The following application of bayes rule often occurs in actual medical practice. suppose you have tested positive for a disease. what is the probability you actually have the disease? it depends on the sensitivity and specificity of the test, and on the prevalence (prior probability) of the disease.
we’ll denote:
a positive test as: test = pos
a negative test as: test = neg
presence of disease as: disease = true
absence of disease as: disease = false
we know from clinical studies done on the test before fda approval that the sensitivity and specificity of the test are:
p (test = pos | disease = true) = 0.95 (true positive rate, or sensitivity)
p (test = neg | disease = false) = 0.90 (true negative rate, or specificity)
from which we can also deduce:
p (test = neg | disease = true) = 0.05 (false negative rate)
p (test = pos | disease = false) = 0.10 (false positive rate)
we also know from public health surveys that the disease is relatively rare. the prevalence in the general population is:
p (disease = true) = 0.01
from which we can deduce:
p (disease = false) = 0.99
a) use bayes rule to calculate p (disease = true | test = pos), i. e. the probability you actually have the disease, given the test was positive.
b) calculate the ratio p (disease = true | test = pos) / p (disease = true). [ in bayesian statistics, a ratio like this is interpreted as the effect of new evidence on our beliefs about a probability. in this case, we are concerned with the probability of disease, and the new evidence is the test result.]