1. **State the problem:** We have a system with seven components, each with an exponential failure rate $\lambda_i$. We want to find the system reliability after 1000 hours and the mean time to failure (MTTF) of the system. 2. **Given failure rates:** $$\lambda_A = 0.0005, \quad \lambda_B = 0.0005, \quad \lambda_C = 0.0003, \quad \lambda_D = 0.0008, \quad \lambda_E = 0.0004, \quad \lambda_F = 0.006, \quad \lambda_G = 0.0064$$ 3. **Assumption:** The time to failure for each component is exponentially distributed, so the reliability of each component at time $t$ is: $$R_i(t) = e^{-\lambda_i t}$$ 4. **System configuration:** Since the figure is not provided, assume the system is a series system (all components must work for system success). The system failure rate $\lambda_{sys}$ is the sum of individual failure rates: $$\lambda_{sys} = \sum_{i} \lambda_i = 0.0005 + 0.0005 + 0.0003 + 0.0008 + 0.0004 + 0.006 + 0.0064 = 0.0149$$ 5. **System reliability after 1000 hours:** $$R_{sys}(1000) = e^{-\lambda_{sys} \times 1000} = e^{-0.0149 \times 1000} = e^{-14.9}$$ Calculate: $$e^{-14.9} \approx 3.37 \times 10^{-7}$$ 6. **Mean time to failure (MTTF) of the system:** For exponential distribution, $$\text{MTTF} = \frac{1}{\lambda_{sys}} = \frac{1}{0.0149} \approx 67.11 \text{ hours}$$ **Final answers:** - Reliability after 1000 hours: $3.37 \times 10^{-7}$ - Mean time to failure: 67.11 hours