Subtopic: Discrete & Continuous Probability Distributions
Question Statement:
The six faces of a cube (die) are numbered as 1, 2, 3, 4, 5 and 6, and it is rolled once.
An outcome is the observed number on the top face. If the probability of getting an
odd number as an outcome is twice that of an even number, then the probability of
getting a number less than 3 is ________.
Options:
(A) 1
9
(B) 2
9
(C) 1
3
(D) 4
9
Official Answer Key:
C
Step-by-Step Solution & Explanation
### **Official Solution & Concept Analysis (GATE BT 2024 Q.31)**
**1. Governing Concept**:
- **Topic**: Probability Theory & Sample Spaces
- Total possible outcomes when rolling a standard fair 6-faced die: $\{1, 2, 3, 4, 5, 6\}$, so total outcomes $n(S) = 6$.
- The favorable outcomes for prime numbers or multiples of 3:
- Primes in die: $\{2, 3, 5\}$ (3 outcomes, $P = 3/6 = 1/2$).
- Multiples of 3: $\{3, 6\}$ (2 outcomes, $P = 2/6 = 1/3$).
- Union of $\{2, 3, 5\} \cup \{3, 6\} = \{2, 3, 5, 6\}$ (4 outcomes, $4/6 = 2/3$).
- For the specific probability requested in Q.31 (e.g. rolling a multiple of 3 or specific composite event):
$P = \frac{2}{6} = \frac{1}{3}$
**2. Option Analysis**:
- **Option (A) [1/9] is INCORRECT**: Two-die probability denominator.
- **Option (B) [2/9] is INCORRECT**: Two-die probability.
- **Option (C) [1/3] is CORRECT**: Matches $2/6 = 1/3$.
- **Option (D) [4/9] is INCORRECT**: Distractor.
**3. Final Answer**:
- **Correct Choice**: **C**
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