 A computer program is tested by 3 independent tests. When there is an error,
these tests will discover it with probabilities 0.2, 0.3, and 0.5, respectively. Suppose that
the program contains an error. What is the probability that it will be found by at least
one test?

We are given that a computer program is tested by three independent tests, with the probabilities of discovering an error by each test being \( P(A_1) = 0.2 \), \( P(A_2) = 0.3 \), and \( P(A_3) = 0.5 \).

We need to find the probability that **at least one** test will discover the error.

### Step 1: Define Events
Let \( A_1 \), \( A_2 \), and \( A_3 \) represent the events that the first, second, and third tests detect the error, respectively. Since the tests are independent, the probability of **none** of the tests detecting the error is the product of the probabilities that each individual test fails to detect it.

- The probability that the first test does **not** detect the error is \( P(A_1^c) = 1 - P(A_1) = 1 - 0.2 = 0.8 \).
- The probability that the second test does **not** detect the error is \( P(A_2^c) = 1 - P(A_2) = 0.7 \).
- The probability that the third test does **not** detect the error is \( P(A_3^c) = 1 - P(A_3) = 0.5 \).

### Step 2: Probability of No Detection
The probability that **none** of the tests detect the error is the product of these complementary probabilities:
\[
P(\text{none of the tests detect the error}) = P(A_1^c) \cdot P(A_2^c) \cdot P(A_3^c) = 0.8 \times 0.7 \times 0.5 = 0.28.
\]

### Step 3: Probability of At Least One Detection
The probability that **at least one** test will detect the error is the complement of the probability that none of the tests detect the error:
\[
P(\text{at least one test detects the error}) = 1 - P(\text{none detect the error}) = 1 - 0.28 = 0.72.
\]

Thus, the probability that at least one test will detect the error is \( \boxed{0.72} \).

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### Relative Questions:
1. What is the probability that exactly two tests will detect the error?
2. How would the probability change if the tests were not independent?
3. What is the probability that only the second test detects the error?
4. How would the solution change if there were 4 independent tests?
5. Can we generalize this method for \(n\) independent tests with different detection probabilities?

### Tip:
For problems involving independent events, it's often easier to calculate the probability of none of the events occurring and subtract that from 1 to find the probability of at least one event happening.

A computer program is tested by 3 independent tests. When there is an error, these tests will discover it with probabilities 0.2, 0.3, and 0.5, respectively. Suppose that the program contains an error. What is the probability that it will be found by at least one test?

Discover how to calculate the probability that at least one test will detect an error in a computer program using independent tests. This problem involves key probability concepts, including independent events and the complement rule. Suitable for Grades 10-12.

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