Calculating Probability of Winning for Player A

Oct 13, 2025 by wbrum.org 141 views

Introduction

In this article, we will explore the concept of probability and its application in a game where three players, A, B, and C, throw a die one after the other in the same order until one of them gets a '6' and wins the game. We will calculate the respective probability of winning for each player, assuming that player A starts the game.

Understanding Probability

Probability is a measure of the likelihood of an event occurring. It is usually expressed as a number between 0 and 1, where 0 represents an impossible event and 1 represents a certain event. In this case, we want to find the probability of each player winning the game.

Calculating Probability

To calculate the probability of each player winning, we need to consider the following:

The probability of getting a '6' on a single throw of a die is 1/6, since there are six possible outcomes (1, 2, 3, 4, 5, and 6).
The probability of not getting a '6' on a single throw is 5/6, since there are five possible outcomes that are not '6'.
Since the game continues until one of the players gets a '6', we need to consider the probability of each player getting a '6' on their first throw, second throw, third throw, and so on.

Calculating Probability of Winning for Player A

Player A starts the game, so we need to calculate the probability of A winning on their first throw, second throw, third throw, and so on.

The probability of A winning on their first throw is 1/6, since the probability of getting a '6' on a single throw is 1/6.
The probability of A winning on their second throw is (5/6) × (1/6), since A needs to throw a non-'6' on their first throw and a '6' on their second throw.
The probability of A winning on their third throw is (5/6) × (5/6) × (1/6), since A needs to throw two non-'6's on their first two throws and a '6' on their third throw.
We can continue this pattern to calculate the probability of A winning on their fourth throw, fifth throw, and so on.

Calculating Probability of Winning for Player B

Player B throws the die after A, so we need to calculate the probability of B winning on their first throw, second throw, third throw, and so on.

The probability of B winning on their first throw is (5/6) × (1/6), since B needs to throw a non-'6' on A's first throw and a '6' on their first throw.
The probability of B winning on their second throw is (5/6) × (5/6) × (1/6), since B needs to throw two non-'6's on A's first two throws and a '6' on their second throw.
We can continue this pattern to calculate the probability of B winning on their third throw, fourth throw, and so on.

Calculating Probability of Winning for Player C

Player C throws the die after B, so we need to calculate the probability of C winning on their first throw, second throw, third throw, and so on.

The probability of C winning on their first throw is (5/6) × (5/6) × (1/6), since C needs to throw two non-'6's on A's first two throws and B's first throw, and a '6' on their first throw.
We can continue this pattern to calculate the probability of C winning on their second throw, third throw, and so on.

Calculating the Probability of Winning for Each Player

To calculate the probability of winning for each player, we need to sum up the probabilities of winning on each throw.

The probability of A winning is 1/6 + (5/6) × (1/6) + (5/6) × (5/6) × (1/6) + ...
The probability of B winning is (5/6) × (1/6) + (5/6) × (5/6) × (1/6) + (5/6) × (5/6) × (5/6) × (1/6) + ...
The probability of C winning is (5/6) × (5/6) × (1/6) + (5/6) × (5/6) × (5/6) × (1/6) + (5/6) × (5/6) × (5/6) × (5/6) × (1/6) + ...

Simplifying the Calculations

To simplify the calculations, we can use the formula for the sum of an infinite geometric series:

1 + r + r^2 + r^3 + ... = 1 / (1 - r)

where r is the common ratio.

In this case, the common ratio is 5/6, since the probability of not getting a '6' on a single throw is 5/6.

Calculating the Probability of Winning for Each Player

Using the formula for the sum of an infinite geometric series, we can calculate the probability of winning for each player:

The probability of A winning is 1/6 / (1 - 5/6) = 1/6 / (1/6) = 1
The probability of B winning is (5/6) × (1/6) / (1 - 5/6) = (5/6) × (1/6) / (1/6) = 5/6
The probability of C winning is (5/6) × (5/6) × (1/6) / (1 - 5/6) = (5/6) × (5/6) × (1/6) / (1/6) = 25/36

Conclusion

In this article, we calculated the probability of winning for each player in a game where three players, A, B, and C, throw a die one after the other in the same order until one of them gets a '6' and wins the game. We found that the probability of A winning is 1, the probability of B winning is 5/6, and the probability of C winning is 25/36.

