Question

If two dice are rolled, determine the probability that the sum of the numbers on the top faces is a multiple of 3, given that their sum is an odd number.

• A. \(\frac{1}{6}\)
• B. \(\frac{11}{36}\)
• C. \(\frac{1}{3}\)
• D. \(\frac{7}{18}\)

Hint:

When dealing with conditional probabilities in dice rolls, organizing the possible outcomes based on the condition can simplify calculations.

Answer 1

The Correct Option is C

First, calculate the total number of outcomes where the sum is odd. Odd sums can be obtained by the following combinations (first die, second die): - (1, 2), (1, 4), (1, 6), (2, 1), (2, 3), (2, 5), ... - Repeating similarly for other numbers, there are 3 odd combinations for each number on the first die, giving: \[ 18 \text{ total odd combinations} \quad (3 \text{ odd outcomes per die} \times 6 \text{ faces}) \] Next, find the combinations where the sum is both odd and a multiple of 3 (3, 9, 15 possible, but only 3 and 9 can be rolled): - Sums of 3: (1, 2), (2, 1) - Sums of 9: (3, 6), (4, 5), (5, 4), (6, 3) Total favorable outcomes: \[ 6 \text{ outcomes} \] Thus, the probability that the sum is a multiple of 3 given that it is odd: \[ \text{Probability} = \frac{\text{favorable outcomes}}{\text{total odd outcomes}} = \frac{6}{18} = \frac{1}{3} \]