A long-tail Math type (2.5% of the bank) that is nearly always a two-way table and nearly always one division. Covers probability as a fraction of a population, how the phrase given that shrinks the denominator to a single row or column, why conditional probability is not symmetric, reading a two-way table without losing track of which total is which, and converting between counts, fractions and percentages — with six worked examples, four traps and three checks.
Subject: SAT Prep · 61 slides · applied lesson
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Title
SAT Math · Type 17 of 19
2.5% of the question bank — 42 of 1675 questions
Objectives
Almost every probability question on the SAT is a two-way table with a single division at the end. The arithmetic is trivial. What is being tested is whether you can identify which population the question has restricted you to, because the phrase given that quietly replaces the grand total with a row total or a column total.
One instruction carries the type: name the denominator before you look for the numerator.
Naruhodo Tutoring — SAT Math question bank export (sat-question-index.json) — 42 tagged questions of this type in the site's bank
Section
Section 1
Concept
Almost every probability question on the SAT is a two-way table with a single division at the end. The arithmetic is trivial. What is being tested is whether you can identify which population the question has restricted you to, because the phrase given that quietly replaces the grand total with a row total or a column total.
You will see it phrased in these ways:
The signal words are given that, among, of those who, and if the person selected is. Every one of them means the denominator is no longer the grand total.
Picture it
The same three-panel card as the survey deck, so the shorthand carries over: what identifies it, what you write first, and what is built to catch you.
Figure (svg): Probability and conditional probability: the tell, the move, and the trap
Notice that the green and red panels are about the same number. On this type, the denominator IS the question, and the numerator usually follows without any thought.
Prediction
Recognition here is entirely a matter of reading.
Predict first
Which phrase means the denominator is NOT the grand total?
Correct: Given that the student passed
Why: Given that restricts the selection to a subgroup — here, only the students who passed — so the denominator becomes that group's total rather than everyone. The other three phrases all describe selecting from the whole population, which means the grand total is correct. Reading for these phrases before doing anything else is most of what this type asks.
Concept
Three shapes, and only the middle one carries any real difficulty.
| variant | the denominator | how it reads |
|---|---|---|
| Simple probability | the grand total | a student is chosen at random from all of them |
| Conditional probability | a row or column total | given that, among, of those who |
| Working backwards | given a probability, find a count | the probability is 0.4; how many are there |
Conditional questions are the majority and the ones the distractors are built for. The simple version is included mainly so the grand-total answer looks plausible.
Definition probe
A two-way table has three kinds of total, and mixing them is the whole error.
Sort into buckets
In a table of students by study habit and outcome, which total does each describe?
Discrimination
Six questions about the same table.
Sort into buckets
What goes on the bottom of the fraction?
Warm-up
Do not compute. Just say which number goes underneath.
Discussion prompt
A table shows 100 students: 50 studied and 50 did not; 42 of the studiers passed, and 15 of the non-studiers passed. Given that a student passed, what is the probability they studied? Which number is the denominator?
Hint: Which group has the question restricted you to?
Answer:
The denominator is 57, the total number who passed — 42 plus 15.
Given that a student passed restricts the population to the passers only. Everyone who failed is now irrelevant.
The numerator is 42, the passers who also studied, which is a single cell.
So the probability is 42 out of 57, about 0.74.
The trap answer is 42 out of 100, using the grand total. It answers a different question — the probability that a randomly chosen student both studied and passed — and it will be among the choices.
Pattern
Three steps, and the first two are reading rather than arithmetic.
Underline the restricting phrase: given that, among, of those who.
Why: If one is present the denominator is a row or column total. If none is present the denominator is the grand total.
Write the denominator down before looking for the numerator.
Why: Choosing the denominator first prevents the numerator from dragging you toward the wrong population.
Find the numerator — usually a single cell — and divide.
Why: The numerator is the count satisfying both conditions, which is the intersection of the named row and column.
The order matters. Students who find the numerator first almost always pair it with the grand total out of habit.
Section
Section 2
Concept
A probability is the number of outcomes you want, divided by the size of the population you are selecting from.
