# Contingency table on Bayers Rule

The question is this:

A medical test for the disease StatsLove will correctly state that a diseased student suffers from StatsLove 95% of the time. The medical test correctly states that a non-diseased student does not suffer from StatsLove 99% of the time. From prior terms it is known that approximately 80% of students suffer from StatsLove.

My understanding is:

             STATSLOVE      HEALTHY
Infected        *95%                 80%
Not Infected                 *99%    20%


The *95% is $80\%\times95\%$, means the positive percentage; similarily, *99 is $99\%\times20\%$.

Any other thoughs? Thanks.

• The typo "Bayers" was probably a Freudian slip, in view of the headaches such problems cause. Oct 26, 2013 at 1:30

It looks as if you essentially know what to do. If you prefer to use percentages, you may do so. But where you have written $\ast95\%$, and indicate that you know it is $95\%\times 80\%$, I would write $(0.95)(0.8)$, and perhaps even compute the product as $0.76$.
The entry next to the right is for the probability that the person is diseased but is diagnosed as healthy. Given that you are diseased, the probability that you are falsely called healthy is $1-0.95$, that is, $0.05$. So the probability that you are diseased but declared healthy is $(0.8)(0.05)=0.04$. That is the required entey. Note that the two entries $0.76$ and $0.04$ add up to $0.80$, so once you knew the $0.76$, you could have computed the $0.04$ as $0.8-0.76$.
For the bottom row, note that the probability that someone who is not diseased is falsely called diseased is $1-0.99=0.01$. The entry you have labelled $\ast 99\%$ is indeed $(0.2)(0.99)=0.198$.
The entry at the bottom right should therefore be $0.2-0.198=0.002$, or equivalently can be computed as $(0.2)(0.01)$.
Remark: The "heart" of the table is: $$\begin{matrix} 0.76 & 0.04 \\0.002& 0.198 \end{matrix}$$ However, you may want also to have the marginal sums, both for the rows and the columns, like this: $$\begin{matrix} 0.76 & 0.04&0.8 \\0.002& 0.198&0.2\\ 0.762 &0.238 & \end{matrix}$$ If you have your accounting hat on, you may want to insert a $1$ in the bottom right corner.