https://file.aiquickdraw.com/mathbot/file/17262412231318clbic6m.jpg

Let's analyze the given system and calculate the probability that the system works.

### Problem Breakdown
1. **Subsystem 1 and 2** are connected in **parallel**, meaning the subsystem will work if at least one of the components (1 or 2) works.
2. **Subsystem 3 and 4** are connected in **series**, meaning the subsystem will work only if **both** components (3 and 4) work.

We are given the following probabilities:
- \( P(\text{component 1 works}) = P(\text{component 2 works}) = 0.77 \)
- \( P(\text{component 3 works}) = P(\text{component 4 works}) = 0.71 \)

### Step-by-Step Solution

#### 1. Calculate the Probability of Subsystems

- **Subsystem 1 (Parallel):**
  The probability that the parallel subsystem (1 or 2) works can be calculated using the formula for the union of two independent events:
  
  \[
  P(\text{Subsystem 1 works}) = P(1 \text{ works}) + P(2 \text{ works}) - P(1 \text{ works}) \times P(2 \text{ works})
  \]

  Substituting the values:

  \[
  P(\text{Subsystem 1 works}) = 0.77 + 0.77 - (0.77 \times 0.77)
  \]

#### 2. Calculate the Probability of Subsystem 2

- **Subsystem 2 (Series):**
  The probability that the series subsystem (3 and 4) works is the product of the probabilities that each component works since both must work:

  \[
  P(\text{Subsystem 2 works}) = P(3 \text{ works}) \times P(4 \text{ works})
  \]

  Substituting the values:

  \[
  P(\text{Subsystem 2 works}) = 0.71 \times 0.71
  \]

#### 3. Calculate the Probability of the Entire System Working

The entire system will work if **both subsystems work**. Since they are independent, the probability that the system works is the product of the probabilities of each subsystem working:

\[
P(\text{System works}) = P(\text{Subsystem 1 works}) \times P(\text{Subsystem 2 works})
\]

Let's compute these values step-by-step.

### Calculation
Now, we will compute the probability that the system works. 

First, we calculate the individual subsystem probabilities, and then use these to find the overall system probability.Here are the computed probabilities:

1. **Probability that Subsystem 1 (parallel) works:**
   \[
   P(\text{Subsystem 1 works}) = 0.9471
   \]

2. **Probability that Subsystem 2 (series) works:**
   \[
   P(\text{Subsystem 2 works}) = 0.5041
   \]

3. **Probability that the entire system works:**
   \[
   P(\text{System works}) = 0.4774
   \]

Thus, the probability that the system works is **0.4774** (rounded to four decimal places).

Would you like a more detailed explanation, or do you have any questions?

### Related Questions:
1. What is the probability that only Subsystem 1 works?
2. What is the probability that only Subsystem 2 works?
3. How does the probability change if one of the components' reliability increases?
4. What is the probability that none of the subsystems work?
5. How would the overall probability change if all components were connected in series?

**Tip:** In parallel systems, increasing the reliability of any one component can significantly increase the overall system reliability.

Consider the system of components connected as in the accompanying picture. Components 1 and 2 are connected in parallel, so that subsystem works if and only if either 1 or 2 works; since 3 and 4 are connected in series, that subsystem works if and only if both 3 and 4 work. If components work independently of one another and P(component i works) = 0.77 for i = 1, 2 and = 0.71 for i = 3, 4, calculate P(system works). (Round your answer to four decimal places.)

Learn how to calculate the probability of a system working when components are connected in parallel and series. This problem includes step-by-step calculations using probability formulas and covers key concepts such as the union and intersection of independent events. Suitable for advanced high school students.

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