Question

Two dice are rolled together. The probability of getting an outcome (a, b) such that b = 2a, is

• A. \(\frac{1}{6}\)
• B. \(\frac{1}{12}\)
• C. \(\frac{1}{36}\)
• D. \(\frac{1}{9}\)

Answer 1

The Correct Option is B

When two dice are rolled together:
- Total number of possible outcomes = \(6 \times 6 = 36\).
- Let the outcome be \((a, b)\) where:
    - \(a\) = number on the first die
    - \(b\) = number on the second die

Step 1: Condition given
We want the probability that:
\[ b = 2a \]

Step 2: Find favorable outcomes
List possible values of \(a\) from 1 to 6 and calculate \(b = 2a\):

• If \(a = 1\), \(b = 2 \times 1 = 2\). Outcome: (1, 2) (valid since \(b \leq 6\))
• If \(a = 2\), \(b = 2 \times 2 = 4\). Outcome: (2, 4) (valid)
• If \(a = 3\), \(b = 2 \times 3 = 6\). Outcome: (3, 6) (valid)
• If \(a = 4\), \(b = 8\) (invalid, since \(b > 6\))
• If \(a = 5\), \(b = 10\) (invalid)
• If \(a = 6\), \(b = 12\) (invalid)

Step 3: Number of favorable outcomes
There are 3 favorable outcomes: (1, 2), (2, 4), (3, 6).

Step 4: Calculate probability
\[ P(b = 2a) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{3}{36} = \frac{1}{12} \]

Final Answer:
\[ \boxed{\frac{1}{12}} \]