# Probability of a test result for a randomly selected patient

A pharmaceutical company is testing a new chemical compound that can be used to detect the presence of a disease. In the clinical trial, the company finds that $3\%$ of the patients test positive and the rest test negative. Among the patients who get a positive test result, $90\%$ have the disease. Among the patients who get a negative test result, only $1\%$ have the disease. Randomly select a patient, what is the probability that he or she has the disease?

\begin{align} P(D|+)&=0.9\\ P(D|+')&=0.01\\ P(+)&=0.03 \end{align}

$$P(+')=1-P(+)= 0.03$$

\begin{align} P(D)&=P(D|+)P(+)+P(D|+')P(+')\\ &=(0.9)(0.03)+(0.01)(0.97)\\ &=0.0367 \end{align}

Is my though correct?

• I would say - is it right. – georg Sep 25 '15 at 16:49

Your answer of $0.0367$ appears right to me, I think that a typo exchanged $6$ and $7$ !

Here is how is I would do that. Imagine $10000$ patients.

the company finds that $3\%$ of the patients test positive and the rest test negative.

So $(10000)(.03)=300$ test positive and $10000-300=9700$ test negative.

Among the patients who get a positive test result, $90\%$ have the disease.

So, of the $300$ who is test positive, $300(.9)=270$ have the disease, $300-270=30$ do not.

Among the patients who get a negative test result, only $1\%$ have the disease.

So, of the $2700$ who is test negative, $2700(.01)=27$ have the disease, $2700- 27=2673$ do not.

Randomly select a patient, what is the probability that he or she has the disease?

Of the $10000$ patients, $270+27=297$ have the disease. The probability a randomly chosen patient has the disease is $297/10000=0.0297$ or $2.97\%$.

Did you round to $3\%$?

• Among the possibles answer there isn't 2.97 or 3%. Only 0.0376, 0.0382 0.04 and 0.045. I belive is a typo in my paper. – JulietaR Sep 25 '15 at 16:59
• I think $2700$ testing negative is a typo. In your first para, you have (correctly) stated $9700$ test negative. – true blue anil Sep 25 '15 at 17:24