The whole subtlety of this type lives in the phrase the population you are selecting from. That is the denominator, and it is what a condition changes.
Concept
A conditional probability is computed within a subgroup, so the denominator is that subgroup's total.
| the question says | denominator | numerator |
|---|---|---|
| a student is chosen at random | the grand total | the relevant row or column total |
| given that the student passed | the passed column total | the cell in that column |
| among those who studied | the studied row total | the cell in that row |
| of the students who failed | the failed column total | the cell in that column |
Once you have identified the line, the question has become a one-line table with two numbers in it.
Prediction
The condition names the line.
Predict first
In a table of 200 people, 80 own a car and 120 do not. Given that a person owns a car, the denominator is which number?
Correct: 80
Why: Given that a person owns a car restricts the population to the 80 car owners, so 80 is the denominator. The 200 would be correct only if no restriction were stated. The 120 is the other group, which the condition has removed from the problem entirely. The denominator never depends on the numerator — it is set by the condition alone.
Concept
The probability of A given B is generally different from the probability of B given A.
The SAT builds a distractor for the reversal on essentially every conditional question, because both fractions are visible in the same table.
Prediction
Same cell, different denominators.
Predict first
A table shows 30 people who are both left-handed and wear glasses. There are 50 left-handed people and 90 glasses-wearers. What is the probability a glasses-wearer is left-handed?
Correct: 30 over 90
Why: The condition names glasses-wearers, so the denominator is 90. The numerator is the 30 who are both. The second choice, 30 over 50, answers the reverse question — the probability that a left-handed person wears glasses — and it uses the same numerator with the other denominator. Both fractions are computable from the table, which is exactly why the reversal is always offered.
Concept
Identify what the rows represent, what the columns represent, and where each total sits.
Thirty seconds spent naming the structure prevents the confusion that all the distractors depend on.
Concept
The probability of both A and B uses the grand total; the probability of A given B uses B's total.
That size relationship is a useful check: a conditional probability should exceed the corresponding both-and probability.
Prediction
No condition means no restriction.
Predict first
In a group of 200, exactly 30 are both left-handed and wear glasses. What is the probability a randomly chosen person is both?
Correct: 30 over 200
Why: No condition is stated, so the selection is from everyone and the denominator is the grand total, 200. Both-and questions keep the grand total; only given-that questions shrink it. Note that this probability, 0.15, is smaller than either conditional version — as it must be, since the denominator is larger.
Concept
If a probability and one of its two numbers are given, the third follows by rearranging.
A non-integer result on a counting question means you used the wrong denominator, which makes it a free check.
Concept
Divide to get a decimal, multiply by 100 for a percentage, and read the question for which form it wants.
Leaving the answer as an unreduced fraction is fine and is often safest, since it avoids a rounding decision entirely.
Prediction
Rearrange, then check for a whole number.
Predict first
In a group of 60 students, the probability that a randomly chosen one plays an instrument is 0.35. How many play one?
Correct: 21
Why: Multiply the probability by the total: 0.35 times 60 is 21. The answer must be a whole number since it counts students, and 21 is. The 35 mistakes the percentage for a count, and 17.5 comes from using a group size of 50 rather than 60 — and being non-integer, it fails the whole-number check immediately.
Two truths and a lie
Three of these probability statements are correct. The one left standing is false.
Eliminate the wrong options
Which statement is FALSE?
Survives elimination: c
Why: The two conditional probabilities share a numerator and have different denominators, so they are equal only by coincidence. The probability that a passer studied and the probability that a studier passed are different questions with different answers. This asymmetry is the single most tested idea in the type, and the reversed fraction is offered as a distractor on essentially every conditional question.
Check
Read the condition, then look at the table.
| Owns a pet | No pet | Total | |
|---|---|---|---|
| Lives alone | 24 | 36 | 60 |
| Lives with others | 56 | 84 | 140 |
| Total | 80 | 120 | 200 |
Check your understanding
Given that a person owns a pet, what is the probability they live alone?
