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Two players put one dollar into a pot. They decide to throw a pair of dice alternatively. The first one who throws a sum of 5 wins the pot. How much should the player who starts, add to the pot to make this a fair game?
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Two players put one dollar into a pot. They decide to throw a pair of ...
Understanding the Game
In this game, players alternate throwing a pair of dice. The first to roll a sum of 5 wins the pot. Each player contributes one dollar initially, so the pot starts at two dollars.
Winning Probability
- The possible outcomes when rolling two dice are 36.
- The combinations that sum to 5 are: (1,4), (2,3), (3,2), (4,1) — a total of 4 favorable outcomes.
- Thus, the probability of rolling a sum of 5 is 4/36 = 1/9.
Player Probabilities
- Player 1's Turn: Player 1 goes first.
- Probability of winning on the first roll: 1/9.
- Probability of not winning: 8/9.
- Player 2's Turn: If Player 1 does not win, Player 2 rolls.
- Player 2's probability of winning is also 1/9, but only if Player 1 didn’t win.
- The game can continue in rounds until one wins.
Calculating Fairness
- Let P1 be the probability that Player 1 wins eventually.
- P1 = (1/9) + (8/9)(8/9)P1, where (8/9)(8/9) represents the scenario where both players fail to win in their first turns.
- Solving gives P1 = 9/17 and P2 = 8/17 for Player 2.
Fair Game Contribution
- The pot is currently $2.
- To make it fair, Player 1 should add an additional amount to balance the expected winnings.
Expected Value Calculation
- Player 1’s expected value: (9/17) * (2 + x) - (8/17) * x = 0.
- Solving gives x = $1. Therefore, Player 1 should add $1.
Conclusion
To ensure fairness in this game, Player 1 should add $1 to the pot, making it a total of $3.
In this game, players alternate throwing a pair of dice. The first to roll a sum of 5 wins the pot. Each player contributes one dollar initially, so the pot starts at two dollars.
Winning Probability
- The possible outcomes when rolling two dice are 36.
- The combinations that sum to 5 are: (1,4), (2,3), (3,2), (4,1) — a total of 4 favorable outcomes.
- Thus, the probability of rolling a sum of 5 is 4/36 = 1/9.
Player Probabilities
- Player 1's Turn: Player 1 goes first.
- Probability of winning on the first roll: 1/9.
- Probability of not winning: 8/9.
- Player 2's Turn: If Player 1 does not win, Player 2 rolls.
- Player 2's probability of winning is also 1/9, but only if Player 1 didn’t win.
- The game can continue in rounds until one wins.
Calculating Fairness
- Let P1 be the probability that Player 1 wins eventually.
- P1 = (1/9) + (8/9)(8/9)P1, where (8/9)(8/9) represents the scenario where both players fail to win in their first turns.
- Solving gives P1 = 9/17 and P2 = 8/17 for Player 2.
Fair Game Contribution
- The pot is currently $2.
- To make it fair, Player 1 should add an additional amount to balance the expected winnings.
Expected Value Calculation
- Player 1’s expected value: (9/17) * (2 + x) - (8/17) * x = 0.
- Solving gives x = $1. Therefore, Player 1 should add $1.
Conclusion
To ensure fairness in this game, Player 1 should add $1 to the pot, making it a total of $3.
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Two players put one dollar into a pot. They decide to throw a pair of dice alternatively. The first one who throws a sum of 5 wins the pot. How much should the player who starts, add to the pot to make this a fair game?
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Two players put one dollar into a pot. They decide to throw a pair of dice alternatively. The first one who throws a sum of 5 wins the pot. How much should the player who starts, add to the pot to make this a fair game? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Two players put one dollar into a pot. They decide to throw a pair of dice alternatively. The first one who throws a sum of 5 wins the pot. How much should the player who starts, add to the pot to make this a fair game? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Two players put one dollar into a pot. They decide to throw a pair of dice alternatively. The first one who throws a sum of 5 wins the pot. How much should the player who starts, add to the pot to make this a fair game?.
Two players put one dollar into a pot. They decide to throw a pair of dice alternatively. The first one who throws a sum of 5 wins the pot. How much should the player who starts, add to the pot to make this a fair game? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Two players put one dollar into a pot. They decide to throw a pair of dice alternatively. The first one who throws a sum of 5 wins the pot. How much should the player who starts, add to the pot to make this a fair game? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Two players put one dollar into a pot. They decide to throw a pair of dice alternatively. The first one who throws a sum of 5 wins the pot. How much should the player who starts, add to the pot to make this a fair game?.
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