Question 1170302
Absolutely! Let's break down this problem step by step.

**a. Constructing the Tree Diagram**

* **Branches:**
    * The first set of branches will represent whether a person has the disease or not.
    * The second set of branches will represent the test result (positive or negative).
* **Notation:**
    * D = Has the disease
    * D' = Does not have the disease
    * + = Test is positive
    * - = Test is negative
* **Probabilities:**
    * P(D') = 0.95 (95% do not have the disease)
    * P(D) = 1 - P(D') = 1 - 0.95 = 0.05 (5% have the disease)
    * P(+|D) = 0.96 (probability of positive test given the disease)
    * P(-|D) = 1 - P(+|D) = 1 - 0.96 = 0.04 (probability of negative test given the disease)
    * P(+|D') = 0.03 (probability of positive test given no disease)
    * P(-|D') = 1 - P(+|D') = 1 - 0.03 = 0.97 (probability of negative test given no disease)

Here's the tree diagram:

```
                                  / + (0.96)
                             D (0.05)
                           /      \ - (0.04)
                          /
                         /
                        /
Start -------------------
                        \
                         \
                          \      / + (0.03)
                             D' (0.95)
                               \ - (0.97)
```

**b. Probability of a Positive Test**

To find the probability of a positive test, we need to consider both cases: having the disease and not having the disease.

* P(+) = P(D and +) + P(D' and +)
* P(+) = P(D) * P(+|D) + P(D') * P(+|D')
* P(+) = (0.05 * 0.96) + (0.95 * 0.03)
* P(+) = 0.048 + 0.0285
* P(+) = 0.0765

Therefore, the probability that the test is positive is 0.0765.

**c. Probability of Having the Disease Given a Negative Test**

We need to find P(D|-). We'll use Bayes' Theorem:

* P(D|-) = [P(-|D) * P(D)] / P(-)

First, let's find P(-):

* P(-) = P(D and -) + P(D' and -)
* P(-) = P(D) * P(-|D) + P(D') * P(-|D')
* P(-) = (0.05 * 0.04) + (0.95 * 0.97)
* P(-) = 0.002 + 0.9215
* P(-) = 0.9235

Now, let's apply Bayes' Theorem:

* P(D|-) = [(0.04) * (0.05)] / 0.9235
* P(D|-) = 0.002 / 0.9235
* P(D|-) ≈ 0.0021656

Therefore, the probability that a person has the disease given a negative test is approximately 0.0021656.