 # 3% of the population has disease X. A laboratory blood test has (a) 96% effective at detecting dis bacfrancaiso0j 2022-04-30 Answered
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?
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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."
$\mathsf{P}\left(D\right)=0.03$
"A laboratory blood test has (a) 96% effective at detecting disease X, given that the person actually has it."
$\mathsf{P}\left(T\mid D\right)=0.96$
"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."
$\mathsf{P}\left(T\mid {D}^{\complement }\right)=0.01$
"What is the probability a person has the disease given that the test result is positive?"
Now find $\mathsf{P}\left(D\mid T\right)$ using what you know of conditional probability (hint: Bayes' Rule) and the Law of Total Probability.