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In probability, a sample space is the collection of all possible outcomes of an experiment or study. For the given exercise, the study observes people according to three different characteristics: gender, drinking habits, and marital status. Each characteristic can take multiple values, leading to different combinations.

Specifically, gender can be either female \( G_1 \) or male \( G_2 \). Drinking habits include abstaining \( K_1 \), drinking occasionally \( K_2 \), and drinking frequently \( K_3 \). Marital status can be married \( M_1 \), single \( M_2 \), divorced \( M_3 \), or widowed \( M_4 \).

All possible combinations of these characteristics form the sample space. There are 24 such combinations. For example, one combination is a female who abstains from drinking and is married, represented as \( (G_1, K_1, M_1) \). Understanding sample space is crucial because it helps us organize and keep track of all the possible scenarios we might encounter in an experiment.

Set theory provides the foundation for understanding probability concepts like unions and intersections. In this study, we're dealing with three sets:

• Set \( A \) which includes all males \( G_2 \).
• Set \( B \) which represents those who drink, occasionally or frequently \( K_2, K_3 \).
• Set \( C \) which includes all singles \( M_2 \).

Understanding the operations between these sets helps us interpret complex events in probability. For instance:

*The union (\( A \cup B \)) encompasses anyone who is male or drinks. This means any event that belongs to either \( A \) or \( B \) is included in this set.

On the other hand, the intersection (\( A \cap C \)) includes only males who are single, highlighting the overlap between \( A \) and \( C \).

Set operations provide clarity and structure, allowing us to address probability problems systematically.

In probability, an event is a collection of outcomes that share a common feature, and these outcomes are the specific results you observe from an experiment.

Let's consider some events from the study:

• Event \( A \) entails the outcome where the observed person is male. The outcomes include all combinations where gender is male.
• Event \( B \) involves outcomes where the person drinks, visible in both occasional and frequent drinking patterns.
• Event \( C \) consists of outcomes where the individual is single.

Each defined event consists of specific outcomes within the sample space.

When we talk about the event \( A \cap B \cap C \), we mean the intersection of all three events, where we find outcomes of someone who is male, drinks, and is single.

Outcomes are the fundamental units within an event, while events can categorize and group these outcomes into meaningful categories for analysis. These concepts together help us calculate probabilities and understand the likelihood of different scenarios.