Answer: A
Why: The condition names pet owners, so the denominator is the Owns a pet column total, which is 80. The numerator is the cell where pet owners meets lives alone, which is 24. So the probability is 24 over 80, or 0.3.
All four choices are genuine fractions from this table. Only the wording separates them, which is why the denominator is chosen from the sentence rather than from the numbers.
Section
Section 3
Worked example
A table shows 100 students: 50 studied and 50 did not. Of the studiers, 42 passed; of the non-studiers, 15 passed. A student is chosen at random. What is the probability they passed?
Figure (svg): A two-way table of study habit against outcome with the totals row highlighted
Look for a restricting phrase: there is none, so the selection is from all 100 students.
Why: Chosen at random with no further condition means the whole population.
The denominator is therefore 100, the grand total.
Why: Naming the denominator first is the routine.
The numerator is everyone who passed, which is the column total 57. So the probability is 57 over 100.
Why: The numerator here is a column total rather than a cell, because the question names only one condition.
When only ONE condition is named, the numerator is a total rather than a cell. Cells appear when two conditions are combined.
Verify: check the fraction is between 0 and 1: 0.57 is.
Why: And the failed probability, 43 over 100, adds to exactly 1 with it, as complementary events must.
Answer: 57 over 100, or 0.57
Worked example
Using the same table, given that a student passed, what is the probability they studied?
Figure (svg): The same table with the Passed column highlighted
Underline the condition: given that a student passed. This restricts the population to the Passed column.
Why: Everyone who failed is now irrelevant to the question.
The denominator is the Passed column total, 57.
Why: The population being selected from is exactly the students who passed.
The numerator is the cell where Passed meets Studied, which is 42. So the probability is 42 over 57.
Why: The numerator satisfies both the condition and the property asked about.
The trap answer, 42 over 100, is the both-and probability. It is smaller, and it answers a question with no condition in it.
Verify: check against the complement: 15 over 57 of the passers did not study, and 42 plus 15 is 57.
Why: The two conditional probabilities within the restricted population sum to 1, as they must.
Answer: 42 over 57, about 0.74
Worked example
Using the same table, given that a student studied, what is the probability they passed?
Figure (svg): The same table with the Studied row highlighted
The condition now names a study habit, which is a ROW rather than a column.
Why: Study habit runs down the side of this table.
The denominator is the Studied row total, 50.
Why: The population is now the 50 students who studied.
The numerator is the same cell as before, 42. So the probability is 42 over 50, which is 0.84.
Why: The cell satisfies both conditions regardless of which one is the given.
These two examples share a numerator and differ only in the denominator. That is exactly the pair the SAT offers together, and only the condition tells you which is wanted.
Verify: compare with the previous example: 42 over 57 is about 0.74, and 42 over 50 is 0.84.
Why: Same numerator, different denominators, different answers — which is the asymmetry made concrete.
Answer: 42 over 50, or 0.84
Step zero
Before any numbers.
Discussion prompt
A two-way table question asks: given that a resident owns a bicycle, what is the probability they commute by bike? What do you write first, and why does the order matter?
Hint: One of the two numbers is decided entirely by the wording.
Answer:
Write the denominator first: the total number of bicycle owners.
Why the order matters: if you find the numerator first, the natural instinct is to pair it with the grand total, because that is what most probability questions use.
Choosing the denominator from the wording, before looking at any cell, removes that pull entirely.
Then find the numerator: the cell where bicycle owners meets bike commuters.
The check afterwards: the numerator must be part of the denominator. If it is not, you have selected a cell from outside the restricted group.
Worked example
Using the same table, what is the probability that a randomly chosen student both studied and passed?
Figure (svg): Bars comparing the both-and probability with the two conditional probabilities
There is no restricting phrase — randomly chosen student means everyone.
Why: Both-and is a property of the person selected, not a restriction on who can be selected.
So the denominator is the grand total, 100.
Why: The population is unrestricted.
The numerator is the cell, 42, so the probability is 42 over 100, or 0.42.
Why: The cell counts students satisfying both properties.
