Having a bit of a problem with this question:
A screening test for Meningitis is known to provide a positive result 95% of the time when a patient with Meningitis is tested, while it gives a negative result for 70% of patients tested who are not suffering from the disease. National statistics suggest that 5% of the population get the disease.
- Given that a patient has tested positive, use Bayes Rule to decide whether the patient has Meningitis or not.
- If the doctor orders a second test that returns a negative result, how would this affect the probabilities associated with Meningitis and ¬Meningitis?
Attempted solution:
Let $p(+)$ denote probability of a positive test result and let $p(m)$ denote probability of having meningitis.
$p(+|m)=0.95$
$p(-|¬m)=0.70$
$p(m)=0.05$
$p(m|+)=p(+|m)*\frac{p(m)}{p(+)}$
$p(m|+)=\frac{0.95*0.05}{p(+)}$
$p(+)=p(+|m)*p(m)+p(+|¬m)*p(¬m)$
$p(+)=0.95*0.05+ p(+|¬m)*0.95$
This is where I get stuck, I do not know what $p(+|¬m)$ is.
Not knowing how to get false positive rates is a reoccurring theme in the questions I try to solve.
I would be really grateful if someone would explain to me how to calculate $p(+|¬m)$ and false positive/negative rates in general, when there is no data. Is there a simple rule?
Thanks very much.
