E\.g\. in the Diseasitis example, $~$\\mathbb P(sick \\mid blackened)$~$ is calculated by dividing the 18% students who are sick and have blackened tongue depressors \($~$\\mathbb P(sick \\wedge blackened)$~$\), by the total 42% students who have blackened tongue depressors \($~$\\mathbb P(blackened)$~$\)\.
Just reiterating that it's 18% of all students (sick and healthy). That's because it's a 90% (0.9) chance the blackened tongue depressor belonged to a sick student, out of all the sick students (20% of total student population).
Sorry if this is really obvious to others, it just took me a while.