You Play A Game In Which Two Coins Are Flipped. If Both Coins Turn Up Tails, You Win 1 Point. How Many Points Would You Need To Lose For Each Of The Other Outcomes So That The Game Is Fair?A. 1 4 \frac{1}{4} 4 1 B. 1 3 \frac{1}{3} 3 1 C. 1 D.

Mar 9, 2025 by ADMIN 248 views

**Fairness in Coin Flipping Games: Determining the Number of Points to Lose**

Introduction

In the world of probability and game theory, fairness is a crucial concept. A game is considered fair when the expected value of winning is equal to the expected value of losing. In this article, we will explore a simple game involving coin flipping and determine the number of points a player would need to lose for each of the other outcomes to make the game fair.

The Game

The game involves flipping two coins. If both coins turn up tails, the player wins 1 point. The other possible outcomes are:

Both coins turn up heads: no points are awarded or lost
One coin turns up heads and the other turns up tails: the player loses 1 point

Determining Fairness

To determine the number of points a player would need to lose for each of the other outcomes to make the game fair, we need to calculate the expected value of winning and the expected value of losing.

Expected Value of Winning

The expected value of winning is the probability of winning multiplied by the number of points won. In this case, the probability of winning is 1/4 (since there is only one outcome where both coins turn up tails), and the number of points won is 1.

Expected value of winning = (1/4) x 1 = 1/4

Expected Value of Losing

The expected value of losing is the sum of the probabilities of losing multiplied by the number of points lost. In this case, there are two outcomes where the player loses 1 point: when one coin turns up heads and the other turns up tails. The probability of each of these outcomes is 1/2 (since there are two possible combinations: HT or TH).

Expected value of losing = (1/2) x 1 + (1/2) x 1 = 1

Making the Game Fair

To make the game fair, the expected value of winning must be equal to the expected value of losing. Since the expected value of winning is 1/4, the expected value of losing must also be 1/4.

Let x be the number of points a player would need to lose for each of the other outcomes to make the game fair. Then, the expected value of losing is:

Expected value of losing = (1/2) x x + (1/2) x x = 2x/2 = x

Since the expected value of losing must be 1/4, we can set up the equation:

x = 1/4

Conclusion

In conclusion, to make the game fair, a player would need to lose 1/4 point for each of the other outcomes. This means that if the player loses 1 point when one coin turns up heads and the other turns up tails, the game is not fair. However, if the player loses 1/4 point in this situation, the game is fair.

Answer

The correct answer is A. $\frac{1}{4}$ .

References

[1] Probability and Statistics by Jim Henley
[2] Game Theory by Steven J. Brams

Additional Information

The concept of fairness in games is crucial in probability and game theory.
The expected value of winning and the expected value of losing are used to determine fairness in games.
The number of points a player would need to lose for each of the other outcomes to make the game fair can be calculated using the expected value of losing.

Frequently Asked Questions

Q: What is the expected value of winning in the game? A: The expected value of winning is 1/4.
Q: What is the expected value of losing in the game? A: The expected value of losing is 1.
Q: How many points would a player need to lose for each of the other outcomes to make the game fair? A: A player would need to lose 1/4 point for each of the other outcomes to make the game fair.
Q&A: Fairness in Coin Flipping Games =====================================

Q: What is the concept of fairness in games?

A: The concept of fairness in games is crucial in probability and game theory. A game is considered fair when the expected value of winning is equal to the expected value of losing.

Q: How is the expected value of winning calculated?

A: The expected value of winning is calculated by multiplying the probability of winning by the number of points won. In the case of the coin flipping game, the probability of winning is 1/4 (since there is only one outcome where both coins turn up tails), and the number of points won is 1.

Q: How is the expected value of losing calculated?

A: The expected value of losing is calculated by summing the probabilities of losing multiplied by the number of points lost. In the case of the coin flipping game, there are two outcomes where the player loses 1 point: when one coin turns up heads and the other turns up tails. The probability of each of these outcomes is 1/2 (since there are two possible combinations: HT or TH).

Q: What is the expected value of losing in the coin flipping game?

A: The expected value of losing in the coin flipping game is 1.

Q: How many points would a player need to lose for each of the other outcomes to make the game fair?

A: A player would need to lose 1/4 point for each of the other outcomes to make the game fair.

Q: What is the relationship between the expected value of winning and the expected value of losing in a fair game?

A: In a fair game, the expected value of winning is equal to the expected value of losing.

Q: Can you provide an example of a game that is not fair?

A: Yes, consider a game where a player wins 1 point if both coins turn up tails, but loses 1 point if one coin turns up heads and the other turns up tails. In this case, the expected value of winning is 1/4, but the expected value of losing is 1, making the game unfair.

Q: How can the fairness of a game be determined?

A: The fairness of a game can be determined by calculating the expected value of winning and the expected value of losing, and comparing them to determine if they are equal.

Q: What is the significance of the concept of fairness in games?

A: The concept of fairness in games is significant because it ensures that players have an equal chance of winning and losing, and that the game is not biased towards one player or outcome.

Q: Can you provide a real-world example of a game that is fair?

A: Yes, consider a game of chance where a player wins 1 point if a coin turns up heads, and loses 1 point if the coin turns up tails. In this case, the expected value of winning is 1/2, and the expected value of losing is also 1/2, making the game fair.

Q: How can the fairness of a game be improved?

A: The fairness of a game can be improved by adjusting the rules or probabilities of the game to ensure that the expected value of winning and the expected value of losing are equal.

Q: What is the relationship between probability and fairness in games?

A: Probability and fairness are closely related in games. A game is considered fair when the probabilities of winning and losing are equal, and the expected value of winning and the expected value of losing are equal.