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In probability theory, the sample space is the set of all possible outcomes of a random experiment. Let's imagine tossing a single six-sided die. The sample space for this experiment is comprised of six simple events: \( \{1, 2, 3, 4, 5, 6\} \). This means that there are six potential results you can observe in this scenario.

When defining a sample space, it's crucial to ensure that each outcome is distinct and the sample space covers every possible outcome, no matter how improbable. For a die, these are simple to list, but in more complex experiments, identifying all potential results can be tricky.

• Each outcome should be mutually exclusive, meaning no two outcomes can happen simultaneously.
• The completeness of a sample space indicates that it has accounted for every possible outcome.

Understanding the full scope of possible outcomes in a sample space is a foundational step in evaluating probabilities.

Event probability is what most people think of when they hear 'probability'—it's the chance that a particular event will occur. In the context of tossing a die, an event is a specific set of outcomes from the sample space. For example, observing a '2' is a simple event, while observing an even number is a compound event.

The probability of an event is calculated by adding up the probabilities of all the simple events in that set. When each outcome is equally likely, as in the case of a fair die, the probability of each simple event is the total number of favorable outcomes divided by the total number of outcomes in the sample space.

• For Event A, observing a '2', there's one favorable outcome: \(P(A) = \frac{1}{6}\).
• For Event B, observing an even number, there are three favorable outcomes (2, 4, and 6), thus \(P(B) = \frac{3}{6} = \frac{1}{2}\).

Event probability allows you to quantify the chance that something specific will happen when you perform the experiment.

In probability, the concepts of intersection and union of sets help describe how events can occur together or separately.

The intersection of two sets, denoted by \(\cap\), represents outcomes that two or more events share. For instance, if Event D is defined as observing both Events A (observe a 2) and B (observe an even number), then the intersection \(A \cap B\) involves looking for outcomes common to both events, which here is \( \{2\} \). If no common outcome exists, the intersection is an empty set, meaning there are no shared outcomes.

• If two sets have no overlap, their intersection is the empty set, and the probability for such an intersection is 0.

The union of two sets, denoted by \(\cup\), includes all outcomes that are in either set or in both. For example, Event E (observe A or B or both) combines all outcomes from A and B: \( \{2, 4, 6\} \). The key point with a union is to include everything from both events without double-counting elements.

Understanding these concepts is crucial for finding probabilities when dealing with multiple related events.