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In probability theory, an event is a set of outcomes from a specific experiment. Think of an event as a scenario that we are interested in. For example, when rolling two dice, we might be interested in the event where the sum of the numbers is odd. Events can be simple, containing just one outcome, or compound, containing multiple outcomes.
When you roll two dice:

• A simple outcome could be rolling a (1, 2).
• The event of having an odd sum (like event E in our exercise) includes several outcomes, such as (1, 2), (2, 3), or (5, 4), where each pair consists of a roll on each die.

Various outcomes combine to form events, shaping the basis of probability calculations.

The intersection and union of events help to explore combined scenarios. The intersection, denoted as \(E \cap F\), refers to the outcomes that occur in both events simultaneously. For instance, if event E is having an odd sum and event F is at least one die landing on 1, then \(E \cap F\) includes outcomes like (1, 2), where both conditions are satisfied.
The union, represented by \(E \cup F\), includes outcomes that occur in either event or in both. So for events E and F, \(E \cup F\) consists of any outcome where the sum is odd or at least one die shows 1, like (1, 4) or (2, 1). These concepts help us to understand possibilities when we look at overlapping and inclusive combinations of events.

Complementary events include all possible outcomes that are not in the given event. For every event A, its complement \(A^c\) consists of all outcomes that do not belong to A. This plays a crucial role when considering what something is not in probability.
For instance, if event F is that at least one die lands on 1, then \(F^c\)—the complement—includes all outcomes where neither die shows a 1, such as (2, 2), (3, 4), and so on.
This idea is especially useful to find probabilities indirectly, as often, it can be easier to calculate the probability of the complement and subtract it from 1, giving us the probability of the original event.

A sample space is the set of all possible outcomes of an experiment. Each possible outcome in the sample space is called a sample point. When throwing two dice, the sample space includes 36 sample points, considering all combinations from (1, 1) to (6, 6).

• The sample space helps to ensure that we're considering every potential scenario.
• It provides a comprehensive view from which events can be formed.

Understanding the full sample space and its sample points is essential as it lays the groundwork for calculating probabilities. It allows us to visualize and correctly enumerate the condition-specific outcomes that define each event.