Introduction

Relevance of the Topic

Probability is a central concept in Mathematics and is used in virtually all fields of study. Probability calculations are fundamental for estimating risks, predicting outcomes, and making decisions in many everyday situations. Understanding the sample space, the basis of probability theory, is essential to comprehend and apply concepts such as events, randomness, independence, and intersection of events.

Contextualization

In the vast universe of Mathematics, probability theory is categorized as part of Discrete Mathematics. Understanding the sample space is a prerequisite for learning quantum mechanics, decision theory, game theory, and is deeply linked to issues of statistics and data analysis.

Within the high school mathematics curriculum, probability theory focusing on the sample space is commonly introduced after the basic concepts of sets, as the sample space is essentially a set. Thus, studying the sample space is a crucial step for students' progression to more advanced topics in probability and statistics.

Theoretical Development

Components

• Defined Sample Space: is the set of possible outcomes of a random experiment. It is crucial to understand that the sample space is a set that contains all possible outcomes, not just those you might imagine to be likely. For example, in the toss of a coin, the sample space is {heads, tails}, any other outcome, such as the coin standing on edge, if it falls on its side, is not contemplated in the sample space.
• Sample Space Element: each element of the sample space is a possible outcome of the experiment. In a dice roll, the numbers from 1 to 6 are the six elements of the sample space.
• Subsets of the Sample Space: These subsets define the events. An event is simply a set of one or more elements of the sample space. Events can be disjoint (have no elements in common), mutually exclusive (only one can occur in a single experiment) or independent (the occurrence of one does not influence the occurrence of the other).

Key Terms

• Random Experiment: An experiment whose outcome is not known in advance. Examples include rolling a die, flipping a coin, drawing a card from a shuffled deck.
• Elementary Event: An event that consists of a single outcome. For example, in the toss of a coin, "heads" and "tails" are elementary events.
• Composite Event: An event that consists of more than one outcome. For example, in the roll of a die, the event of rolling a prime number is a composite event.

Examples and Cases:

• Example of Dice Roll: In the roll of a six-sided die, the sample space is {1, 2, 3, 4, 5, 6}. Each face of the die represents a possible outcome of the experiment.
• Example of Drawing a Card from a Deck: In a deck of 52 cards, if we select a card randomly, the sample space is the set of all 52 cards in the deck.
• Example of Coin Toss: In the toss of a coin, the sample space is {heads, tails}. Note that if the coin stops standing or leaning against an object, these results are not part of the sample space, as they are not possible outcomes in this experiment.

Detailed Summary

Relevant Points

• Definition of Sample Space: The sample space is the set of all possible outcomes of an experiment. Each possible outcome is known as an element of the sample space.
• Properties of Sample Space: The sample space needs to be exhaustive (include all possible outcomes) and mutually exclusive (not include any impossible outcomes). It must be defined based on the experiment under consideration.
• Differentiation between Results and Sample Space: It is essential to understand that the sample space represents all potential outcomes of an experiment, not just those that seem likely.
• Events and Sample Space: Events are subsets of the sample space. They can be disjoint, mutually exclusive, or independent - concepts that will be deepened in later studies.
• Elementary and Composite Events: Elementary events consist of only one result, while composite events involve more than one result.

Conclusions

• Understanding the sample space is fundamental to the theory of probabilities and its subsequent unfolding in practical applications.
• The sample space is an interdisciplinary concept, being essential in other fields of Mathematics, including statistics and game theory.
• A well-defined sample space is crucial for estimating the probability of an event, which is one of the most common applications of probability theory.

Exercises

1. Exercise 1: In the roll of a die, what is the sample space?
2. Exercise 2: If we choose a card from a deck of 52 cards, what is the sample space?
3. Exercise 3: In the toss of a biased (non-fair) coin that has a 60% chance of landing heads, what is the sample space?
4. Exercise 4: In the roll of two dice, what is the sample space? Remember that the order in which the results appear matters in this case.