A  parallel system  of three components functions whenever at least one of its components works. Suppose that each component independently  works with probability 0.85.  What is the probability that the system is functioning?  Enter your answer with at least two digits after the decimal, may use standard rounding rule. 

To determine the probability that the parallel system is functioning, we need to calculate the probability that at least one of its components is working. We can do this by first calculating the probability that none of the components are working and then subtracting this from 1.

Let \( P(A) \) be the probability that a component works, which is given as 0.85.

The probability that a component does not work is:
\[
P(A^c) = 1 - P(A) = 1 - 0.85 = 0.15
\]

Since the components are independent, the probability that none of the three components work is:
\[
P(\text{none work}) = P(A^c) \times P(A^c) \times P(A^c) = 0.15 \times 0.15 \times 0.15 = 0.15^3 = 0.003375
\]

The probability that at least one component is working is:
\[
P(\text{at least one works}) = 1 - P(\text{none work}) = 1 - 0.003375 = 0.996625
\]

Thus, the probability that the system is functioning is approximately:
\[
0.9966
\]

Would you like more details or have any questions? Here are some related questions you might find useful:

1. What is the difference between series and parallel systems in reliability engineering?
2. How do you calculate the reliability of a system with components in series?
3. What is the impact of component independence on system reliability?
4. Can you explain the concept of redundancy in engineering systems?
5. How would the probability change if the components were not independent?
6. What are some real-life applications of parallel systems?
7. How can you improve the reliability of a system with multiple components?
8. What is the difference between "at least one" and "exactly one" in probability calculations?

**Tip:** In reliability engineering, understanding the difference between series and parallel configurations is crucial. In a series system, all components must work for the system to function, while in a parallel system, only one component needs to work.

Learn how to calculate the probability of a parallel system with three components, where each component operates independently with a given probability. Understand the concept of independent events and complementary probability in reliability engineering.

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