2. A disease afflicts 1 person in 10 in a population. A test for this disease results in 5%
false positives. Assume that a random person is tested for the disease (this person is not
selected for the test because there are other symptoms indicating the presence of the
disease nor are there any special signs of being disease free). Assume that there are no
false negatives from this test. A person tests positive. What is the probability that the
person truly has the disease?
B. Now assume that false negatives are 15%. What is the probability that the person truly
has the disease?

### Part A: No False Negatives
Let’s define the given data first:

- The disease afflicts 1 person in 10, i.e., the probability that a person has the disease (prior probability) is:
  \[
  P(D) = \frac{1}{10} = 0.1
  \]
- The probability that a person does **not** have the disease is:
  \[
  P(\neg D) = 1 - P(D) = 1 - 0.1 = 0.9
  \]
- The test results in 5% false positives, i.e., the probability of testing positive given the person does **not** have the disease is:
  \[
  P(T^+ | \neg D) = 0.05
  \]
- There are no false negatives, meaning if a person has the disease, the test will always come back positive:
  \[
  P(T^+ | D) = 1
  \]

We need to find the probability that a person has the disease given that they test positive, i.e., \( P(D | T^+) \).

Using **Bayes’ Theorem**:
\[
P(D | T^+) = \frac{P(T^+ | D) P(D)}{P(T^+)}
\]
Where \( P(T^+) \) (the total probability of testing positive) is given by:
\[
P(T^+) = P(T^+ | D)P(D) + P(T^+ | \neg D)P(\neg D)
\]

Substitute the values:
\[
P(T^+) = (1)(0.1) + (0.05)(0.9) = 0.1 + 0.045 = 0.145
\]

Now, calculate \( P(D | T^+) \):
\[
P(D | T^+) = \frac{(1)(0.1)}{0.145} = \frac{0.1}{0.145} \approx 0.6897
\]

Thus, the probability that the person truly has the disease given that they tested positive is approximately **68.97%**.

---

### Part B: With 15% False Negatives
Now, we are given that there are 15% false negatives, meaning the probability that a person tests negative given that they have the disease is:
\[
P(T^- | D) = 0.15
\]
Thus, the probability that a person tests positive given that they have the disease is:
\[
P(T^+ | D) = 1 - 0.15 = 0.85
\]

Using Bayes’ Theorem again:
\[
P(D | T^+) = \frac{P(T^+ | D) P(D)}{P(T^+)}
\]

We need to recalculate \( P(T^+) \) with the new information:
\[
P(T^+) = P(T^+ | D)P(D) + P(T^+ | \neg D)P(\neg D)
\]
Substitute the values:
\[
P(T^+) = (0.85)(0.1) + (0.05)(0.9) = 0.085 + 0.045 = 0.13
\]

Now, calculate \( P(D | T^+) \):
\[
P(D | T^+) = \frac{(0.85)(0.1)}{0.13} = \frac{0.085}{0.13} \approx 0.6538
\]

Thus, the probability that the person truly has the disease given that they tested positive with 15% false negatives is approximately **65.38%**.

---

Would you like more details on any step, or do you have any questions?

Here are some related questions to consider:
1. How does the sensitivity of a test impact the probability of disease?
2. What happens to the probability if the false positive rate is reduced?
3. How would the prior probability of the disease (prevalence) affect the results?
4. Can Bayes' Theorem be applied to other real-world scenarios like drug testing?
5. How does the test accuracy affect public health policies?

**Tip**: False negatives are critical in disease testing because missing a case could lead to further spread of the disease or delayed treatment!

A disease afflicts 1 person in 10 in a population. A test for this disease results in 5% false positives. Assume that a random person is tested for the disease (this person is not selected for the test because there are other symptoms indicating the presence of the disease nor are there any special signs of being disease free). Assume that there are no false negatives from this test. A person tests positive. What is the probability that the person truly has the disease? Now assume that false negatives are 15%. What is the probability that the person truly has the disease?

Learn how to use Bayes' Theorem to calculate the probability of a person having a disease after testing positive, considering 5% false positives and no false negatives. Explore how the probability changes when false negatives are introduced. Ideal for undergraduate and advanced high school students, this problem covers key concepts in probability and medical testing scenarios.

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