91Ó°ÊÓ

Many of you probably played the game "Rock, Paper, Scissors" as a child. Consider the following variation of that game. Instead of two players, suppose three players play this game, and let us call these players \(\mathrm{A}, \mathrm{B}\), and \(\mathrm{C}\). Each player selects one of these three items- Rock, Paper, or Scissors-independent of each other. Player A will win the game if all three players select the same item, for example, rock. Player B will win the game if exactly two of the three players select the same item and the third player selects a different item. Player \(\mathrm{C}\) will win the game if every player selects a different item. If Player B wins the game, he or she will be paid \(\$ 1\). If Player \(\mathrm{C}\) wins the game, he or she will be paid \(\$ 3\). Assuming that the expected winnings should be the same for each player to make this a fair game, how much should Player A be paid if he or she wins the game?

Step by step solution

01

Calculate Total Possible Outcomes

In each game, every player can choose one of three items: Rock, Paper, or Scissors. Therefore, the total number of possible outcomes is \(3^3 = 27\) since each of the three players has three choices.

02

Calculate Probability of Player A Winning

Player A wins the game if all three players choose the same item. There are only three such cases (all choosing Rock, all choosing Paper, or all choosing Scissors). Thus, the probability of player A winning is \(3/27 = 1/9\).

03

Calculate the Probability of Player B Winning

Player B wins the game if exactly two of the three players choose the same item and the third player chooses a different one. Each of the three players can be the one to choose a different item and for each of these three cases, there are three choices for the item that two players select, and two choices for the item that the third player (who selects a different item) chooses. This gives us \((3)*(3)*(2) = 18\) possible outcomes. Therefore, the probability of player B winning is \(18/27 = 2/3\).

04

Calculate the Probability of Player C Winning

Player C wins the game if each player chooses a different item. There are three choices for the item player A chooses, two remaining choices for player B, and one remaining choice for player C, giving \(3*2*1=6\) possible outcomes. Hence, the probability of player C winning is \(6/27 = 2/9\).

05

Calculate Player A's Winning

Since Player B being paid $1 and Player C being paid $3, the expected winnings for each player should be the same for the game to be fair. Let's denote by X the amount that Player A gets paid when he/she wins. The expected value for each player should be equal so: \( (1/9)*X = (2/3)*1 + (2/9)*3 \) Solving this equation for X gives us \(X = 9* [(2/3) + (2/3)] = 6\). Therefore, Player A should be paid $6 if he or she wins the game.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability Calculations

Probability is all about figuring out how likely something is to happen. In this Rock, Paper, Scissors variation, each player has three possible choices. With three players, that's a total of \(3 \times 3 \times 3 = 27\) possible combinations of choices. When calculating probabilities, we look at the specific outcomes that lead to a win and divide that by the total number of possible outcomes.

For instance, Player A wins if all three players choose the same item. There are three such cases out of the 27 possible outcomes. So, the probability of Player A winning is \( \frac{3}{27} \), which simplifies to \( \frac{1}{9} \). Similarly, Player B and Player C have different sets of outcomes leading to their wins, calculated in similar fashion but with different criteria.

Understanding the probabilities of different outcomes helps define how the game can be balanced and fair for all players.

Fair Games

A fair game ensures that every participant has the same chance of winning or, at least, getting the same expected return over time. This assurance comes from balancing the probabilities of winning against what each player wins. In our game, fairness means calculating not just the chance of each player winning, but also aligning the payouts (or prizes) accordingly.

In this game, Player B and Player C have set payouts: $1 and $3, respectively. To make the game fair, Player A's payout must be such that their expected value is equal to the expected value of the other players' winnings. Balancing these payouts ensures nobody has an unfair advantage or disadvantage, maintaining excitement and unpredictability in each game round.

Expected Value

Expected value is a core concept which helps quantify what a player can anticipate winning on average when probabilities and payouts are considered together. It's a way to measure the fairness of a game.

For our game, the formula to calculate expected value is:\[ \text{Expected Value} = (\text{Probability of Winning}) \times (\text{Payout}) \] Each player’s expected value should equal to ensure fairness. For Player A, setting this equal to B's and C's combined expected values means solving for X in the equation:\[ \frac{1}{9} \times X = \frac{2}{3} \times 1 + \frac{2}{9} \times 3 \]
The computation leads to a payout of $6 for Player A, thus equating the expected value across the board and ensuring game fairness.

Rock Paper Scissors variation

This variation of Rock, Paper, Scissors is unique as it involves three players instead of the usual two, which changes the strategies and probabilities involved.

• Player A wins if all players choose the same item.
• Player B wins if two players choose the same item but the third player chooses differently.
• Player C wins if all three players choose different items.

The different win conditions for each player not only change how you might play but also how the game is mathematically evaluated for fairness and balance. Unlike the classic version where outcomes are simpler, this version offers more complexity, making the calculation of each player's probability crucial. With precise calculations, we ensure that despite the added complexity, each player has an equal opportunity to benefit, keeping the game engaging and balanced.