That s Not Fair! ASSESSMENT # Benchmark Grades: 9-12 Summary: Students consider the difference between fair and unfair games, using probability to analyze games. The probability will be used to find ways to make unfair games fair. Keywords: Probability Problem solving Page 1 Copyright 1999 Center for Performance Assessment
That s Not Fair! ASSESSMENT # Information for the Teacher Task Description Task 1: Students are asked to evaluate the fairness of a game involving a coin and a die by conducting an experiment of 30 trials. Task 2: Students evaluate the fairness of the coin and die game by calculating the theoretical probability of winning. Task 3: Students choose a game from the three listed and evaluate the fairness of the game through experimentation and theoretical probability. Students find a way to alter the game to make it fair so that all players have an equal chance of winning. Task 4: Students design a fair and unfair version of a game. They will present the experimental and theoretical probability of winning the game. For the students who complete Tasks 1 through 4 ahead of the rest of the class, there is an enrichment task provided at the end of this assessment. Required Materials Each student will need one coin and one die. Some students will need a paperclip to complete Task 3. Page 2 Copyright 1999 Center for Performance Assessment
Scoring key for the teacher Tasks 1 & 2: The options for winning and losing can be described in a table: Die 1 1 2 2 3 3 4 4 5 5 6 6 Coin H T H T H T H T H T H T W/L W W W L W L W L W L W W Inspection of the table shows that, as each possibility is equally likely, 8 of the 12 chances yield a win. The probability of winning is 2/3 or 67%. In 30 trials, you would expect 20 wins. However, in an actual trial, results could vary significantly. Task 3: Rock, Paper, Scissors Player A Player B Player C Wins 1 point R R R A R R P B R R S B R P R B R P P B R P S C R S R B R S P C R S S B P R R B P R P B P R S C P P R B P P P A P P S B P S R C P S P B P S S B S R R B S R P C S R S B S P R C S P P B S P S B S S R B S S P B S S S A Page 3 Copyright 1999 Center for Performance Assessment
The probability of Player A winning is 3/27 or 1/9. The probability of Player B winning is 18/27 or 2/3 The probability of Player C winning is 6/27 or 2/9. One way to make this a fair game is to award Player A 6 points when all of the signs match, to give Player B 1 point when 2 signs match and to assign Player C 3 points when none of the signs match. Multiplying Dice Game: 1 2 3 4 5 6 1 1 2 3 4 5 6 2 2 4 6 8 10 12 3 3 6 9 12 15 18 4 4 8 12 16 20 24 5 5 10 15 20 25 30 6 6 12 18 24 30 36 The probability of Player A winning is 27/36 or ¾. The probability of Player B winning is 9/36 or ¼. One way to make this game fair is to add the dice rather than multiply. Another way is to award Player B 3 points for odd products and to give Player A only one point for even products. Spinners 1 2 3 4 5 6 7 8 2 3 4 5 6 7 8 9 10 3 4 5 6 7 8 9 10 11 4 5 6 7 8 9 10 11 12 5 6 7 8 9 10 11 12 13 6 7 8 9 10 11 12 13 14 The probability of Player A winning is 15/40. The probability of Player B winning is 25/40. One way to make the game fair is for Player A to win 1 point if the sum is less than 9 and for Player B to win 1 point if the sum is 9 or greater. Another way to make the game fair is for Player A to win 1 point if the sum is even and Player B to win 1 point if the sum is odd. Page 4 Copyright 1999 Center for Performance Assessment
That s Not Fair! Assessment Introduction: Here is what you will do... Your teacher begins class by saying, To grade your last test, I stood at the top of a staircase and threw your tests down the stairs. Wherever your test landed determined your grade if your test landed on a stair towards the top, you received a high grade, if your test landed closer to the bottom, you got a low grade. How would you feel about this method of grading? Would you say that it is fair or unfair? Why? Discuss with your classmates a good definition for fair. Page 5 Copyright 1999 Center for Performance Assessment
Task 1: Analyze a game by doing an experiment A game called Coin and Die is played by flipping a coin and rolling a die. You win if a head on the coin and/or a 1 or 6 on the die appear. You lose under all other situations. Write a paragraph in which you predict whether this game is fair or not. Include your definition of fair and why you think that the game is fair or unfair and a reason for your prediction. Play the game 30 times, recording the results of the coin and die and whether you win or lose in a table. Write another paragraph in which you explain if playing the game has altered your opinion on whether it is fair or not. Review your work for errors in spelling and grammar. Page 6 Copyright 1999 Center for Performance Assessment
Scoring Guide - Task 1 4 Exemplary Criteria for the Proficient category have been successfully completed. Advanced work is included. For example, the student provides more than one type of data display or chart, does more than thirty trials of the experiment, or compares the data from multiple runs of the experiment. Other examples of advanced work include: 3 Proficient The student conducts 30 trials of the game. The student records the events of each trial in a chart or table. The response includes two paragraphs; one which describes the student s definition of fair and whether the game appears fair before playing and one which describes whether the game appears fair after playing, both with rational explanations. Spelling and grammatical errors do not significantly affect the communication of ideas. 2 Progressing Three of the criteria in the Proficient category have been met. More work is needed. 1 Not meeting the standard(s) Less than three of the criteria in the Proficient category have been met. The task should be repeated. Page 7 Copyright 1999 Center for Performance Assessment
Task 2: Analyze a game using logic You have experimented with the Coin and Die game to test its fairness. A more formal way to analyze the game is to look at the mathematical probability of winning. Create a chart or diagram that illustrates all of the possible outcomes of the coin and the die. How many possible outcomes are there using a coin and die? Determine the probability of each event. What is the probability of winning? You are playing on a game show. The final prize, your dream car, can be yours if you win at a game of chance. Your choices are to play The Coin and Die game described in Task 1, or you can flip a coin, choosing heads or tails. The game show host asks you for your choice: Which will it be? Keep in mind that your dream car is one the line. IF YOU WIN, YOU GET TO DRIVE THE CAR HOME TODAY! It is loaded with all of the extras, and we will even throw in free insurance and gas for a year! If you lose, well, you just get to go home... Which will you choose The Coin and Die or Flipping a Coin?!!? Write out your answer to the game show host, keeping in mind that like all game show contestants, you are nervous, and therefore feel the need to explain your choice in GREAT detail. Describe how you have studied The Coin and Die game. Explain why your experimental results may have differed from the mathematical probability, and give your choice and the reason for it. Make sure you include whether you won or not and if you won, describe the dream car you drove home in. Review your work for errors in spelling and grammar. Page 8 Copyright 1999 Center for Performance Assessment
