"The following would be simpler and more consist..."

Rearranging the definition of conditional probability, $\\mathbb P(X \\wedge Y) \= \\mathbb P(Y) \\cdot \\mathbb P(X|Y).$ E\.g\. to find "the fraction of all patients that are sick and get a positive result", we multiply "the fraction of patients that are sick" times "the probability that a sick patient blackens the tongue depressor"\.

The following would be simpler and more consistent with the beginning of the sentence: “the fraction of sick patients that got a positive result”