Subtopic: Discrete & Continuous Probability Distributions
Question Statement:
Two fair six-sided dice are thrown. The probability of getting 12 as the product of
the numbers on the dice (rounded off to two decimal places) is ____________.
Official Answer Key:
0.10 to 0.12
Step-by-Step Solution & Explanation
### **Step-by-Step Solution & Mathematical Derivation (GATE BT 2023 Q.33)**
**1. Governing Concept & Definition**:
- **Topic**: Probability & Product of Independent Dice Rolls
- Total possible outcomes when rolling two fair six-sided dice:
$n(S) = 6 \times 6 = 36$
- Probability is:
$P(E) = \frac{n(E)}{n(S)}$
**2. Detailed Step-by-Step Calculation**:
- **Condition**: Product of numbers on the two dice equals 12 ($x \times y = 12$).
- **Possible pairs $(x, y)$ where $x, y \in \{1, 2, 3, 4, 5, 6\}$**:
1. $(2, 6)$ since $2 \times 6 = 12$
2. $(3, 4)$ since $3 \times 4 = 12$
3. $(4, 3)$ since $4 \times 3 = 12$
4. $(6, 2)$ since $6 \times 2 = 12$
- **Total favorable outcomes**: $n(E) = 4$
- **Probability Calculation**:
$P(E) = \frac{4}{36} = \frac{1}{9} \approx 0.1111 \approx 0.11$
**3. Final Answer**:
- **Calculated Value**: **0.11**
- **Official GATE Accepted Range**: **0.10 to 0.12**
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