allows us to measure how likely an event is to occur by assigning a number between 0 and 1. This number is called the probability of the event.
To understand probability, consider a simple example: tossing a coin. There are two possible outcomes, heads or tails, each with a probability of 0.5. This framework can be expanded to more complex situations, such as rolling dice or drawing a card from a deck.
In probability theory, understanding how different types of events, such as mutually exclusive events, interact is crucial. This is where concepts like mutually exclusive events become important for calculating probabilities.

In probability, an event is a set of outcomes from a random experiment. For example, rolling a die and getting an even number is an event. It encompasses the outcomes where the die shows a 2, 4, or 6.
Events can be categorized based on certain characteristics:

• Independent Events: These are events that do not affect each other’s outcomes. For example, the probability of rolling a die and flipping a coin.
• Mutually Exclusive Events: These events cannot happen at the same time. If one occurs, the other cannot. In the provided exercise, events A and B are mutually exclusive, meaning the occurrence of event A excludes the possibility of event B happening simultaneously.

Understanding the nature of events is key to applying the correct probability rules and ensures that calculations are accurate.

Probability rules help us to systematically calculate the likelihood of various outcomes. By knowing these rules, we can handle even the most complex probabilistic scenarios with ease. For mutually exclusive events like those described in the exercise, the rules simplify our calculations significantly.
Here are some of the key probability rules:

• Addition Rule for Mutually Exclusive Events: If two events, say A and B, are mutually exclusive, then the probability of either A or B occurring is the sum of their probabilities: \(P(A \text{ or } B) = P(A) + P(B)\).
• Multiplication Rule: If we want to calculate the probability of two independent events both occurring, we multiply their probabilities. However, for mutually exclusive events, this rule tells us \(P(A \text{ and } B) = 0\).

These laws provide a toolkit for solving real-world problems involving chance and uncertainty, such as the one presented in the exercise.