Problem #$3$: A doctor knows that the disease meningitis causes the patient to have a stiff neck $50\%$ of the time. The doctor also knows that the probability that the patient has meningitis is $1$ in $50,000.$ And the probability that any patient has stiff neck is $1$ in $20.$ Find the probability that a patient with the stiff neck has meningitis.

$Pstif{f}_n$eck $|$ meningitis $|=0\mathrm{.}5$ Pmeningitis $=\frac{1}{50},000Pstif{f}_n$eck $=\frac{1}{20}$