# Game Show

In the problem Game Show, students use mathematical concepts of game theory, probability, and expected value. The mathematical topics that underlie this problem are knowledge of game strategies, reasoning, graph theory, sample spaces, fairness, probability ratios, experimental and theoretical probability, counting principles/strategies, and expected value.In each level, students must make sense of the problem and persevere in solving it (MP.1). Each problem is divided into five levels of difficulty, Level A through Level E, to allow access and scaffolding for students into different aspects of the problem and to stretch students to go deeper into mathematical complexity.

PRE-K
In this task, students play a simple game where the game is structured such that the same player wins every time. Students determine whether the game is fair for all players and why.

LEVEL A
In this level, students play a simple game where the game is structured such that the same player wins every time. Students determine whether the game is fair for all players and who is favored. They explain why the outcome is always in favor of one player.

Students can address this level using skills from Common Core standard 4.OA.C.5. In this standard, students generate a number pattern that follows a given rule. In the task, the students can use a table to track the pattern following the rule of each player taking 2.

LEVEL B
In this level, the students play and analyze a Nim-type game, then determine a strategy where they can always win. They explain their solution and why it will always work.

This level does not align to any specific standards but could be used as an extension of the level A problem, reinforcing students’ pattern recognition. Students would also reinforce recognizing numbers that are multiples of 3. It also gives students exposure to game theory.

LEVEL C
In this level, students are presented with a more complex Nim-type game that requires strategies to change as the opponent makes his or her moves. Students develop and explain a strategy for winning the game.

This level does not align to any specific standards but could be used as an extension of the level B problem for upper-elementary-aged students. This problem gives students exposure to game theory.

LEVEL D
In this level, the students are asked to analyze the famous Monty Hall game that involves selecting the door with the grand prize. Students apply their knowledge of finding probabilities of compound events to justify their strategy and explain their chances of winning.

Students are asked to analyze the famous Monty Hall game that involves selecting the door with the grand prize. Students apply their knowledge of finding probabilities of compound events (7.SP.C.8a, 7.SP.C.8b) to justify their strategy and explain their chances of winning (SMP 3).

LEVEL E
In this level, students are asked to determine the best strategy for playing a game involving a spinner and payoff using expected value. Students must defend their strategy to maximize their winnings.

Students can solve this problem with an understanding of expected value. Student define the number of spins as a variable (S-MD.A.1) and write an equation for the expected value (S-MD.A.3). They analyze payoff and cost in this equation (S-MD.A.2).

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