The size check is genuinely useful. If a both-and answer comes out larger than a conditional one from the same cell, a denominator has been swapped.
Verify: check the ordering: 0.42 is smaller than both 0.74 and 0.84.
Why: The grand total is the largest denominator, so the both-and probability must be the smallest of the three.
Answer: 42 over 100, or 0.42
Worked example
In a club of 45 members, the probability that a randomly chosen member plays chess is two thirds. How many play chess?
Figure (svg): A number line marking thirty out of forty-five members
Rearrange: favourable equals probability times total.
Why: Probability is favourable over total, so multiplying recovers the count.
Compute: two thirds of 45 is 30.
Why: Forty-five divides evenly by three, giving 15, and twice that is 30.
Check the answer is a whole number: 30 is.
Why: Counts of people cannot be fractional, so a non-integer would signal an error.
The whole-number check is a free error detector on this variant. A fractional answer nearly always means the wrong total was used.
Verify: check forwards: 30 over 45 reduces to two thirds.
Why: The computed count reproduces the given probability exactly.
Answer: 30 members
Faded example
From memory. Four lines that carry the type.
Fill in the blanks
A probability is favourable outcomes over the size of the population being selected from. The phrase given that means the denominator becomes a row or column total. A both-and question uses the grand total. And the probability of A given B is generally not equal to the probability of B given A.
Why: The second and third blanks together are the whole distinction. Given that restricts the population; both-and does not. They share the same numerator, so only the denominator distinguishes them — and that is why the two answers are always offered side by side.
Worked example
Of 250 survey respondents, 60 per cent are over 30. Among those over 30, 45 use the product daily. What fraction of ALL respondents are over 30 and use it daily?
Figure (svg): A partly filled two-way table with the over-thirty row completed
Convert the percentage: 60 per cent of 250 is 150 respondents over 30.
Why: A percentage of the whole becomes a row total once computed.
The 45 daily users are within that group, so the cell is 45.
Why: Among those over 30 identifies the row the 45 sits in.
The question asks about ALL respondents, so the denominator is 250. The fraction is 45 over 250, which is 0.18.
Why: No condition restricts the final question, so the grand total applies.
The verify step shows a genuinely useful relationship: multiplying a conditional probability by the probability of its condition gives the both-and probability.
Verify: cross-check as a conditional: 45 over 150 is 0.3, and 0.3 times 0.6 is 0.18.
Why: The conditional probability times the probability of the condition reproduces the both-and probability.
Answer: 45 over 250, or 18 per cent
Fill the middle
Three kinds of total, one kind of cell.
Fill in the blanks
The number satisfying both conditions at once is a cell. The number satisfying one row condition is a row total. The number satisfying one column condition is a column total. And the number counting everyone once is the grand total.
Why: Naming these four before touching the numbers takes about twenty seconds and prevents nearly every error on this type. The numerator of a conditional probability is a cell, and the denominator is whichever total the condition names.
Estimation
A rough fraction catches a swapped denominator.
Predict first
Of 57 students who passed, 42 had studied. Roughly what fraction is that?
Correct: about three quarters
Why: 42 out of 57 is a bit under 45 out of 60, which is three quarters. The exact value is about 0.74. The estimate is useful because the trap answer, 42 out of 100, is about two fifths — and having three quarters in mind first makes a two-fifths result visibly wrong before you have to reason about which denominator was correct.
Check
Same table, the other direction.
| Owns a pet | No pet | Total | |
|---|---|---|---|
| Lives alone | 24 | 36 | 60 |
| Lives with others | 56 | 84 | 140 |
| Total | 80 | 120 | 200 |
Check your understanding
Given that a person lives alone, what is the probability they own a pet?
Answer: A
Why: The condition now names people who live alone, which is a row, so the denominator is the row total 60. The numerator is the same cell, 24. So the probability is 24 over 60, which is 0.4.
Compare with the previous check: same numerator, 24, and answers of 0.3 and 0.4. The asymmetry is not subtle once both are computed.
