Probability & Two-Way Tables — 700-800

Probability & Two-Way Tables: The 700-800 Patterns

Part 1 of 3 — The Archetypes Hard-Tier Items Are Built From

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

Here is a table of the kind these items use — $500$ shelter animals by species and length of stay:

	Under 1 year	1 to 3 years	Over 3 years	Total
Cats	96	74	40	210
Dogs	120	85	35	240
Rabbits	24	18	8	50
Total	240	177	83	500

Archetype 1: The Three Denominators

One cell, $35$ dogs over 3 years, supports three completely different probabilities:

• Joint — "a randomly chosen animal is a dog AND has been here over 3 years": $\frac{35}{500}$. Denominator is the grand total.
• Conditional on the row — "given that it is a dog, it has been here over 3 years": $\frac{35}{240}$. Denominator is the dog total.
• Conditional on the column — "given that it has been here over 3 years, it is a dog": $\frac{35}{83}$. Denominator is the over-3-years total.

All three are always in the options. The phrase "given that" — or "of the dogs," or "among those who…" — names your denominator.

Archetype 2: The Reversed Conditional

"Of the cats, what fraction have been here over 3 years" and "of the animals here over 3 years, what fraction are cats" are different numbers built from the same cell. "Given that the animal is a cat, the probability it has been here over 3 years" is $\frac{40}{210}$; "given that it has been here over 3 years, the probability it is a cat" is $\frac{40}{83}$. Reading the condition off the wrong side of the sentence is the most common way strong students lose this question.

Archetype 3: The Missing Entry

One cell is blank, and you recover it from the margins: a row must sum to its row total, a column to its column total. Fill it in, then answer the actual question — which is never "what is the missing entry."

Archetype 4: Rate vs. Count

"Which group has the greatest number who prefer tea" and "which group has the greatest proportion who prefer tea" usually have different answers, because the groups are different sizes. The largest group almost always supplies the largest raw count, and that is exactly the trap.

Archetype 5: The Complement Group

"Selected at random from those who do not prefer coffee" builds a new population that appears nowhere in the table. Compute it: grand total minus the coffee column, and the matching pieces of each row.

Probability & Two-Way Tables: Traps & Speed

Part 2 of 3 — Distractor Autopsy

Here is the table these questions work from — beverage preferences of $650$ people at a university:

	Coffee	Tea	Neither	Total
Staff	84	36	30	150
Undergraduates	168	92	90	350
Graduate students	78	42	30	150
Total	330	170	150	650

Distractor Species 1: The Grand Total

Whenever a question conditions on anything, $650$ stops being the denominator — but the option built on $650$ is always offered, because dividing by the biggest number in the table feels like the safe move. It is the correct denominator only when the question describes a single random selection from everyone with no condition attached.

Distractor Species 2: The Reversed Conditional

$\frac{168}{330}$ ("of the coffee drinkers, what share are undergraduates") and $\frac{168}{350}$ ("of the undergraduates, what share prefer coffee") are both defensible-looking numbers near $50\%$. Only the sentence tells you which. Underline the words immediately following "of the" or "given that" — that group is the denominator.

Distractor Species 3: Count Where Rate Belongs

Undergraduates supply the most tea drinkers ($92$) but not the highest tea rate ($92/350 \approx 26.3\%$, behind graduate students at $28\%$). Any time the groups are unequal in size, "most" and "greatest proportion" split apart — and the item exists because of that split.

Distractor Species 4: The Margin That Was Never Computed

"Those who do not prefer coffee" is $650 - 330 = 320$ people, a number printed nowhere in the table. Students who cannot find their denominator in a cell reach for the nearest visible total — usually the row total — and answer a different question.

Distractor Species 5: Percent of the Wrong Whole

A percent answer must state its base. $\frac{240}{510}$ and $\frac{240}{450}$ are both "percent of suburban residents who…" until you decide whether the restriction is suburban or not approving.

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Speed Techniques

Write the denominator first. Before touching a number, write "out of ___" and fill in the population the sentence restricted you to. The numerator then takes care of itself.

Translate "given that" mechanically. Rewrite "given that the person is a staff member, the probability that the person prefers tea" as "of the staff members, what fraction prefer tea?" The group named after "given that" is always the denominator, and the numerator is the members of that group who also have the other trait.

For rate comparisons, use a common yardstick. Convert each group's count to a percent of its own total before comparing. Comparing raw counts across unequal groups is meaningless — and is precisely what the wrong answer does.

Build complement groups on the margin. Total minus the named column gives the complement population; the row pieces of the complement are the row total minus that row's entry in the named column.

Probability & Two-Way Tables: Timed Drill

Part 3 of 3 — Four Items at Test Pace

About 90 seconds each. The first, second, and fourth questions use this table — $1{,}000$ residents' responses to a policy question:

	Approve	Disapprove	No opinion	Total
Urban	168	84	48	300
Suburban	210	150	90	450
Rural	112	98	40	250
Total	490	332	178	1,000

Before every answer, say the denominator out loud:

1. Out of whom? Grand total, a row, a column, or a complement group you had to build?
2. Rate or count? If the groups are different sizes, these disagree.
3. Which way does the conditional run? "Of the approvers, how many are rural" is not "of the rural residents, how many approve."