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Data Interview Question

Three Dice Rolls: Achieving a Sum of Ten

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Solution & Explanation

Problem Statement: Calculate the probability of achieving a total sum of 10 when rolling a six-sided die three times.

Step-by-Step Solution:

  1. Understanding the Problem:

    • Each die has 6 faces, numbered from 1 to 6.
    • We roll the die three times and want the sum of the numbers on the top faces to be 10.
  2. Total Possible Outcomes:

    • Each roll is independent, and there are 6 possible outcomes per roll.
    • Total possible outcomes when rolling three dice = 6×6×6=2166 \times 6 \times 6 = 216.
  3. Desired Outcomes:

    • We need to find all combinations of three numbers (x,y,zx, y, z) such that x+y+z=10x + y + z = 10 where each number is between 1 and 6.
    • Possible combinations are:
      • (1,3,6),(1,4,5),(2,2,6),(2,3,5),(2,4,4),(3,3,4)(1, 3, 6), (1, 4, 5), (2, 2, 6), (2, 3, 5), (2, 4, 4), (3, 3, 4)
  4. Calculating Permutations for Each Combination:

    • (1,3,6)(1, 3, 6): All numbers are different, so there are 3!=63! = 6 permutations.
    • (1,4,5)(1, 4, 5): All numbers are different, so there are 3!=63! = 6 permutations.
    • (2,2,6)(2, 2, 6): Two numbers are the same, so there are 3!2!=3\frac{3!}{2!} = 3 permutations.
    • (2,3,5)(2, 3, 5): All numbers are different, so there are 3!=63! = 6 permutations.
    • (2,4,4)(2, 4, 4): Two numbers are the same, so there are 3!2!=3\frac{3!}{2!} = 3 permutations.
    • (3,3,4)(3, 3, 4): Two numbers are the same, so there are 3!2!=3\frac{3!}{2!} = 3 permutations.
  5. Total Favorable Outcomes:

    • Summing all permutations gives us the total number of favorable outcomes:
      • 6+6+3+6+3+3=276 + 6 + 3 + 6 + 3 + 3 = 27
  6. Probability Calculation:

    • Probability of achieving a sum of 10 = Total favorable outcomesTotal possible outcomes\frac{\text{Total favorable outcomes}}{\text{Total possible outcomes}}
    • 27216=18\frac{27}{216} = \frac{1}{8}
  7. Conclusion:

    • The probability of rolling three six-sided dice and getting a sum of 10 is 18\frac{1}{8} or 12.5%.