Introduction
Probability is the study of the likelihood that an event will occur, or the chance of occurrence of that event. It is a branch of mathematics concerned with numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. In the real world, random events occur all the time, and understanding probability and sample spaces is a useful skill in meteorology, insurance, gambling, and many other areas involving uncertainty.
A sample space is a set of all possible outcomes of a random experiment. For example, if we flip a coin, the sample space is {heads, tails}. The probability of each of these outcomes is 1/2. Sample spaces can be either discrete (a finite or countably infinite number of outcomes) or continuous (an uncountable number of outcomes, such as the possible temperatures on a given day).
Importance of Probability and Sample Spaces
Life is full of uncertainty. Almost every decision we make on a daily basis involves some degree of uncertainty. Will it rain today? Should I bring an umbrella? What are my chances of winning the lottery? Should I buy a lottery ticket? These questions all involve assessing probabilities.
Probability is used in a wide range of other fields including meteorology, where weather forecasts are estimates of the probability of certain weather events occurring. In insurance, companies use probability to determine the risk of an event (such as a car accident) and set insurance prices. In engineering, probability can be used to estimate the reliability of a system, or the chance of failure.
Students can begin learning more about probability at Math is Fun, Khan Academy, and Brilliant.
Learning Activity
Activity Title: Random Events and Probability: The Game of Mathematical Life
Project Goal
The purpose of this project is to create a better understanding of sample spaces and probability by creating a fun, hands-on simulation. Students will create a board game that will reinforce these concepts and allow them to see how these concepts are applied in the real world.
Detailed Project Description
Students will work in groups of 3-5 to create a board game called "The Game of Mathematical Life" in which the events (how players move around the board) are determined by random experiments and sample spaces.
The game of Mathematical Life will include a game board with 30-40 spaces that contain both positive and negative random events (for example, gain money, lose money, move forward spaces, move backward spaces) and properties (such as a house or car) that players can purchase.
The students will roll dice, a classic example of a random experiment, to determine how many spaces to move. They will need to determine what the possible outcomes (sample space) are and calculate the probability of each outcome. Since the events on the board will correspond to the numbers rolled on the dice, the students need to understand probability in order to make good decisions during the game.
Materials
- Dice
- Poster board or cardboard to create the game board
- Paper and markers to create event cards
- Game pieces (these can be from old games or created by the students)
Detailed Step-by-Step Instructions
- Divide students into groups of 3-5.
- Explain the activity and the concepts of sample spaces and probability.
- Have students design their game board and event spaces.
- Next, have them create rules for the game, including how movement will be determined by dice rolls and how they can purchase properties.
- Have the students play the game multiple times to test it out and adjust the rules as needed.
- Finally, have students write a report describing their game, including the rules, the sample space, the events, and the probabilities associated with each event. They should discuss how their understanding of probability influenced their decisions during the game.
Project Deliverables
Students will turn in their completed Game of Mathematical Life along with a report describing their project. The report should be divided into four sections: Introduction, Development, Conclusion, and Bibliography.
In the Introduction, students should provide background on the project, including the importance of probability and sample spaces and the purpose of their project.
In the Development section, students should describe the game they created, including the rules and how they relate to the concepts of probability and sample spaces. They should discuss how they calculated the probabilities for each event in the game and how these probabilities influenced the decisions they made during the game.
In the Conclusion section, students should describe what they learned from the project, how they applied the concepts of probability and sample spaces, and how they could use these concepts in the real world.
In the Bibliography section, students should list any resources they used to help them develop their project, such as books, websites, videos, etc.
