An Ordinary Fair Die Is A Cube With The Numbers 1 Through 6

An Ordinary Fair Die Is A Cube With The Numbers1through6on The Sides

An ordinary (fair) die is a cube with the numbers 1 through 6 on the sides. When such a die is rolled twice in succession, and the face values of the two rolls are added together to produce the outcome of a single trial, the probability of specific events related to the sums obtained needs to be computed. Specifically, the problem asks for the probability that the sum is greater than 7 and the probability that the sum is neither divisible by 3 nor by 6. The calculations are to be expressed as exact fractions.

Paper For Above instruction

To analyze this problem, we first understand that rolling a fair six-sided die twice produces a total of 36 equally likely outcomes since each roll has 6 possible outcomes (6×6=36). Each outcome can be described as an ordered pair (i, j), where i is the result of the first roll and j is the result of the second roll, with each of i and j taking values from 1 to 6.

The sum of the two rolls for each outcome is given by S = i + j, which can range from 2 (when both rolls are 1) to 12 (when both rolls are 6). The probability distributions for sums of two dice are well-established, but we will verify and use them directly for accuracy.

Event A: The Sum is Greater Than 7

To find P(A), the probability that the sum exceeds 7, we first identify all outcomes where S > 7. The possible sums greater than 7 are 8, 9, 10, 11, and 12.

Let's enumerate the outcomes for each sum:

  • S = 8: (2,6), (3,5), (4,4), (5,3), (6,2)
  • S = 9: (3,6), (4,5), (5,4), (6,3)
  • S = 10: (4,6), (5,5), (6,4)
  • S = 11: (5,6), (6,5)
  • S = 12: (6,6)

Counting the total outcomes: 5 + 4 + 3 + 2 + 1 = 15 outcomes.

Since all 36 outcomes are equally likely, the probability that the sum is greater than 7 is

P(A) = 15/36 = 5/12.

Event B: The Sum is Not Divisible by 3 and Not Divisible by 6

To find P(B), we need to consider all outcomes where the sum is neither divisible by 3 nor by 6. First, identify all sums of two dice from 2 to 12 that are divisible by 3 or 6 and then eliminate outcomes with these sums.

Divisible by 3: sums 3, 6, 9, 12

Divisible by 6: sums 6, 12

Therefore, the sums that are divisible by 3 or 6 are 3, 6, 9, 12, with 6 and 12 also being divisible by 6.

Now, list the outcomes for sums that are NOT divisible by 3 or 6:

  • Sums = 2: (1,1)
  • Sums = 4: (1,3), (2,2), (3,1)
  • Sums = 5: (1,4), (2,3), (3,2), (4,1)
  • Sums = 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1)
  • Sums = 8: (2,6), (3,5), (4,4), (5,3), (6,2)
  • Sums = 10: (4,6), (5,5), (6,4)

Counting the total outcomes for these sums:

  • S = 2: 1 outcome
  • S = 4: 3 outcomes
  • S = 5: 4 outcomes
  • S = 7: 6 outcomes
  • S = 8: 5 outcomes
  • S = 10: 3 outcomes

Total outcomes satisfying the condition: 1 + 3 + 4 + 6 + 5 + 3 = 22 outcomes.

Thus, the probability that the sum is neither divisible by 3 nor by 6 is

P(B) = 22/36 = 11/18.

Concluding Remarks

The calculations demonstrate that for two fair six-sided dice, the probability of the sum exceeding 7 is 5/12, and the probability that the sum is not divisible by 3 or 6 is 11/18. These results are useful in understanding probability distributions for sums of dice and can be applied in various gaming and statistical scenarios.

References

  • Diaconis, P. (2000). The mathematics of perfect shuffles. The American Mathematical Monthly, 107(1), 10-24.
  • Grinstead, C. M., & Snell, J. L. (1997). Introduction to Probability. American Mathematical Society.
  • Ross, S. M. (2014). Introduction to Probability Models. Academic Press.
  • Feller, W. (1968). An Introduction to Probability Theory and Its Applications. Wiley.
  • Knuth, D. E. (1997). The Art of Computer Programming, Volume 1: Fundamental Algorithms. Addison-Wesley.
  • Feller, W. (1968). An Introduction to Probability Theory and Its Applications. Wiley.
  • Nelsen, R. B. (2006). An Introduction to Copulas. Springer Science & Business Media.
  • Ross, S. (2010). A First Course in Probability. Pearson Education.
  • Galambos, J. (1972). The Asymptotic Theory of Extreme Order Statistics.

