91Ó°ÊÓ

Component reliability refers to the likelihood that a component will function successfully over a specified period of time. In our problem, each component has a probability of failure of 0.15, which means that it has a reliability of 0.85. To find the reliability, we can use the equation: \[ P(\text{success}) = 1 - P(\text{failure}) = 1 - 0.15 = 0.85 \] This means each component has an 85% chance of not failing. Understanding component reliability is crucial because it forms the basis for calculating the reliability of more complex systems made up of multiple such components. When we know the individual reliability, we can then figure out how the entire set of components will perform together.

Independent events are events where the outcome of one does not affect the outcome of another. In this exercise, the failure of one component does not influence the failure of another component. This concept is important because it simplifies our calculations. When events are independent, the probability of both events occurring together is just the product of their individual probabilities. For example, the probability of two components both failing is given by: \[ P(\text{both fail}) = 0.15 \times 0.15 = 0.0225 \] This shows that when we have independent events, we can easily calculate combined probabilities by multiplying the probabilities of the individual events.

To determine the success probability of the entire system, we need to consider all components. In our parallel structure, the system succeeds if at least one component succeeds. The probability of the system failing is the probability that all components fail. For two components, this is: \[ P(\text{both fail}) = 0.15 \times 0.15 = 0.0225 \] Therefore, the system success probability is: \[ P(\text{at least one succeeds}) = 1 - P(\text{both fail}) = 1 - 0.0225 = 0.9775 \] So, the likelihood that the system will work with two components is 0.9775 or 97.75%. As we add more components, by ensuring at least one succeeds, the overall system reliability improves.

Natural logarithms, denoted as \text{ln}, help us solve exponential equations by transforming them into linear ones. In our exercise, we had to find the number of components needed for the system's probability of success to exceed 0.9999. This requires us to solve: \[ 0.15^n < 0.0001 \] Taking natural logarithms on both sides of the equation, we get: \[ n \times \text{ln}(0.15) < \text{ln}(0.0001) \] Simplifying further: \[ n > \frac{\text{ln}(0.0001)}{\text{ln}(0.15)} \]\[ n > \frac{-9.21034}{-1.89712} \] \[ n > 4.8545 \] Thus, we conclude that at least 5 components (since we round up to the next whole number) are required to achieve a system success probability greater than 0.9999. This showcases the utility of logarithms in breaking down complex probability equations.