Consider the following system:

→ 0.85 → 0.85 →

Determine the probability that the system will operate under each of these conditions:

The system as shown.

Note: Do not round your intermediate calculations. Round your final answer to 4 decimal places.



Each system component has a backup with a probability of .85 and a switch that is 100 percent reliable.

Note: Do not round your intermediate calculations. Round your final answer to 4 decimal places.



Each system component has a backup with a probability of .85 and a switch that is 98 percent reliable.

Note: Do not round your intermediate calculations. Round your final answer to 4 decimal places.



This problem involves calculating the overall probability of system success based on the given conditions. Let's go through each part step by step:

### 1. The system as shown (→ 0.85 → 0.85 →)

For this scenario, assume that the system consists of two independent components, each with a probability of success of 0.85. Since the components are arranged in series, the overall probability of the system operating is the product of their individual probabilities.

\[
P_{\text{system}} = P_1 \times P_2 = 0.85 \times 0.85
\]

Let me calculate this.

\[
P_{\text{system}} = 0.85^2
\]

\[
P_{\text{system}} = 0.7225
\]

Thus, the probability that the system will operate as shown is **0.7225**.

---

### 2. Each system component has a backup with a probability of 0.85 and a 100% reliable switch

In this case, each component has a backup, meaning that the main component and the backup component can both try to operate, increasing the reliability of each component.

For each component, the probability of failure is:
\[
P_{\text{failure of component}} = (1 - P_{\text{main}}) \times (1 - P_{\text{backup}})
\]
Since the probability of both the main and backup failing is the product of their individual failure probabilities:

\[
P_{\text{failure}} = (1 - 0.85) \times (1 - 0.85) = 0.15 \times 0.15 = 0.0225
\]

So, the probability of success for each component (with a backup) is:
\[
P_{\text{component success}} = 1 - P_{\text{failure}} = 1 - 0.0225 = 0.9775
\]

Since the components are in series, the overall probability of the system operating is:
\[
P_{\text{system}} = 0.9775 \times 0.9775
\]

Let me calculate this.

\[
P_{\text{system}} = 0.9775^2 = 0.9555
\]

Thus, the probability that the system will operate under these conditions is **0.9555**.

---

### 3. Each system component has a backup with a probability of 0.85 and a switch that is 98% reliable

In this case, the switch is 98% reliable, meaning that it introduces an additional probability of failure in the system. The probability of each component operating successfully (including the backup and the switch reliability) becomes:

\[
P_{\text{component success}} = 0.9775 \times 0.98
\]

Let's calculate this:

\[
P_{\text{component success}} = 0.9775 \times 0.98 = 0.95895
\]

Since the components are in series, the overall probability of the system operating is:
\[
P_{\text{system}} = 0.95895 \times 0.95895
\]

Let me calculate this.

\[
P_{\text{system}} = 0.95895^2 = 0.9196
\]

Thus, the probability that the system will operate under these conditions is **0.9196**.

---

### Final Results:
1. Probability of the system as shown: **0.7225**
2. Probability with backup and 100% reliable switch: **0.9555**
3. Probability with backup and 98% reliable switch: **0.9196**

Do you have any questions or would you like additional details?

---

Here are 5 related questions:
1. How does adding more components in series affect the overall system reliability?
2. How would a 90% reliable switch affect the system probability in the third case?
3. What is the effect of having a backup system with a lower probability of success (e.g., 0.80)?
4. How would the system probability change if the components were arranged in parallel instead of series?
5. What would the system reliability be if only one component had a backup?

**Tip**: For series systems, the reliability decreases as more components are added, but backups can significantly improve the overall probability of success.

Consider the following system: → 0.85 → 0.85 →. Determine the probability that the system will operate under each of these conditions: The system as shown. Each system component has a backup with a probability of 0.85 and a switch that is 100 percent reliable. Each system component has a backup with a probability of 0.85 and a switch that is 98 percent reliable. Do not round your intermediate calculations. Round your final answer to 4 decimal places.

Learn how to calculate the probability of a system operating under various conditions, including with backup components and different switch reliability levels. This problem focuses on reliability theory and is ideal for advanced high school or college students.

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