Contextualization

Introduction to the Sample Space

The sample space is a fundamental concept in probability theory. It is a collection of all possible outcomes of a random experiment. For example, if we toss a coin, the sample space is {heads, tails}, if we roll a die, the sample space is {1,2,3,4,5,6}. As simplistic as it may seem, understanding and correctly defining the sample space of an experiment is the first step to calculate the probabilities of events involving that experiment.

The algebra of events is a mathematical structure associated with the sample space that allows us to define and work with combinations of events, such as union, intersection, and complement of events. It is important to understand that, although all events are subsets of the sample space, not all subsets of the sample space are events.

Finally, probability is a function defined on the algebra of events that assigns a numerical value between 0 and 1 to each event, so that the most likely events have higher values than the less likely events. Probability is a quantitative way to express the uncertainty associated with an event.

The Importance of the Sample Space

Probability theory and, in particular, the concept of sample space have applications in a wide variety of areas of knowledge. In mathematics, it is used to solve problems of combinations and permutations. In physics, it is used to model uncertainty in experimental measurements. In medicine, it is used to calculate the risk associated with different treatments.

This discipline of mathematics goes beyond solving problems in the classroom, it is used to make decisions in the real world. It is used by companies to analyze risks and make predictions. It is used by governments to plan public policies. Therefore, understanding probability theory and learning how to calculate sample spaces is an important skill that will be useful in many aspects of your life.

Practical Activity: Simulating the Sample Space

Project Objective

The main objective of this project is to solidify the concept of sample space and its application in solving probability problems. Students will be challenged to perform random experiments, identify sample spaces, and calculate the probabilities associated with different events.

Project Description

Groups will be divided into 3 to 5 students. Each group will receive a set of materials and instructions for conducting a series of experiments involving coin tosses, dice rolls, and drawing cards from a deck. In addition, the groups must record their results, identify the sample spaces, and calculate the probabilities of different events.

Required Materials

Each group will need the following materials:

1. Coin
2. 6-sided die
3. Complete deck of cards (52 cards)

Activity Steps

1. Coin Toss: Toss the coin 50 times and record the results. Identify the sample space and calculate the probability of getting heads or tails.
2. Dice Roll: Roll the die 60 times and record the results. Identify the sample space and calculate the probability of getting each face of the die.
3. Drawing Cards from the Deck: Shuffle the deck and draw a card 52 times (until the deck runs out) and record the results. Identify the sample space and calculate the probability of getting each suit (spades, hearts, diamonds, or clubs).

Recording Results

Each group must record the results of the experiments in a table, identifying the sample space and calculating the probabilities of the events. This record must be made in a document shared among the group members.

Project Delivery

After completing the practical part, the groups must prepare a report on the experience, which should contain the following elements:

1. Introduction: Provide context on the topic and explain the concept of Sample Space and its relevance. Explain the purpose of the project.

2. Development: Describe the step-by-step of the experiment. Explain how the sample spaces were identified and how the probabilities were calculated. Present the results obtained, including frequency tables and graphs.

3. Conclusions: Discuss the results obtained and analyze if they are in line with what would be expected theoretically. For example, the probability of getting heads or tails in a coin toss is 50%, are the results obtained close to this value? Explain which technical and socio-emotional skills were developed or improved during the project.

4. Bibliography: List the sources used to support the work.

The report must be submitted in digital format in the school's management system within one month from the start date of the project.