Probability & Two-Way Tables at the 700-800 Level

Every hard item in this bank reduces to one question: what is the denominator? The numerator is almost always in plain sight. The entire difficulty is deciding which population the question has restricted you to.

One cell supports three completely different probabilities. The joint probability divides by the grand total; the row conditional divides by that row's total; the column conditional divides by that column's total. All three are always among the options, and the phrase 'given that' — or 'of the...', or 'among those who...' — names your denominator.

The reversed conditional is where strong students lose the question: $P(A \mid B)$ and $P(B \mid A)$ are built from the same cell and are different numbers. Read which side of the sentence the condition sits on. Missing entries are recovered from the margins, then discarded — the question is never 'what is the missing entry.' Rate versus count splits apart whenever groups are unequal in size, and the largest group reliably supplies the largest raw count. Complement groups ('those who do not prefer coffee') appear nowhere in the table and must be built on the margin.

Before touching a number, write 'out of ___' and fill in the restricted population.