SOLUTION: MAT 145: Topics In Contemporary Math More Probability 3) A disease has an incidence rate of 1.3%. A test for the disease has a false positive rate of 7% and a fa

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Question 1191593: MAT 145: Topics In Contemporary Math

More Probability
3) A disease has an incidence rate of 1.3%. A test for the disease has a false positive rate of 7% and a false negative rate of 2%. Find each of the following probabilities:
a) A person who tests positive has the disease.
b) A person who tests negative does not have the disease

Answer by CPhill(1959) About Me  (Show Source):
You can put this solution on YOUR website!
Here's how to calculate the probabilities, using Bayes' Theorem:
**a) Probability of having the disease given a positive test result:**
* **P(Disease):** Incidence rate = 0.013 (1.3%)
* **P(Positive | Disease):** True positive rate (sensitivity) = 1 - False negative rate = 1 - 0.02 = 0.98
* **P(Positive | No Disease):** False positive rate = 0.07
We want to find P(Disease | Positive). Using Bayes' Theorem:
P(Disease | Positive) = [P(Positive | Disease) * P(Disease)] / [P(Positive | Disease) * P(Disease) + P(Positive | No Disease) * P(No Disease)]
P(Disease | Positive) = (0.98 * 0.013) / (0.98 * 0.013 + 0.07 * (1 - 0.013))
P(Disease | Positive) = 0.01274 / (0.01274 + 0.06851)
P(Disease | Positive) = 0.01274 / 0.08125
P(Disease | Positive) ≈ 0.156 or 15.6%
**b) Probability of not having the disease given a negative test result:**
* **P(No Disease):** 1 - Incidence rate = 1 - 0.013 = 0.987
* **P(Negative | No Disease):** True negative rate (specificity) = 1 - False positive rate = 1 - 0.07 = 0.93
* **P(Negative | Disease):** False negative rate = 0.02
We want to find P(No Disease | Negative). Using Bayes' Theorem:
P(No Disease | Negative) = [P(Negative | No Disease) * P(No Disease)] / [P(Negative | No Disease) * P(No Disease) + P(Negative | Disease) * P(Disease)]
P(No Disease | Negative) = (0.93 * 0.987) / (0.93 * 0.987 + 0.02 * 0.013)
P(No Disease | Negative) = 0.91791 / (0.91791 + 0.00026)
P(No Disease | Negative) = 0.91791 / 0.91817
P(No Disease | Negative) ≈ 0.9997 or 99.97%