In this system, we have two subsystems of components connected differently: one in parallel and one in series. Let’s break the problem into smaller parts to find the probability that the entire system works.

Subsystem 1 (Parallel Connection of Components 1 and 2)

For components connected in parallel, the subsystem works if either component 1 or component 2 works. The probability that at least one of them works can be found using the formula:

$P(\text{Parallel subsystem works}) = 1 - P(\text{both fail})$

The probability that both components fail is the product of their individual failure probabilities:

$P(\text{both fail}) = (1 - P(\text{Component 1 works})) \times (1 - P(\text{Component 2 works}))$

Given that $P(\text{Component 1 works}) = 0.86$ and $P(\text{Component 2 works}) = 0.86$, the failure probability for each component is $1 - 0.86 = 0.14$. So,

$P(\text{both fail}) = 0.14 \times 0.14 = 0.0196$

Thus, the probability that the parallel subsystem works is:

$P(\text{Parallel subsystem works}) = 1 - 0.0196 = 0.9804$

Subsystem 2 (Series Connection of Components 3 and 4)

For components connected in series, the subsystem works only if both components 3 and 4 work. The probability that both components work is the product of their individual working probabilities:

$P(\text{Series subsystem works}) = P(\text{Component 3 works}) \times P(\text{Component 4 works})$

Given that $P(\text{Component 3 works}) = 0.75$ and $P(\text{Component 4 works}) = 0.75$, we have:

$P(\text{Series subsystem works}) = 0.75 \times 0.75 = 0.5625$

System Probability

The entire system works if both subsystems work, i.e., the parallel subsystem (components 1 and 2) works and the series subsystem (components 3 and 4) works. Therefore, the probability that the entire system works is the product of the two subsystem probabilities:

$P(\text{System works}) = P(\text{Parallel subsystem works}) \times P(\text{Series subsystem works})$

Substitute the calculated values:

$P(\text{System works}) = 0.9804 \times 0.5625 = 0.5511$

Thus, the probability that the system works is approximately:

$\boxed{0.5511}$

Let me know if you would like more details or further clarification.

Here are some related questions to explore:

1. How would the system probability change if components 1 and 2 had different probabilities?
2. What if components 3 and 4 were connected in parallel instead of series?
3. How does the system reliability change if the probability of each component working increases?
4. What is the system failure probability, given the same setup?
5. How would the calculation change if the components were not independent?

Tip: For series connections, the reliability is always lower because all components must work, whereas in parallel connections, having more components increases the chance of success.