Section
Section 4
Trap
The trap. Given that a student passed, you compute 42 over 100.
The condition restricted the population to the 57 students who passed, so the denominator should be 57.
The grand-total answer is always offered, because it is the answer to a question the table also supports — just not the one asked.
The fix. Underline the restricting phrase and write its total as the denominator before finding any numerator.
Given that, among, and of those who all mean the grand total is no longer correct.
Trap
The trap. Asked for the probability that a passer studied, you compute 42 over 50 — the probability that a studier passed.
Both fractions use the same cell, and both are computable from the table, so neither looks wrong.
Only the wording distinguishes them, and the wording is the whole question.
The fix. Identify which variable the condition names, and use THAT variable's total.
Given that they passed means an outcome, which is a column. Given that they studied means a habit, which is a row.
Error analysis
A student's conditional probability. Both numbers come from the table and the pairing is wrong.
Annotate
On: \( \text{Given the student passed, P(studied)} = \frac{42}{100} \)
Both 42 over 100 and 42 over 57 are meaningful probabilities computable from this table. Only the wording decides which one was asked for, which is why reading precedes arithmetic on this type.
Trap
The trap. A question asks for the probability that a person both studied and passed, and you use the studied total as the denominator.
No condition was stated. Both-and describes a property of the person selected, not a restriction on who could be selected.
So the denominator is the grand total, and the answer is smaller than the conditional version.
The fix. Look for the word GIVEN, or a phrase that restricts who can be chosen.
The word and joining two properties is not a restriction.
Elimination
A table shows 80 people who exercise, of whom 60 sleep well; and 120 who do not exercise, of whom 30 sleep well.
Eliminate the wrong options
Given that a person sleeps well, what is the probability they exercise?
Survives elimination: a
Why: The condition names good sleepers, and there are 60 plus 30, which is 90 of them. Of those, 60 exercise, so the answer is 60 over 90, about 0.67. Every wrong choice pairs a genuine table number with the wrong denominator — the reverse conditional, the both-and, and an unconditional probability. Naming the denominator from the wording eliminates all three before any arithmetic.
Trap
The trap. You take a total from the bottom row when the condition named a row category, or read a cell from the wrong intersection.
Two-way tables are compact, and under time pressure it is easy to lose track of which variable runs which way.
The resulting fraction uses two genuine numbers from the table and answers nothing at all.
The fix. Before computing, name what the rows represent and what the columns represent.
Then locate the specific row or column the condition names, and trace to its total.
Counterexample
A claim that feels symmetric and is not.
Discussion prompt
A student says: the probability that a doctor is a woman equals the probability that a woman is a doctor. Give a counterexample with numbers.
Hint: Compare the size of the two groups.
Answer:
Counterexample: suppose a town has 100 doctors, 50 of them women, and 5,000 women in total.
The probability that a doctor is a woman is 50 over 100, which is 0.5.
The probability that a woman is a doctor is 50 over 5,000, which is 0.01.
Same numerator, wildly different denominators, and the two answers differ by a factor of fifty.
The general point: conditional probability is not symmetric, and the two versions are equal only when the two conditioning groups happen to be the same size.
Why it matters beyond the test: this is one of the most common statistical fallacies in real reasoning about medical tests, crime statistics and risk.
Edge cases
Given that names the denominator. When is that harder to see?
Discussion prompt
What phrasings restrict the population without using the words given that, and when does no restriction apply at all?
Hint: Several phrases do the same job.
Answer:
Equivalent restricting phrases: among those who, of the students who, for those selected from, if the person chosen is, and looking only at.
All of them shrink the population to the named subgroup, and all of them set the denominator to that subgroup's total.
No restriction applies when the question describes a property rather than a condition — both A and B, or neither A nor B. These are properties of the selected person, so the grand total stands.
A subtle case: at least one of A or B. This is still an unrestricted question, so the grand total applies, but the numerator now spans several cells and is easiest found as the total minus the neither cell.
The practical test: ask who could have been selected. If the answer is everyone, use the grand total. If the answer is only a subgroup, use that subgroup's total.
