3% of the population has disease X. A laboratory blood test has (a) 96% effective at detecting dis

bacfrancaiso0j

bacfrancaiso0j

Answered question

2022-04-30

3% of the population has disease X.
A laboratory blood test has
(a) 96% effective at detecting disease X, given that the person actually has it.
(b) 1% “false positive” rate. i.e, a person who does not have disease X has a probability of 0.01 of obtaining a test result implying they have the disease.
What is the probability a person has the disease given that the test result is positive?

Answer & Explanation

Genesis Reilly

Genesis Reilly

Beginner2022-05-01Added 12 answers

A "False positive" means just what it sounds like: the test gives a positive result and this is wrong (ie: the patient does not actually have the disease). The false positive rate is the probability of a positive result for patients without the disease.
So let D be the event of having the disease, and T be the event of the test being positive.
"3% of the population has disease X."
P ( D ) = 0.03
"A laboratory blood test has (a) 96% effective at detecting disease X, given that the person actually has it."
P ( T D ) = 0.96
"A laboratory blood test has (b) 1% “false positive” rate. i.e, a person who does not have disease X has a probability of 0.01 of obtaining a test result implying they have the disease."
P ( T D ) = 0.01
"What is the probability a person has the disease given that the test result is positive?"
Now find P ( D T ) using what you know of conditional probability (hint: Bayes' Rule) and the Law of Total Probability.

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