Reliability
For a parallel structure of identical components, the system can succeed if at least one of the components succeeds. Assume that components fail independently of each other and that each component has a 0.15 probability of failure.
c. How many components would be needed in the structure so that the probability the system will succeed is greater than 0.999999?

Verified step by step guidance

1. Identify the probability of failure for each component, which is given as 0.15. Therefore, the probability of success for each component is \$1 - 0.15 = 0.85$.
2. Since the components are arranged in parallel and fail independently, the system fails only if all components fail simultaneously. If there are \(n\) components, the probability that all fail is \$0.15^n$.
3. The probability that the system succeeds is the complement of the system failing, so it is \$1 - 0.15^n$.
4. Set up the inequality to find the minimum number of components \(n\) such that the system success probability is greater than 0.999999: \$1 - 0.15^n > 0.999999$.
5. Solve the inequality for \(n\): \$0.15^n < 1 - 0.999999 = 0.000001\( Take the natural logarithm of both sides to isolate \)n\(: \)n \cdot \ln(0.15) < \ln(0.000001)\( Then solve for \)n\(: \)n > \frac{\ln(0.000001)}{\ln(0.15)}$.

Key Concepts

Reliability of Parallel Systems

In a parallel system, the overall system succeeds if at least one component works. This increases reliability compared to individual components because failure occurs only if all components fail simultaneously. Understanding this helps calculate system success probability based on component reliability.

Independent Component Failures

When components fail independently, the failure of one does not affect the others. This allows the use of multiplication of probabilities to find the joint failure probability, which is essential for calculating the system failure or success probability in parallel configurations.

Probability Inequalities and Solving for Number of Components

To find the number of components needed for a desired system reliability, set up an inequality comparing system success probability to the target value. Use algebraic manipulation and logarithms to solve for the minimum number of components that meet the reliability threshold.