Ohio Resource Center
Content Supports - Activities and rich problems
How to Fix an Unfair Game
Discipline
Mathematics
7, 8, 9, 10, 11, 12
Professional Commentary

Michelle and Jim wager on picking colored chips out of a bag. In the bag are two black chips and one red one. Someone takes out a chip, notes its color, and puts it back. Then he or she reaches in again and takes out a chip. If it is the same color as the first one, Michelle wins a dollar from Jim. If not, Jim wins a dollar from Michelle. Is the game fair to both Michelle and Jim? Extensions of the problem and a complete discussion of the underlying mathematical ideas are included. This mathematically rich problem was originally developed for the Project Discovery Mathematics by Inquiry institutes for middle grades teachers taught in 1992 - 1994 at The Ohio State University. Project Discovery was co-funded by the Ohio Board of Regents and the Statewide Systemic Initiative (SSI) program of the National Science Foundation. (author/sw)

Common Core State Standards for Mathematics
Statistics and Probability
Investigate chance processes and develop, use, and evaluate probability models.
7.SP.C.7
Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy.
1. Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events. For example, if a student is selected at random from a class, find the probability that Jane will be selected and the probability that a girl will be selected.
2. Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process. For example, find the approximate probability that a spinning penny will land heads up or that a tossed paper cup will land open-end down. Do the outcomes for the spinning penny appear to be equally likely based on the observed frequencies?
7.SP.C.8
Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.
1. Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs.
2. Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., “rolling double sixes”), identify the outcomes in the sample space which compose the event.
3. Design and use a simulation to generate frequencies for compound events. For example, use random digits as a simulation tool to approximate the answer to the question: If 40% of donors have type A blood, what is the probability that it will take at least 4 donors to find one with type A blood?
High School - Statistics and Probability
Using Probability to Make Decisions
Calculate expected values and use them to solve problems
HSS-MD.A.3
(+) Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value. For example, find the theoretical probability distribution for the number of correct answers obtained by guessing on all five questions of a multiple-choice test where each question has four choices, and find the expected grade under various grading schemes.
HSS-MD.A.4
(+) Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value. For example, find a current data distribution on the number of TV sets per household in the United States, and calculate the expected number of sets per household. How many TV sets would you expect to find in 100 randomly selected households?
Use probability to evaluate outcomes of decisions
HSS-MD.B.5
Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.
1. Find the expected payoff for a game of chance. For example, find the expected winnings from a state lottery ticket or a game at a fast-food restaurant.
2. Evaluate and compare strategies on the basis of expected values. For example, compare a high-deductible versus a low-deductible automobile insurance policy using various, but reasonable, chances of having a minor or a major accident.
HSS-MD.B.6
(+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).
HSS-MD.B.7
(+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).
Ohio Mathematics Academic Content Standards (2001)
Data Analysis and Probability Standard
Benchmarks (5–7)
H.
Find all possible outcomes of simple experiments or problem situations, using methods such as lists, arrays and tree diagrams.
I.
Describe the probability of an event using ratios, including fractional notation.
K.
Make and justify predictions based on experimental and theoretical probabilities.
Benchmarks (8–10)
J.
Compute probabilities of compound events, independent events, and simple dependent events.
K.
Make predictions based on theoretical probabilities and experimental results.
7.
Compute probabilities of compound events; e.g., multiple coin tosses or multiple rolls of number cubes, using such methods as organized lists, tree diagrams and area models.
10.
Use theoretical and experimental probability, including simulations or random numbers, to estimate probabilities and to solve problems dealing with uncertainty; e.g., compound events, independent events, simple dependent events.
10.
Understand and use the concept of random variable, and compute and interpret the expected value for a random variable in simple cases.
Mathematical Processes Standard
Benchmarks (5–7)
B.
Apply and adapt problem-solving strategies to solve a variety of problems, including unfamiliar and non-routine problem situations.
H.
Use representations to organize and communicate mathematical thinking and problem solutions.
Benchmarks (8–10)
A.
Formulate a problem or mathematical model in response to a specific need or situation, determine information required to solve the problem, choose method for obtaining this information, and set limits for acceptable solution.
E.
Use a variety of mathematical representations flexibly and appropriately to organize, record and communicate mathematical ideas.
H.
Locate and interpret mathematical information accurately, and communicate ideas, processes and solutions in a complete and easily understood manner.
Principles and Standards for School Mathematics
Data Analysis and Probability Standard
Understand and apply basic concepts of probability
Expectations (6–8)
use proportionality and a basic understanding of probability to make and test conjectures about the results of experiments and simulations;
compute probabilities for simple compound events, using such methods as organized lists, tree diagrams, and area models.
Expectations (9–12)
compute and interpret the expected value of random variables in simple cases;