# Bayesian statistics problem

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.

1. Given that a patient has tested positive, use Bayes Rule to decide whether the patient has Meningitis or not.
2. 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.

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You can use $\LaTeX$ for mathematics on this site. –  cardinal Apr 30 '12 at 16:35
What is p(-|¬m)? What relationship (if any) does it have to p(+|¬m)? –  jbowman Apr 30 '12 at 16:35
The denominator in question (1) should be $p(m|+)=\frac{p(+|m)p(m)}{p(+|m)p(m)+p(+|¬m)p(¬m)}=\frac{(.95)(.05)}{(.95)(.05)+‌​(.30)(.95)}$ –  Ken Apr 30 '12 at 16:43
sorry didn't know about letex @jbowman ahh does p(+|¬m) = 1-p(-|¬m)? –  RRs_Ghost Apr 30 '12 at 16:45
Right. Now you can fill in the blank in the p(+) equation. Note also Ken's comment; a good way to check your results is to make sure, e.g., that p(m|+) and p(not-m|+) add to one (where you haven't just calculated the latter by 1 - the former.) –  jbowman Apr 30 '12 at 16:56