The NIH is creating a test for the Zika virus. The incidence of a Zika is low, 11 percent in the general population.  There is a trial version, but it sometimes misses the disease.  The probability of a false negative is 3 percent.  Similarly, someone who does not have Zika, will sometimes get a positive reading.  The probability of this is 30 percent.   Fill in the tree diagram and ...
Answer in percent to 2 decimal places. using tree diagram to solve

We can solve this problem using a **tree diagram** approach. The tree diagram will help us break down the different conditional probabilities and find the desired outcomes. The basic structure of the tree involves two primary branches: whether a person has the Zika virus or not, and from those, we consider the outcomes of the test being positive or negative.

Here are the steps:

### Step 1: Define the Events
Let:
- \( P(Z) \) = Probability that a person has Zika = 11% = 0.11
- \( P(\text{no } Z) \) = Probability that a person does not have Zika = 89% = 0.89

The probabilities of false negatives and false positives:
- \( P(\text{false negative}) \) = Probability that the test is negative even though the person has Zika = 3% = 0.03
- \( P(\text{false positive}) \) = Probability that the test is positive even though the person does not have Zika = 30% = 0.30

The correct test results:
- \( P(\text{true positive}) \) = Probability that the test is positive given the person has Zika = \( 1 - P(\text{false negative}) = 0.97 \)
- \( P(\text{true negative}) \) = Probability that the test is negative given the person does not have Zika = \( 1 - P(\text{false positive}) = 0.70 \)

### Step 2: Tree Diagram Structure

The tree diagram has two initial branches:
1. **Has Zika** (11%)
   - True Positive (97%)
   - False Negative (3%)

2. **Does not have Zika** (89%)
   - False Positive (30%)
   - True Negative (70%)

### Step 3: Calculate the Probabilities for Each Branch

1. **Has Zika (11%)**
   - True Positive: \( 0.11 \times 0.97 = 0.1067 \) or **10.67%**
   - False Negative: \( 0.11 \times 0.03 = 0.0033 \) or **0.33%**

2. **Does not have Zika (89%)**
   - False Positive: \( 0.89 \times 0.30 = 0.267 \) or **26.70%**
   - True Negative: \( 0.89 \times 0.70 = 0.623 \) or **62.30%**

### Step 4: Summarize the Results in a Tree Diagram

| **Branch**               | **Probability** | **Outcome**               |
|--------------------------|-----------------|---------------------------|
| Has Zika, True Positive   | 10.67%          | Correct positive result    |
| Has Zika, False Negative  | 0.33%           | Incorrect negative result  |
| No Zika, False Positive   | 26.70%          | Incorrect positive result  |
| No Zika, True Negative    | 62.30%          | Correct negative result    |

### Step 5: Interpret the Results

- The **true positive rate** is 10.67%, meaning 10.67% of the general population will correctly test positive for Zika.
- The **false negative rate** is 0.33%, meaning 0.33% of the general population will have Zika but test negative.
- The **false positive rate** is 26.70%, meaning 26.70% of the general population will incorrectly test positive for Zika.
- The **true negative rate** is 62.30%, meaning 62.30% of the general population will correctly test negative for Zika.

### Step 6: Concluding Thoughts

This tree diagram and the corresponding probabilities highlight that the test is not perfect, particularly because the **false positive rate (26.70%)** is quite high. In contrast, the **false negative rate (0.33%)** is much lower, which is a positive aspect of the test.

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Would you like further details or clarification on any part of the solution?  

Here are five related questions to explore:
1. What is the overall accuracy of the test based on this data?
2. What is the probability that a person tests positive for Zika, regardless of whether they have it or not?
3. How could we improve the sensitivity or specificity of the test?
4. What is the probability that a person who tested positive actually has Zika (positive predictive value)?
5. How would changing the prevalence of Zika affect the false positive rate?

**Tip:** Understanding false positives and false negatives is crucial when interpreting medical test results. Consider how these rates affect patient outcomes.

The NIH is creating a test for the Zika virus. The incidence of Zika is low, 11 percent in the general population. There is a trial version, but it sometimes misses the disease. The probability of a false negative is 3 percent. Similarly, someone who does not have Zika will sometimes get a positive reading. The probability of this is 30 percent. Fill in the tree diagram and answer in percent to 2 decimal places.

Explore how to calculate probabilities for a Zika virus test using a tree diagram. Learn how to find true positive, false positive, false negative, and true negative rates with detailed step-by-step solutions. This problem covers key probability concepts and is suitable for Grades 11-12.

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