Scoring Guide - Task 2 4 Exemplary Criteria for the Proficient category have been successfully completed. Advanced work is included. For example, additional tests are run, or the student explains how much difference you might expect between playing the game and the mathematical probability. Other examples of advanced work include: 3 Proficient A table, tree-diagram or other chart is assembled using the game from Task 1. The student correctly calculates the probability of winning the game. The student explains which game he or she would choose to play for the dream car and gives a rational explanation for the choice. Included in the answer is an analysis of the Coin and the Die game and an explanation for why the experimental and theoretical probabilities may not have given the same results. Spelling and grammatical errors do not significantly affect the communication of ideas. 2 Progressing Three of the criteria in the Proficient category have been met. More work is needed. 1 Not meeting the standard(s) Less than three of the criteria in the Proficient category have been met. The task should be repeated. Page 9 Copyright 1999 Center for Performance Assessment
Task 3: Making unfair games fair Choose one of the games described below. For the game you choose, do the following: Conduct an experiment to test the fairness of the game. Present the results of your experiment in a table and analyze the results, indicating what fraction of the trials each player won. Analyze the probability of the game by providing a chart or diagram and calculating the mathematical probability that each player will win. Alter the game so that the game is fair. Assume that fair means that all players have an equal chance of winning. Rock, Paper, Scissors Players A, B, and C simultaneously display the rock, paper, or scissors sign of their choice. If all of the signs match, player A gets 1 point. If two of the signs match, player B gets 1 point, and if none of the signs match, player C gets 1 point. Multiplying Dice Game Roll two dice and multiply the two numbers. If the product is even, Player A wins 1 point. If it is odd, player B wins 1 point. Spinner Use the spinners below with a paperclip and pencil. Add the number from each spinner. If the sum is prime, Player A wins. If the sum is not prime, Player B wins. 5 6 7 8 1 2 4 3 2 6 5 4 3 Page 10 Copyright 1999 Center for Performance Assessment
Scoring Guide - Task 3 4 Exemplary Criteria for the Proficient category have been successfully completed. Advanced work is included. For example, the student evaluates more than one game or provides multiple ways to make one of the games fair. Other examples of advanced work include: 3 Proficient The student selects one of the games and presents the results of an experiment with at least 30 trials in a table or chart. The mathematical probability is calculated and the supporting diagram or chart is accurately presented. A suggestion is provided that would alter the game so that all players have an equal chance of winning. 2 Progressing Two of the criteria in the Proficient category have been met. More work is needed. 1 Not meeting the standard(s) Less than two of the criteria in the Proficient category have been met. The task should be repeated. Page 11 Copyright 1999 Center for Performance Assessment
Task 4: Design your own game Your task is to design two versions of a game: an unfair version, where all players do not have an equal chance of winning, and a fair version, where all players have an equal chance of winning. Take into account that if the unfair version of the game is obviously unfair, then no one will be willing to play. For each version of the game, provide the following: A description of how the game is played and the rules. Experimental results from playing the game. An analysis of the mathematical probability of the game. Review your work for errors in spelling and grammar. Page 12 Copyright 1999 Center for Performance Assessment
Scoring Guide - Task 4 4 Exemplary Criteria for the Proficient category have been successfully completed. Advanced work is included. For example, the student creates and analyzes more than two versions of the game or an experiment is created and run to determine whether others can identify the fair game. Other examples of advanced work include: 3 Proficient The student clearly presents two versions of a game and correctly identifies that one is fair and one is not. Experimental results from playing both the fair and unfair game are provided and analyzed. The mathematical probability of winning each game is correctly calculated and presented with supporting work. Spelling and grammatical errors do not significantly affect the communication of the ideas. 2 Progressing Three of the criteria in the Proficient category have been met. More work is required. 1 Not meeting the standard(s) Less than three of the criteria in the Proficient category have been met. The task should be repeated. Page 13 Copyright 1999 Center for Performance Assessment
Enrichment Task: A Happy Ending? A long time ago in a land far, far away, lived a king who loved games. If a prisoner were sentenced to be executed, but there was some hint that the prisoner might be innocent, then the king played the following game with the prisoner. The prisoner is given three empty boxes, 100 red stones and 100 blue stones. The prisoner places the stones in the boxes in any way he chooses. The prisoner is blindfolded and the boxes rearranged. The prisoner draws a stone out of one of the boxes. If the stone is red he is set free. If the stone is blue, he is executed. Write a short story that involves the situation above. One of the characters in the story must, at some point, describe a good method for distributing the stones and provide a mathematical analysis of the method and the probability of being set free. Page 14 Copyright 1999 Center for Performance Assessment
Scoring Guide-Enrichment Here is your opportunity to decide what makes up a good response. Think of what your response should include to be proficient. 4 Exemplary 3 Proficient 2 Progressing 1 Not meeting the standard(s) Page 15 Copyright 1999 Center for Performance Assessment