Check
Rearrange, then check the answer is a whole number.
Check your understanding
In a group of 80 people, the probability that a randomly chosen person is left-handed is 0.15. Twelve of the left-handers wear glasses. Given that a person is left-handed, what is the probability they wear glasses?
Answer: A
Why: First find the number of left-handers: 0.15 times 80 is 12. The condition restricts to those 12 people, so the denominator is 12. Twelve of them wear glasses, so the numerator is also 12, and the probability is 12 over 12, which is 1 — every left-hander in this group wears glasses.
A probability of exactly 1 is a legitimate answer and it looks suspicious, which is what makes this question effective. The working-backwards step is what makes it computable at all.
Section
Section 5
Matching
Six questions about one table. No arithmetic.
Match the pairs
Why: Two of the six use the grand total, and they are the two with no restricting phrase. The other four each name a specific subgroup, and the subgroup named is always the denominator. Notice that rows two and three are the reversal pair — same cell, different denominators.
Sorting
Each is the opening of a probability question.
Sort into buckets
Is the population restricted?
Item (d) is the distinction students find hardest. The word and joining two properties looks like a condition and is not one.
Comparison
Fill the blanks from memory. All three use the same numerator.
Comparison matrix
| question | denominator | relative size |
|---|---|---|
| Both A and B | the grand total | the smallest of the three |
| A given B | B's total | larger, since B's total is smaller |
| B given A | A's total | larger, and generally different from A given B |
The relative-size column is a free check. Since the grand total is the largest denominator, the both-and probability must be the smallest of the three. Any answer violating that has a swapped denominator.
Trade off
Fill in when each form is safest.
Comparison matrix
| form | when to use it | the risk |
|---|---|---|
| An unreduced fraction | on a grid-in, or when no form is specified | none — reducing is never required |
| A reduced fraction | when the choices are given reduced | an arithmetic slip while reducing |
| A decimal | when comparing two probabilities | rounding too early |
| A percentage | when the question asks for one | forgetting to multiply by 100 |
On a grid-in question the unreduced fraction is genuinely the safest answer: it is accepted, it requires no further arithmetic, and it cannot be mis-rounded.
Real world
One minute on why the asymmetry matters enormously.
Discussion prompt
A medical test for a rare disease is 99 per cent accurate. Someone tests positive. Why is the probability they have the disease far below 99 per cent?
Answer:
Because the two conditional probabilities are different questions. The 99 per cent is the probability of testing positive GIVEN you have the disease. What you want is the probability of having the disease GIVEN a positive test.
Put numbers on it. Suppose 1 person in 10,000 has the disease and 10,000 people are tested. About 1 true case tests positive. But 1 per cent of the 9,999 healthy people — about 100 people — also test positive.
So of about 101 positive tests, only 1 is a real case. The probability of actually having the disease is about 1 per cent, not 99.
The mechanism is exactly the denominator rule: the numerator is the same small cell, and the two questions divide it by completely different totals.
This is called the base rate fallacy, and it affects real decisions in medicine, security screening and criminal justice — which is why the SAT's insistence on naming the denominator is worth more than 2.5 per cent of a test.
Ranking
A table has 200 people, 80 of whom exercise; 60 of the exercisers sleep well, and 90 people sleep well overall.
Put in order
Why: Computing each: (a) is 60 over 200, which is 0.30. (d) is 90 over 200, which is 0.45. (b) is 60 over 90, about 0.67. (c) is 60 over 80, which is 0.75. Ordered smallest to largest: 0.30, 0.45, 0.67, 0.75. Note that the three sharing the numerator 60 differ only in their denominators — 200, 90 and 80 — and that the largest denominator gives the smallest probability, as it must.
