bacfrancaiso0j

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?

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?

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."

$\mathsf{P}(D)=0.03$

"A laboratory blood test has (a) 96% effective at detecting disease X, given that the person actually has it."

$\mathsf{P}(T\mid D)=0.96$

"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."

$\mathsf{P}(T\mid {D}^{\complement})=0.01$

"What is the probability a person has the disease given that the test result is positive?"

Now find $\mathsf{P}(D\mid T)$ using what you know of conditional probability (hint: Bayes' Rule) and the Law of Total Probability.

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."

$\mathsf{P}(D)=0.03$

"A laboratory blood test has (a) 96% effective at detecting disease X, given that the person actually has it."

$\mathsf{P}(T\mid D)=0.96$

"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."

$\mathsf{P}(T\mid {D}^{\complement})=0.01$

"What is the probability a person has the disease given that the test result is positive?"

Now find $\mathsf{P}(D\mid T)$ using what you know of conditional probability (hint: Bayes' Rule) and the Law of Total Probability.

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