The number of micro-meteorite strikes hitting a spacecraft hull follows a Poisson process with an average rate of $\lambda = 2.0$ impacts per year. What is the probability (in %) that the spacecraft experiences exactly 1 impact in a given year? (Take $e^{-2} \approx 0.1353$ and round off to one decimal place)

Answer Key:

27.1

### 5-Stage Derivation & Analytical Solution: 1. **Stage 1: Poisson Probability Law:** $P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!}$. 2. **Stage 2: Parameter Substitution:** * Average rate $\lambda = 2.0$. * Number of occurrences $k = 1$. 3. **Stage 3: Numerical Computation:** * $P(X = 1) = \frac{e^{-2} (2)^1}{1!} = 2 e^{-2}$. * $e^{-2} \approx 0.13533528$. * $P(X = 1) = 2 \times 0.13533528 = 0.27067$. 4. **Stage 4: Percentage Conversion:** In percent: $0.27067 \times 100 = 27.067\% \approx 27.1\%$. 5. **Stage 5: Official Reference:** Sheldon Ross, *A First Course in Probability*, Chapter 4.

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