Notes on Tree Diagrams and Bayes’ Method

Problem #3.47

(a) There are two ways to test positive.
A true positive happens with probability Pr(disease)Pr(test positive | disease) = (.001)(.995) = .000995.
A false positive happens with probability Pr(no disease)Pr(test positive | no disease) = (.999)(.002) = .001998.
Thus Pr(test positive) = .000995 + .001998 = .002993

PV+ = Pr(have disease | test positive) = Pr(have disease and test positive)/Pr(test positive) = .000995/.002993. The denominator here comes from above.

(b) There are two ways to test negative.
A true negative happens with probability Pr(no disease)Pr(test negative | no disease) = (.999)(.998) = .997002.
A false negative happens with probability Pr(disease)Pr(test negative | disease) = (.001)(.005) = .000001.
Thus Pr(test positive) = .997002 + .000001 = .997003

PV- = Pr(no disease | test negative) = Pr(no disease and test negative)/Pr(test negative) = .997002/.997003 = .999998997 The denominator here comes from above.

Problem #3.48

(a) There are two ways to test positive.
A true positive happens with probability Pr(disease)Pr(test positive | disease) = (1/25)(.995) = .0398.
A false positive happens with probability Pr(no disease)Pr(test positive | no disease) = (24/25)(.002) = .00192.
Thus Pr(test positive) = .0398 + .00192 = .04172

PV+ = Pr(have disease | test positive) = Pr(have disease and test positive)/Pr(test positive) = .0398/.04172 = .953978907. The denominator here comes from above.

(b) There are two ways to test negative.
A true negative happens with probability Pr(no disease)Pr(test negative | no disease) = (24/25)(.998) = .95808.
A false negative happens with probability Pr(disease)Pr(test negative | disease) = (1/25)(.005) = .0002.
Thus Pr(test positive) = .95808 + .0002 = .95828

PV- = Pr(no disease | test negative) = Pr(no disease and test negative)/Pr(test negative) = .95808/.95828 = .9997912927 The denominator here comes from above.