Probability of Winning and Losing

Oct 13, 2025 by wbrum.org 76 views

Introduction

In this article, we will explore the probability of going bankrupt when playing a simple coin game. The game involves flipping a coin, where if Heads is flipped, the player receives their bet back plus the same in winnings, and if Tails is flipped, the player loses their bet. We will use the concept of probability and stochastic processes to analyze the game and determine the likelihood of going bankrupt.

The Game Rules

The game rules are as follows:

The player starts with a bankroll of $10.
The player can place a bet on each flip of the coin.
If the coin lands on Heads, the player receives their bet back plus the same in winnings.
If the coin lands on Tails, the player loses their bet.
The game continues until the player's bankroll reaches $0 or the player chooses to stop playing.

Probability of Winning and Losing

To analyze the game, we need to calculate the probability of winning and losing. Since the coin is fair, the probability of Heads is 0.5, and the probability of Tails is also 0.5.

Calculating the Probability of Winning

The probability of winning is the probability of the coin landing on Heads, which is 0.5.

Calculating the Probability of Losing

The probability of losing is the probability of the coin landing on Tails, which is 0.5.

Expected Value

The expected value of the game is the average amount of money the player can expect to win or lose per bet. To calculate the expected value, we need to multiply the probability of winning by the amount of money won and the probability of losing by the amount of money lost.

Expected Value Formula

The expected value formula is:

E = P(W) * W - P(L) * L

Where:

E is the expected value
P(W) is the probability of winning
W is the amount of money won
P(L) is the probability of losing
L is the amount of money lost

Plugging in the Values

We can plug in the values as follows:

E = 0.5 * 2 - 0.5 * 10 E = 1 - 5 E = -4

Interpretation

The expected value of -4 means that the player can expect to lose $4 per bet on average.

Probability of Going Bankrupt

To calculate the probability of going bankrupt, we need to use the concept of stochastic processes. We can model the game as a random walk, where the player's bankroll increases or decreases by the amount of the bet on each flip of the coin.

Random Walk Model

The random walk model is a mathematical model that describes the movement of a particle in a random environment. In this case, the particle is the player's bankroll, and the environment is the coin flips.

Calculating the Probability of Going Bankrupt

To calculate the probability of going bankrupt, we need to use the concept of first passage time. The first passage time is the time it takes for the particle to reach a certain level, in this case, the level of $0.

First Passage Time Formula

The first passage time formula is:

P(FPT ≤ t) = 1 - (1 - p)^t

Where:

P(FPT ≤ t) is the probability of first passage time less than or equal to t
p is the probability of winning
t is the time

Plugging in the Values

We can plug in the values as follows:

P(FPT ≤ t) = 1 - (1 - 0.5)^t

Solving for t

To solve for t, we need to find the time it takes for the probability of first passage time to reach 1, which means the player has gone bankrupt.

Solving the Equation

We can solve the equation as follows:

1 - (1 - 0.5)^t = 1 (1 - 0.5)^t = 0 t = ∞

Interpretation

The result shows that the probability of going bankrupt is 1, which means that the player will eventually go bankrupt with probability 1.

Conclusion

In conclusion, the probability of going bankrupt on a coin game is 1, which means that the player will eventually go bankrupt with probability 1. This is because the expected value of the game is negative, and the random walk model shows that the player's bankroll will eventually reach $0.

Recommendation

Based on the analysis, we recommend that the player should not play the game, as the probability of going bankrupt is 1.

Limitations

The analysis assumes that the coin is fair and that the player starts with a bankroll of $10. In reality, the coin may not be fair, and the player may start with a different bankroll. Additionally, the analysis does not take into account the player's risk tolerance and other factors that may affect the game.

Future Research

Future research could involve analyzing the game with different coin probabilities and bankrolls, as well as incorporating the player's risk tolerance and other factors into the analysis.

References

[1] Feller, W. (1968). An introduction to probability theory and its applications. John Wiley & Sons.
[2] Karlin, S., & Taylor, H. M. (1975). A first course in stochastic processes. Academic Press.
[3] Ross, S. M. (2014). Introduction to probability models. Academic Press.
Probability of Going Bankrupt on a Coin Game: Q&A =====================================================

Introduction

In our previous article, we explored the probability of going bankrupt when playing a simple coin game. We used the concept of probability and stochastic processes to analyze the game and determine the likelihood of going bankrupt. In this article, we will answer some frequently asked questions about the game and provide additional insights.

Q: What is the probability of going bankrupt on a coin game?

A: The probability of going bankrupt on a coin game is 1, which means that the player will eventually go bankrupt with probability 1.

Q: Why is the probability of going bankrupt 1?

A: The probability of going bankrupt is 1 because the expected value of the game is negative, and the random walk model shows that the player's bankroll will eventually reach $0.

Q: What is the expected value of the game?

A: The expected value of the game is -4, which means that the player can expect to lose $4 per bet on average.

Q: How does the random walk model work?

A: The random walk model is a mathematical model that describes the movement of a particle in a random environment. In this case, the particle is the player's bankroll, and the environment is the coin flips.

Q: What is the first passage time?

A: The first passage time is the time it takes for the particle to reach a certain level, in this case, the level of $0.

Q: How is the first passage time calculated?

A: The first passage time is calculated using the formula:

P(FPT ≤ t) = 1 - (1 - p)^t

Where:

P(FPT ≤ t) is the probability of first passage time less than or equal to t
p is the probability of winning
t is the time

Q: What is the probability of winning on a coin game?

A: The probability of winning on a coin game is 0.5, which means that the player has a 50% chance of winning on each flip of the coin.

Q: What is the probability of losing on a coin game?

A: The probability of losing on a coin game is 0.5, which means that the player has a 50% chance of losing on each flip of the coin.

Q: Can the player's bankroll ever reach a certain level?

A: Yes, the player's bankroll can reach a certain level, but it will eventually reach $0 with probability 1.

Q: What is the implication of the probability of going bankrupt being 1?

A: The implication of the probability of going bankrupt being 1 is that the player should not play the game, as the probability of going bankrupt is 1.

Q: Are there any limitations to the analysis?

A: Yes, the analysis assumes that the coin is fair and that the player starts with a bankroll of $10. In reality, the coin may not be fair, and the player may start with a different bankroll. Additionally, the analysis does not take into account the player's risk tolerance and other factors that may affect the game.

Q: What are some potential future research directions?

A: Some potential future research directions include analyzing the game with different coin probabilities and bankrolls, as well as incorporating the player's risk tolerance and other factors into the analysis.

Conclusion

In conclusion, the probability of going bankrupt on a coin game is 1, which means that the player will eventually go bankrupt with probability 1. We hope that this Q&A article has provided additional insights into the game and its analysis.

References

[1] Feller, W. (1968). An introduction to probability theory and its applications. John Wiley & Sons.
[2] Karlin, S., & Taylor, H. M. (1975). A first course in stochastic processes. Academic Press.
[3] Ross, S. M. (2014). Introduction to probability models. Academic Press.