Multiple choice

Two dice are rolled simultaneously. Find the probability that the sum of the numbers on the faces is neither divisible by 3 nor by 4.

  1. $\dfrac{4}{7}$
  2. $\dfrac{4}{9}$
  3. $\dfrac{2}{9}$

  4. None of these
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B Correct answer
Explanation

Total outcomes = 36. Sums divisible by 3: 3, 6, 9, 12 (counts: 2, 5, 4, 1 = 12). Sums divisible by 4: 4, 8, 12 (counts: 3, 5, 1 = 9). Sums divisible by both 3 and 4 (i.e., 12): 1. Union = 12 + 9 - 1 = 20. Neither divisible = 36 - 20 = 16. Probability = 16/36 = 4/9.

AI explanation

When two dice are rolled, the sample space contains 36 outcomes, and we must find the probability of sums that are neither divisible by 3 nor by 4. The sums divisible by 3 are 3, 6, 9, and 12, yielding 12 outcomes; the sums divisible by 4 are 4, 8, and 12, yielding 9 outcomes. Since the sum of 12 is counted in both groups, we use the inclusion-exclusion principle to find 12 + 9 - 1 = 20 total unfavorable outcomes. This leaves 16 favorable outcomes, making the probability 16/36, which simplifies to 4/9.

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