Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2024 Q.29)**
**1. Probability Space for Rolling Two Fair 6-Sided Dice**:
- Total possible outcomes when rolling two fair dice:
$$N(S) = 6 \times 6 = 36$$
**2. Favorable Outcomes (Odd Numbers on Both Dice)**:
- Odd numbers on a single die: $\{1, 3, 5\}$ (3 outcomes).
- Probability of getting an odd number on the first die: $P(O_1) = \frac{3}{6} = \frac{1}{2}$.
- Probability of getting an odd number on the second die: $P(O_2) = \frac{3}{6} = \frac{1}{2}$.
Because the outcomes of rolling two fair dice are mutually independent:
$$P(O_1 \cap O_2) = P(O_1) \times P(O_2) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} = 0.25$$
Alternatively:
- Number of favorable pairs $(d_1, d_2) \in \{1, 3, 5\} \times \{1, 3, 5\} = 3 \times 3 = 9$ pairs.
- $P = \frac{9}{36} = 0.25$.
**3. Official Master Key**:
- **Official Answer Key**: **0.25** (Acceptable Range: `[0.24, 0.26]`)
- **Exam Takeaway**: For independent events $A$ and $B$, $P(A \cap B) = P(A) P(B) = 0.5 \times 0.5 = 0.25$.