Multiple choice

A pair of $12$-sided fair dice with faces numbered $1, 2, 3,..., 12$ is rolled. The probability that the sum of the numbers appearing has remainder $2$ when divided by $9$ is?

  1. $\displaystyle\frac{7}{72}$
  2. $\displaystyle\frac{5}{48}$
  3. $\displaystyle\frac{11}{144}$
  4. $\displaystyle\frac{1}{9}$
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D Correct answer
Explanation

Total outcomes = 12 * 12 = 144. We want sum S such that S mod 9 = 2. Possible sums: 2, 11, 20. Sum = 2: (1,1) [1 way]. Sum = 11: (1,10), (2,9), (3,8), (4,7), (5,6), (6,5), (7,4), (8,3), (9,2), (10,1) [10 ways]. Sum = 20: (8,12), (9,11), (10,10), (11,9), (12,8) [5 ways]. Total = 1 + 10 + 5 = 16. Probability = 16/144 = 1/9.

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