Two policemen, A and B, fire once each at the same time at an escaping convict. The probability that A hits the convict is three times the probability that B hits the convict. If the probability of the convict not getting injured is 0.5, the probability that B hits the convict is:
Options:
(A) 0.14
(B) 0.22
(C) 0.33
(D) 0.40
Official Answer Key:
A
Step-by-Step Solution & Explanation
### **Dedicated Step-by-Step Solution (GATE AE 2012 Q.61)**
**1. Problem Statement & Parameters**:
- **Module**: Aerospace Structures (2 Mark, MCQ)
- **Topic**: Thin-Walled Structures (Torsion of Multi-Cell Thin-Walled Tubes)
- **Organizing Institute**: IIT Delhi
- **Problem Statement**:
> Two policemen, A and B, fire once each at the same time at an escaping convict. The probability that A hits the convict is three times the probability that B hits the convict. If the probability of the convict not getting injured is 0.5, the probability that B hits the convict is:
**2. Mathematical Derivation & Physical Analysis**:
1. **Probability Model Setup**:
Let $P(B) = p$ be the probability that policeman B hits the convict.
Given that A's hit probability is three times B's: $P(A) = 3p$.
2. **Non-Injury (Both Miss) Condition**:
The convict is not injured if and only if both independent shots miss:
P(\text{not injured}) = (1 - P(A))(1 - P(B)) = (1 - 3p)(1 - p) = 0.5
1 - 4p + 3p^2 = 0.5 \implies 3p^2 - 4p + 0.5 = 0 \implies 6p^2 - 8p + 1 = 0
3. **Quadratic Solution**:
p = \frac{8 \pm \sqrt{(-8)^2 - 4(6)(1)}}{2(6)} = \frac{8 - \sqrt{40}}{12} = \frac{8 - 6.3246}{12} = \frac{1.6754}{12} \approx 0.1396 \approx 0.14
(The root $p = \frac{8 + 6.32}{12} > 1$ is physically extraneous since $3p \le 1$).
Matches **Option (A)** (0.14).
**3. Comprehensive Options Analysis**:
- **(A)** *"0.14"*: ✅ **CORRECT**.
- **(B)** *"0.22"*: ❌ **INCORRECT**.
- **(C)** *"0.33"*: ❌ **INCORRECT**.
- **(D)** *"0.40"*: ❌ **INCORRECT**.
**4. Official Answer Verification**:
- **Official Master Answer**: **A**
- **Evaluation Status**: Matches official GATE 2012 (IIT Delhi) master answer key with 100% precision.