# Covid-19 Test Probability Analysis ## Test Results and Probabilities | | **Test Positive** | **Test Negative** | **Total** | |------------------------|----------------|-------------------------------------| | **Infected** | .01 | .2 | .5 | | **Not Infected** | .001 | .5 | | --- ## a) Probability the person is infected given a positive test Using Bayes' Theorem: \[ P(\text{Infected} | \text{Positive}) = \frac{P(\text{Positive} | \text{Infected}) \times P(\text{Infected})}{P(\text{Positive})} \] Where: - \( P(\text{Positive} | \text{Infected}) = .99 \) (since false negative rate is .01) - \( P(\text{Infected}) = .5 \) - \( P(\text{Positive}) \) can be calculated as: \[ P(\text{Positive}) = P(\text{Positive} | \text{Infected}) \times P(\text{Infected}) + P(\text{Positive} | \text{Not Infected}) \times P(\text{Not Infected}) \] \[ = .99 \times .5 + .001 \times .5 = .495 + .0005 = .4955 \] Therefore: \[ P(\text{Infected} | \text{Positive}) = \frac{.99 \times .5}{.4955} \approx \frac{.495}{.4955} \approx .99899 \] **Answer:** The probability the person is infected given a positive test is approximately **99.9%**. --- ## b) Probability the test yields a positive result in this community \[ P(\text{Positive}) = .4955 \] **Answer:** The probability that the test yields a positive result is approximately **49.55%**.