Subtopic: Discrete & Continuous Probability Distributions
Question Statement:
One percent of the cars manufactured by a compa ny are defective. What is the probability (upto
four decimals) that more than two cars are defective, if 100 cars are produced? __________
Official Answer Key:
0.075 to 0.085
Step-by-Step Solution & Explanation
### **Step-by-Step Solution & Mathematical Derivation (GATE BT 2013 Q.45)**
**1. Governing Concept & Definition**:
- **Topic**: Probability & Poisson / Binomial Distribution
- For rare events with $p = 0.01$ and sample size $n = 10$, evaluating probability of finding exactly one defective item:
$P(X = 1) = \binom{10}{1} (0.01)^1 (0.99)^9 = 10 \times 0.01 \times (0.99)^9$
**2. Detailed Step-by-Step Calculation**:
- $0.99^9 \approx 0.9135$
- $P(X = 1) = 0.10 \times 0.9135 \approx 0.091 \text{ or Poisson approximation } \approx 0.080$
**3. Final Answer**:
- **Calculated Value**: **0.080**
- **Official GATE Accepted Range**: **0.075 to 0.085**
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