Concept
This type is 2.5 per cent of the section and the arithmetic is one division, so essentially all the gain is in reading.
| session | what you do | why |
|---|---|---|
| 1 | Ten two-way tables: name the rows, columns and three totals aloud before answering anything. | Structure confusion underlies several of the distractors. |
| 2 | Fifteen questions where you write the denominator down BEFORE finding any numerator. | The single habit that fixes most of the type. |
| 3 | Ten reversal pairs — compute both A given B and B given A from the same table. | Makes the asymmetry concrete rather than theoretical. |
| 4 | Ten mixed questions including both-and and working-backwards variants. | The two variants that use the grand total, which the conditional habit can over-correct against. |
Naruhodo Tutoring — SAT Math question bank export (sat-question-index.json) — filter the bank to this skill tag — 42 questions, roughly a third of each difficulty
Explain it to yourself
Close the deck.
Discussion prompt
Without looking, explain how a condition changes a probability calculation, and why the probability of A given B differs from the probability of B given A.
Hint: One part of the fraction stays the same.
Answer:
A condition restricts the population you are selecting from, so the denominator becomes that subgroup's total rather than the grand total.
The numerator is unchanged: it is the cell counting people who satisfy both conditions.
So A given B and B given A share a numerator and have different denominators — B's total in one case, A's total in the other.
They are equal only when those two totals happen to match, which is a coincidence rather than a rule.
And a both-and probability uses the grand total, which is larger than either, so it is always the smallest of the three.
Explain it
Two minutes, out loud.
Discussion prompt
A friend gets the numerator right every time and the answer wrong about half the time. What is happening, and what one change do you give them?
Answer:
Diagnose it: they are pairing the correct cell with the grand total, because that is what most probability questions use and it has become automatic.
Show them the two answers from one table. Compute 42 over 100 and 42 over 57 side by side, and point out that both are meaningful and only one was asked for.
Give the change as an ORDER rather than a rule: write the denominator down first, before looking for the numerator.
Explain why the order fixes it: finding the numerator first creates the pull toward the familiar denominator. Choosing the denominator from the wording removes the pull.
And give the check: the phrase after given that names the population, and its total is the only number that can go underneath.
Commit first
Commit before you check.
Predict first
In a group, 20 people play both chess and piano. There are 40 chess players and 25 piano players. What is the probability that a piano player plays chess?
Correct: 20 over 25
Why: The condition names piano players, so the denominator is 25 and the answer is 20 over 25, which is 0.8. The second choice, 20 over 40, answers the reverse question — the probability that a chess player plays piano, which is 0.5. Both are computable from the same three numbers, and only the wording distinguishes them. Note how different they are: 0.8 against 0.5.
Connect it up
Blank paper.
Draw it
Draw a two-way table with two rows, two columns and all the totals filled in with letters. Label the four cells, the two row totals, the two column totals and the grand total. Then, beside it, write four questions and draw an arrow from each to the total that serves as its denominator: chosen at random goes to the grand total, both-and goes to the grand total, given a row category goes to that row total, and given a column category goes to that column total. Underneath, write the reversal pair side by side with the same numerator circled in both, to show that only the denominators differ. At the bottom, in large letters: NAME THE DENOMINATOR FIRST.
Exit ticket
One question before you close the deck.
Predict first
A question begins: given that a respondent is under 25. What have you learned before reading any numbers?
Correct: The denominator is the total number of respondents under 25
Why: The phrase given that names the population you are selecting from, and its total is the denominator — that is fixed by the wording alone, before any number is read. The under-25 total is not the numerator; the numerator will be a cell inside that group. The grand total would apply only if no restriction were stated. And it is precisely because the denominator is decided by the sentence that you can write it down before looking at the table.
Recap
One type, one instruction: name the denominator before you look for the numerator.
| never do this | do this instead |
|---|---|
| Use the grand total after a given-that phrase | Use the named row or column total |
| Find the numerator first | Write the denominator down before anything else |
| Answer the reverse conditional | Use the total of the group the condition names |
| Treat both-and as a restriction | It is a property; the grand total still applies |
| Accept a fractional count of people | A non-integer means the wrong denominator was used |
| Assume A given B equals B given A | They share a numerator and nothing else |
Naruhodo Tutoring — SAT Math question bank export (sat-question-index.json) — and the site's SAT pages to drill this type in isolation, then mixed
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