Successful implementation of a new system is based on three independent modules. Module 1 
works properly with probability 0.96. For modules 2 and 3, these probabilities equal 0.95 and 
0.90. Compute the probability that at least one of these three modules fails to work properly.

To solve this problem, we need to calculate the probability that at least one of the three modules fails to work properly.

### Step-by-Step Approach:

Let:
- \( P(M_1) = 0.96 \) be the probability that module 1 works properly.
- \( P(M_2) = 0.95 \) be the probability that module 2 works properly.
- \( P(M_3) = 0.90 \) be the probability that module 3 works properly.

The problem asks for the probability that **at least one module fails**, which is equivalent to \( 1 \) minus the probability that **all modules work properly**.

#### Step 1: Compute the probability that all three modules work properly
Since the modules are independent, the probability that all three modules work properly is the product of their individual probabilities:

\[
P(\text{all work}) = P(M_1) \times P(M_2) \times P(M_3)
\]
\[
P(\text{all work}) = 0.96 \times 0.95 \times 0.90
\]

#### Step 2: Compute the result
Let's calculate this product:

\[
P(\text{all work}) = 0.96 \times 0.95 \times 0.90 = 0.8208
\]

#### Step 3: Compute the probability that at least one module fails
The probability that at least one module fails is the complement of the probability that all modules work properly:

\[
P(\text{at least one fails}) = 1 - P(\text{all work})
\]
\[
P(\text{at least one fails}) = 1 - 0.8208 = 0.1792
\]

### Final Answer:
The probability that at least one of the three modules fails is \( 0.1792 \).

---

Would you like more details or have any questions? 

Here are 5 related questions to deepen your understanding:
1. What is the probability that exactly one of the three modules fails?
2. What is the probability that none of the modules fail?
3. How would the result change if the modules were not independent?
4. If the modules were dependent, how would you approach solving the problem?
5. What is the probability that exactly two of the three modules work properly?

**Tip**: When dealing with independent events, multiplying individual probabilities helps you find the probability of all events occurring together.

Successful implementation of a new system is based on three independent modules. Module 1 works properly with probability 0.96. For modules 2 and 3, these probabilities equal 0.95 and 0.90. Compute the probability that at least one of these three modules fails to work properly.

This problem explores the probability that at least one of three independent system modules fails, given probabilities of 0.96, 0.95, and 0.90 for successful operation. Learn the step-by-step calculation using the complement rule and the multiplication rule for independent events. Suitable for students in Grades 10-12.

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