    An Ordinary Fair Die Is A Cube With The Numbers1through6on The Sides

    An ordinary (fair) die is a cube with the numbers 1 through 6 on the sides. When such a die is rolled twice in succession, and the face values of the two rolls are added together to produce the outcome of a single trial, the probability of specific events related to the sums obtained needs to be computed. Specifically, the problem asks for the probability that the sum exceeds 7 and the probability that the sum is neither divisible by 3 nor by 6. The calculations are to be expressed as exact fractions.

    Paper For Above instruction

    To analyze this problem, we first understand that rolling a fair six-sided die twice produces a total of 36 equally likely outcomes since each roll has 6 possible outcomes (6×6=36). Each outcome can be described as an ordered pair (i, j), where i is the result of the first roll and j is the result of the second roll, with each of i and j taking values from 1 to 6.

    The sum of the two rolls for each outcome is given by S = i + j, which can range from 2 (when both rolls are 1) to 12 (when both rolls are 6). The probability distributions for sums of two dice are well-established, but we will verify and use them directly for accuracy.

    Let’s begin by computing the probability of Event A: the sum is greater than 7.

    Possible sums greater than 7 are 8, 9, 10, 11, and 12.

    Enumerate the outcomes for each sum:

    • S = 8: (2,6), (3,5), (4,4), (5,3), (6,2)
    • S = 9: (3,6), (4,5), (5,4), (6,3)
    • S = 10: (4,6), (5,5), (6,4)
    • S = 11: (5,6), (6,5)
    • S = 12: (6,6)

    Total outcomes for sums greater than 7: 5 + 4 + 3 + 2 + 1 = 15 outcomes.

    Since all possible outcomes are equally likely, the probability P(A) that the sum exceeds 7 is:

    P(A) = 15/36 = 5/12.

    Next, the probability of Event B: the sum is not divisible by 3 and not divisible by 6.

    For this, first identify all sums between 2 and 12 that are divisible by 3 or 6:

    • Divisible by 3: 3, 6, 9, 12
    • Divisible by 6: 6, 12

    Therefore, sums to exclude are 3, 6, 9, and 12.

    Remaining sums are 2, 4, 5, 7, 8, 10, which are not divisible by 3 or 6.

    Now, enumerate outcomes for these sums:

    • S = 2: (1,1)
    • S = 4: (1,3), (2,2), (3,1)
    • S = 5: (1,4), (2,3), (3,2), (4,1)
    • S = 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1)
    • S = 8: (2,6), (3,5), (4,4), (5,3), (6,2)
    • S = 10: (4,6), (5,5), (6,4)

    Count the number of outcomes:

    • S=2: 1 outcome
    • S=4: 3 outcomes
    • S=5: 4 outcomes
    • S=7: 6 outcomes
    • S=8: 5 outcomes
    • S=10: 3 outcomes

    Total outcomes satisfying the condition: 1 + 3 + 4 + 6 + 5 + 3 = 22 outcomes.

    Given the total of 36 equally likely outcomes, the probability P(B) is:

    P(B) = 22/36 = 11/18.

    In summary,

    The probability that the sum is greater than 7 is 5/12, and the probability that the sum is neither divisible by 3 nor by 6 is 11/18. These calculations help us understand the likelihood of different sum outcomes when rolling two fair dice and can inform applications in gaming, probability theory, and statistical analysis.

    References

    • Diaconis, P. (2000). The mathematics of perfect shuffles. The American Mathematical Monthly, 107(1), 10-24.
    • Grinstead, C. M., & Snell, J. L. (1997). Introduction to Probability. American Mathematical Society.
    • Ross, S. M. (2014). Introduction to Probability Models. Academic Press.
    • Feller, W. (1968). An Introduction to Probability Theory and Its Applications. Wiley.
    • Knuth, D. E. (1997). The Art of Computer Programming, Volume 1: Fundamental Algorithms. Addison-Wesley.
    • Nelsen, R. B. (2006). An Introduction to Copulas. Springer Science & Business Media.
    • Galambos, J. (1972). The Asymptotic Theory of Extreme Order Statistics.
    • Feller, W. (1968). An Introduction to Probability Theory and Its Applications. Wiley.
    • Nelsen, R. B. (2006). An Introduction to Copulas. Springer Science & Business Media.
    • Ross, S. (2010). A First Course in Probability. Pearson Education.

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