Victory on the Initial Dice Roll

Solution & Explanation

To determine the probability of winning on the initial roll in a dice game where you win if the sum of two six-sided dice is 7 or 11, we can follow these steps:

1. Understand the Total Possible Outcomes

• Each die has 6 sides, and since we are rolling two dice, the total number of possible outcomes is calculated by multiplying the number of sides on each die:

  $\text{Total possible outcomes} = 6 \times 6 = 36$

2. Identify the Winning Outcomes

• To win, the sum of the dice must be either 7 or 11. We will list down all combinations that result in these sums:

  • Sum of 7:

    • (1, 6)

    • (2, 5)

    • (3, 4)

    • (4, 3)

    • (5, 2)

    • (6, 1)

    • Total combinations for a sum of 7 = 6

  • Sum of 11:

    • (5, 6)

    • (6, 5)

    • Total combinations for a sum of 11 = 2

• Total winning outcomes = 6 (for sum of 7) + 2 (for sum of 11) = 8

3. Calculate the Probability of Winning

• The probability of winning on the first roll is the ratio of the number of winning outcomes to the total number of possible outcomes:

  $P(\text{Win}) = \frac{\text{Number of winning outcomes}}{\text{Total number of outcomes}} = \frac{8}{36}$

• Simplifying the fraction gives:

  $P(\text{Win}) = \frac{2}{9} \approx 0.2222$

4. Conclusion

• The probability of winning on your first roll by achieving a sum of 7 or 11 is approximately 0.2222, or 22.22%. This means that, on average, you can expect to win roughly 22.22% of the time on the initial roll under these rules.

Understanding this probability helps in strategic decision-making and expectation management when engaging in games of chance involving dice.