My 15 year old has miffed tabulating Bayes' Theorem at least twice. Let D = a person is diseased, and + = the test is positive.
We're befuddled because we don't know how to intuit or remember where $D, D^C$ and $+, -$ must be written. E.g. for the (+, D) entry, we know that the probability must contain just + and D. But how can we intuit if this must be $\Pr(+)\Pr(D|+)$ or $\Pr(D)\Pr(+|D)$? My daughter keeps tabulating Bayes Theorem wrongly like this ...
\begin{array}{r|cc|c} \text{WRONG}&D &\lnot D &\text{Total}\\ \hline H_0, +&\color{green}{\Pr(+)\Pr(D|+)}&\color{red}{\Pr(+)\Pr(D^C|+)}&\text{add the 2 left entries}\\ H_a, - &\color{red}{\Pr(-)\Pr(D|-)}&\color{green}{\Pr(-)\Pr(D^C|-)}&\color{black}{\text{add the 2 left entries}}\\ \hline \text{Total} \end{array}
But the correct table is
\begin{array}{r|cc|c} \text{Correct} &D &\lnot D &\text{Total}\\ \hline H_0, +&\color{green}{\Pr(D)\Pr(+|D)}&\color{red}{\Pr(D^C)\Pr(+|D^C)}&\text{add the 2 left entries}\\ H_a, - &\color{red}{\Pr(D)\Pr(-|D)}&\color{green}{\Pr(D^C)\Pr(-|D^C)}&\color{black}{\text{add the 2 left entries}}\\ \hline \text{Total} \end{array}
Can you please approve this picture? https://i.stack.imgur.com/mXJGw.png
First comment: As @Rushy pointed out, the "wrong" solution is actually not wrong; it's equivalent to the "correct" solution. So the "wrong" answer is in fact perfectly correct.
One way to check your answer is to use concrete numbers.
Imagine there are 1000 people, and 100 of them have the disease. Suppose that the probability of testing positive given that you have the disease is 90%, and the probability of testing positive given that you don't have the disease is 10%.
Now start filling in the table with actual numbers. The upper left entry is the number of people who have the disease and who test positive. Well, 100 people have the disease, and 90% of them test positive, so that means we have 90 people in the upper left corner. The fraction of the population that ends up in the upper left corner is $90/1000 = (100/1000) \cdot (90/100)$. This agrees with the correct table, which says that the fraction of the population that ends up in the upper left corner is $P(+) P(D \mid +) = (100/1000) \cdot (90/100)$.
The other entries of the table can be checked and understood in a similar way.
Another comment: The lower left entry of the table, for example, is the fraction of the population that has the disease but also tests negative for the disease. To compute that fraction, take the fraction of the population with the disease (that is, $P(D)$) and multiply by the fraction of the diseased population which tests negative (that is, $P(-\mid D)$). Thus, the lower left entry of the table is $P(-\mid D) P(D)$.