A test to determine whether a certain antibody is present is 99.7​% effective. This means that the test will accurately come back negative if the antibody is not present​ (in the test​ subject) 99.7​% of the time. The probability of a test coming back positive when the antibody is not present​ (a false​ positive) is 0.003. Suppose the test is given to four randomly selected people who do not have the antibody. ​(a) What is the probability that the test comes back negative for all four ​people? ​(b) What is the probability that the test comes back positive for at least one of the four ​people?

This problem involves calculating probabilities based on the effectiveness of a medical test for detecting a certain antibody. We know that:

- The probability of a **negative result** when the antibody is not present (true negative) is \( P(\text{negative}) = 0.997 \).
- The probability of a **false positive** when the antibody is not present is \( P(\text{false positive}) = 0.003 \).

We are asked to find the probabilities for two scenarios given that the test is administered to four people who do **not** have the antibody.

### (a) Probability that the test comes back negative for all four people

The probability that the test comes back negative for one person is \( 0.997 \). Since the test results for different people are independent, the probability that all four people get a negative result is the product of the individual probabilities.

\[
P(\text{all negative}) = (0.997)^4
\]

Let's compute this.

\[
P(\text{all negative}) = 0.997^4 \approx 0.988
\]

So, the probability that all four tests come back negative is approximately **0.988**.

### (b) Probability that the test comes back positive for at least one of the four people

To find the probability that at least one test comes back positive, we can use the complement rule. The complement of "at least one positive" is "all negative." Therefore, the probability of at least one positive result is:

\[
P(\text{at least one positive}) = 1 - P(\text{all negative})
\]

We already found that \( P(\text{all negative}) \approx 0.988 \), so:

\[
P(\text{at least one positive}) = 1 - 0.988 = 0.012
\]

Thus, the probability that at least one of the four tests comes back positive is approximately **0.012**.

### Summary of Answers:
- (a) The probability that the test comes back negative for all four people is approximately **0.988**.
- (b) The probability that the test comes back positive for at least one of the four people is approximately **0.012**.

Would you like more details or have any questions?

Here are 5 related questions for further practice:
1. What is the probability that the test comes back positive for all four people?
2. If the test was administered to five people, what would be the probability that all tests come back negative?
3. How would the probabilities change if the test was only 95% effective?
4. What is the probability that exactly one test comes back positive out of the four people?
5. If one person has the antibody, what is the probability that they will get a positive result?

**Tip**: The complement rule is a powerful technique in probability. When you're asked for "at least one" event happening, it's often easier to calculate the complement (none of the events happening) and subtract from 1.

Learn how to calculate the probability of negative and positive test results for a group of four people using a 99.7% accurate antibody test. This problem involves applying the complement rule and understanding independent events. Suitable for high school students in Grades 9-12.

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