References

[1] Probability Theory and Statistical Inference, 2nd Edition, by Robert V. Hogg and Elliot A. Tanis
[2] A First Course in Probability, 7th Edition, by Sheldon M. Ross

Appendix

The following is a Python code snippet that calculates the probability of winning for each player:

import math

def calculate_probability_of_winning():
    # Define the probability of getting a '6' on a single throw
    probability_of_getting_six = 1/6

    # Define the probability of not getting a '6' on a single throw
    probability_of_not_getting_six = 5/6

    # Calculate the probability of A winning
    probability_of_a_winning = 1 / (1 - probability_of_not_getting_six)

    # Calculate the probability of B winning
    probability_of_b_winning = (probability_of_not_getting_six) * (probability_of_getting_six) / (1 - probability_of_not_getting_six)

    # Calculate the probability of C winning
    probability_of_c_winning = (probability_of_not_getting_six) * (probability_of_not_getting_six) * (probability_of_getting_six) / (1 - probability_of_not_getting_six)

    return probability_of_a_winning, probability_of_b_winning, probability_of_c_winning

# Calculate the probability of winning for each player
probability_of_a_winning, probability_of_b_winning, probability_of_c_winning = calculate_probability_of_winning()

# Print the results
print("The probability of A winning is:", probability_of_a_winning)
print("The probability of B winning is:", probability_of_b_winning)
print("The probability of C winning is:", probability_of_c_winning)
```<br/>
**A, B and C Throw a Die: Q&A**
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**Q: What is the probability of A winning the game?**
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A: The probability of A winning the game is 1, since A has a 1/6 chance of getting a '6' on their first throw, and if they don't get a '6', the game continues until one of them gets a '6'.

**Q: What is the probability of B winning the game?**
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A: The probability of B winning the game is 5/6, since B has a 5/6 chance of not getting a '6' on A's first throw, and then a 1/6 chance of getting a '6' on their first throw.

**Q: What is the probability of C winning the game?**
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A: The probability of C winning the game is 25/36, since C has a 5/6 chance of not getting a '6' on A's first two throws, and then a 5/6 chance of not getting a '6' on B's first throw, and finally a 1/6 chance of getting a '6' on their first throw.

**Q: How does the game continue if none of the players get a '6' on their first throw?**
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A: If none of the players get a '6' on their first throw, the game continues with the next player throwing the die. The game continues until one of the players gets a '6'.

**Q: What is the probability of the game continuing indefinitely?**
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A: The probability of the game continuing indefinitely is 0, since the game will eventually end when one of the players gets a '6'.

**Q: Can the game end in a tie?**
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A: No, the game cannot end in a tie, since the game will continue until one of the players gets a '6'.

**Q: How does the probability of winning change if the players throw the die multiple times?**
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A: The probability of winning does not change if the players throw the die multiple times, since the probability of getting a '6' on a single throw is still 1/6.

**Q: Can the game be simulated using a computer program?**
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A: Yes, the game can be simulated using a computer program. The program can simulate the throwing of the die and calculate the probability of winning for each player.

**Q: What is the expected value of the game?**
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A: The expected value of the game is the average number of throws it takes for one of the players to get a '6'. The expected value can be calculated using the formula:

Expected value = 1 / (1 - probability of not getting a '6')

**Q: How does the expected value change if the players throw the die multiple times?**
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A: The expected value does not change if the players throw the die multiple times, since the probability of getting a '6' on a single throw is still 1/6.

**Q: Can the game be generalized to more than three players?**
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A: Yes, the game can be generalized to more than three players. The probability of winning for each player can be calculated using the same formula.

**Q: What is the probability of all players getting a '6' on their first throw?**
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A: The probability of all players getting a '6' on their first throw is (1/6)^3, since each player has a 1/6 chance of getting a '6' on their first throw.

**Q: What is the probability of all players not getting a '6' on their first throw?**
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A: The probability of all players not getting a '6' on their first throw is (5/6)^3, since each player has a 5/6 chance of not getting a '6' on their first throw.

**Q: Can the game be used to model real-world situations?**
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A: Yes, the game can be used to model real-world situations where there is a small probability of success. For example, the game can be used to model the probability of a company going bankrupt or the probability of a person getting a certain disease.