Bayes Problem

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September 21st 2010, 10:36 AM
Ruferto

Bayes Problem

Can't seem to figure this one out. Any help would be appreciated:

A positive result on a test for diabetes is correct 90% of the time, and a negative result is correct 95% of the time. If 10% of the population has diabetes, what is the probability that a randomly selected person will have a false positive test result?
September 21st 2010, 11:09 AM
Pim

I'm not sure you stated the problem correctly. In this form it's not a Bayesian problem.

As it is now, you are looking for a person who belongs to the non-diabetes group (.9 probability) and then gets an incorrect positive test result. (.1 probability) You then multiply these to get your answer.

It would be a Bayesian problem if the question was: Given a positive result, what is the chance that person has diabetes?
September 21st 2010, 11:09 AM
MathoMan

Hypothesis are
$H_1:=\{person\, has \, diabetes\} \Rightarrow P(H_1)=0.1$
$H_2:=\{person\, doesn't\, have \, diabetes\} \Rightarrow P(H_1)=0.9$

Define events
$A=\{test\, is \, positive\}$ and $B=\{test\, is \, negative\}.$

Then
$P(A|H_1)=0.9$
$P(A|H_2)=0.1$
$P(B|H_1)=0.05$
$P(B|H_2)=0.95$

You have to find the probability that person has a false positive result hence you have to find $P(A\cap H_2)=P(A|H_2)P(H_2)=0.1 \cdot 0.9=0.09=9\%$
September 21st 2010, 11:18 AM
Ruferto

Thank you both. I found the wording of the problem somewhat confusing.

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