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 Based on the reliability diagram below showing the probability of success for each component, what is the probability of success for the whole set? 

The image shows a reliability block diagram with two groups of components connected in series and parallel configurations. To calculate the probability of success for the entire system, we need to:

1. Calculate the probability of success for each subsystem (group of components).
2. Combine them according to their configuration (series or parallel).

### Step 1: Left subsystem
The left side has two sets of components:
- The first set has two components: 0.20 and 0.60, arranged in **parallel**.
- The second set has two components: 0.75 and 0.60, also arranged in **parallel**.

For parallel systems, the probability of failure for the group is the product of the probabilities of failure of each component. Then, the probability of success is 1 minus this result.

#### First pair (0.20 and 0.60 in parallel):
- Failure probability = (1 - 0.20) * (1 - 0.60) = 0.80 * 0.40 = 0.32
- Success probability = 1 - 0.32 = 0.68

#### Second pair (0.75 and 0.60 in parallel):
- Failure probability = (1 - 0.75) * (1 - 0.60) = 0.25 * 0.40 = 0.10
- Success probability = 1 - 0.10 = 0.90

Now, these two parallel subsystems are connected **in series**. For series systems, the overall success probability is the product of the success probabilities of each part.

- Series success probability (left side) = 0.68 * 0.90 = 0.612

### Step 2: Right subsystem
The right side also has two sets of components:
- The first set has two components: 0.15 and 0.35, arranged in **parallel**.
- The second component has a reliability of 0.99, which is directly in **series** with the result of the parallel components.

#### First pair (0.15 and 0.35 in parallel):
- Failure probability = (1 - 0.15) * (1 - 0.35) = 0.85 * 0.65 = 0.5525
- Success probability = 1 - 0.5525 = 0.4475

Now this parallel subsystem is connected **in series** with the 0.99 component.

- Series success probability (right side) = 0.4475 * 0.99 = 0.4430

### Step 3: Combine both subsystems
The left and right subsystems are connected **in series**. So, the overall success probability is the product of the success probabilities of the two subsystems.

- Overall system success probability = 0.612 * 0.4430 = 0.2713

Thus, the probability of success for the whole set is **0.2713** or **27.13%**.

Would you like further details or have any questions?

Here are 5 related questions:
1. How do we calculate the probability of failure for series systems?
2. What would happen to the system reliability if we increase the reliability of one component?
3. Can this method be used for systems with more complex configurations (e.g., mixed series-parallel)?
4. How can redundancy improve system reliability in critical systems?
5. What are other ways to model reliability besides series-parallel networks?

**Tip:** For a parallel system, even if one component fails, the system can still succeed if the other components work, making it more reliable than a series system.

Based on the reliability diagram below showing the probability of success for each component, what is the probability of success for the whole set?

Find out how to calculate the probability of success for a system using a reliability block diagram. This problem involves key concepts in probability theory, with components arranged in both series and parallel configurations. We break down the steps, including the formulas for series and parallel systems, and provide a detailed solution. Suitable for Grades 10-12 students.

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