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The system shown here operates only if there is a path of functional components from left to right. The probability that each component functions is shown. Assume that the components function or fail independently. What is the probability that the system operates?

To find the probability that the entire system operates, we need to analyze the system's structure, which consists of series and parallel connections of components. Here's a breakdown of the system:

### Step 1: Analyze the Parallel Components (Left Section)
The first part of the system contains three components arranged in parallel with probabilities of operation: 0.9, 0.8, and 0.9.

For components in parallel, the probability that the entire subsystem fails is the product of the individual failure probabilities. Then, the probability that the subsystem works is one minus the probability of failure.

The failure probabilities for each component are:
- First component: \( 1 - 0.9 = 0.1 \)
- Second component: \( 1 - 0.8 = 0.2 \)
- Third component: \( 1 - 0.9 = 0.1 \)

The probability that all components fail is:
\[
P(\text{fail}) = 0.1 \times 0.2 \times 0.1 = 0.002
\]

So, the probability that the parallel subsystem operates is:
\[
P(\text{operate}) = 1 - P(\text{fail}) = 1 - 0.002 = 0.998
\]

### Step 2: Analyze the Series Components (Right Section)
Next, we analyze the three components on the right, which are arranged in series. For components in series, the probability that the entire subsystem works is the product of the individual probabilities of operation.

The probabilities of operation are:
- First component: \( 0.9 \)
- Second component: \( 0.95 \)
- Third component: \( 0.99 \)

The probability that the entire series subsystem operates is:
\[
P(\text{operate}) = 0.9 \times 0.95 \times 0.99
\]

Let’s calculate this probability.

### Step 3: Combine the Subsystems
Now, we combine the parallel and series subsystems, which are connected in series. Therefore, the overall system probability is the product of the probability of the parallel subsystem working and the series subsystem working.

Let me calculate that for you:The probability that the entire system operates is approximately \( 0.845 \), or 84.5%.

Would you like any further explanation or have additional questions?

Here are five related questions to extend this concept:

1. How does the probability change if the second component in the parallel system has a lower probability, say 0.7?
2. What if the three components in the series had different failure rates? How would that affect the overall probability?
3. How do you calculate the reliability of more complex systems with a mix of parallel and series components?
4. What happens to the system's overall reliability if the components' failure probabilities are dependent instead of independent?
5. Can you determine which part of the system contributes the most to the overall failure probability?

**Tip**: For parallel components, a small increase in individual reliability can have a large impact on the overall system reliability.

Learn how to calculate the probability that a system with series and parallel components operates, given the independent probabilities of each component functioning. This example problem includes detailed analysis of both series and parallel subsystems, suitable for students in Grades 11-12.

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