## Baye's Rule

A diagnostic test for a certain disease has 95% sensitivity and 95% specificity. Only 1% of the population has the disease in question. If the diagnostic test reports that a person chosen at random from the population test positive, what is the conditional probability that the person does, in fact, have the disease?

Lets agree on some notation.

S = population with disease so $\displaystyle P(S) = .01 \ \ \ \ P(\bar{S})=.99$

D = has disease

+ = test positive

$\displaystyle \displaystyle\begin{bmatrix} & & \text{Diagnoses}& \\ & & + & -\\ \text{Test} & +& a&b\\ \text{Results}& -&c&d\end{bmatrix}$

$\displaystyle \displaystyle\text{Sensitivity}=\frac{a}{a+c}=.95 \ \ \text{and} \ \ \text{Specificity}=\frac{d}{d+b}=.95$

$\displaystyle a=d \ \ \ b=c$

$\displaystyle \displaystyle\begin{bmatrix} & & \text{Diagnoses}& \\ & & + & -\\ \text{Test} & +& a&b\\ \text{Results}& -&b&a\end{bmatrix}$

$\displaystyle \displaystyle P(D| +)=\frac{D\cap +}{+}$

I estimated a and b, but I am not sure if there is a way to do it without arbitrary estimation.