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The image shows a logic circuit where components \( A \), \( B \), \( C \), \( D \), and \( E \) have individual failure probabilities of 0.10, 0.15, 0.10, 0.20, and 0.25, respectively. The problem asks for the system's reliability calculation.

### Circuit Analysis
The system contains:
1. Two parallel branches: 
    - Top branch: \( A \) and \( D \) in series.
    - Bottom branch: \( B \), \( C \), and \( E \) in series.
2. These two branches are connected in **parallel**, meaning the system will fail only if both branches fail.

### Step-by-Step Solution
#### 1. Reliability of Each Branch
- **Reliability of the Top Branch (Series of \( A \) and \( D \)):**
  For components in series, the overall reliability is the product of the individual reliabilities:
  \[
  R_{\text{top}} = (1 - P_A) \times (1 - P_D) = (1 - 0.10) \times (1 - 0.20) = 0.90 \times 0.80 = 0.72
  \]
  
- **Reliability of the Bottom Branch (Series of \( B \), \( C \), and \( E \)):**
  Similarly, the reliability for components in series is the product of the individual reliabilities:
  \[
  R_{\text{bottom}} = (1 - P_B) \times (1 - P_C) \times (1 - P_E) = (1 - 0.15) \times (1 - 0.10) \times (1 - 0.25)
  \]
  \[
  R_{\text{bottom}} = 0.85 \times 0.90 \times 0.75 = 0.57375
  \]

#### 2. System Reliability (Parallel Connection)
The system reliability for parallel branches is:
\[
R_{\text{system}} = 1 - \left( (1 - R_{\text{top}}) \times (1 - R_{\text{bottom}}) \right)
\]
Substitute the values:
\[
R_{\text{system}} = 1 - \left( (1 - 0.72) \times (1 - 0.57375) \right) = 1 - (0.28 \times 0.42625) = 1 - 0.11935 = 0.88065
\]

### Final Answer:
The system reliability is approximately \( 0.881 \) (rounded to three decimal places).

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Do you have any further questions or would you like more details? Here are five related questions for deeper understanding:
1. What is the significance of series and parallel arrangements in reliability problems?
2. How does increasing the number of components in a series affect the system's overall reliability?
3. How can one optimize system reliability when dealing with components of varying failure probabilities?
4. Can you explain how redundancy improves the reliability of a system?
5. What other factors, besides failure probabilities, could influence real-world reliability analysis?

**Tip:** In reliability problems, remember that for series components, the system is only as reliable as the weakest link, while parallel arrangements help improve overall reliability by providing alternative pathways.

Calculate the reliability of each system shown in the diagram. Components A, B, C, D, and E function properly with probabilities of 0.10, 0.15, 0.10, 0.20, and 0.25, respectively. Calculate the overall system reliability.

Learn how to calculate the reliability of a system with components in series and parallel connections. This example uses components A, B, C, D, and E with failure probabilities and demonstrates a step-by-step solution, including important reliability formulas.

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