## False positives and false negatives

29 November 2020

False positive and false negative rates are important to bear in mind in a mass testing scenario.

By way of example: you reside in an area with a 1 in 1,000 background probability of infection with blanket testing. You take a test, get a positive result. You learn the test has a 1% false positives, and 10% false negatives. What is the probability you are infected?

$\text{False negative rate (FNR)} = 0.1$

$\text{False positive rate (FPR)} = 0.01$

$\text{True positive rate (TPR)} = 1-\text{FNR} = 0.9$

Let’s assume $100$ are infected from $100 \times 1000 = 100000$ tests, which accords with the $\frac{1}{1000}$ probability.

$100000-100=99900$ are not infected.

$\text{True positives (TPs)} = 100 \times \text{TPR} = 90$

$\text{False positives (FPs)} = 99900 \times \text{FPR} = 999$

Probability is then true positives over all positive test results

$\frac{\text{TPs}}{\text{TPs}+\text{FPs}} = \frac{90}{90+999} \approx 0.083$

If we then re-test all the positives we can take $\text{c. } 0.083$ as our background probability, which means that a second positive test result will yield a much higher probability of infection.