Question type
Conditional probability from a two-way table on the SAT
In a two-way table question, the condition decides the denominator. "Given that", "of the" or "among" restricts you to one row or column, so divide by that group's total, not the grand total. The numerator is the cell where both conditions hold.
These questions give a table that sorts a group by two categories, with totals, and ask for a probability restricted to one subgroup: "of the students who chose biology, what is the probability that the student is a senior".
The easy version names a row printed in the table; the harder ones make you add two cells or subtract from a total first.
- On 1 of 5 official practice tests
- Desmos partly helps
Why students miss it
Dividing by the grand total. Every question has a choice with the grand total underneath, for the student who misses the condition.
Reversing the condition: the probability that a senior chose biology is a different fraction from the probability that a biology student is a senior.
Missing a combined group. "Not paid with cash" means card plus mobile, which you may have to add up yourself.
How to spot it
- A table with row and column totals.
- "Given that", "of the", "among", or "one of the ... is selected at random".
- Answer choices that are fractions of two table entries, often unreduced.
The method
Denominator first
Find the group the condition restricts you to, and its total.
- The group is a printed row or column
- use its total.
- The group is "not" something
- grand total minus that row.
- Two groups combined
- add their totals.
- Numerator
- the cells inside that group that also meet the question.
With Desmos
Desmos does the division; the table reading is yours.
- Once you have the cells, type (66+78)/(150+140) to check against a decimal choice.
Choices are often written as unreduced fractions; match the numerator and denominator rather than the decimal.
Two real questions
From the 1500prep bank, the way the app shows them. Answer first, then open the explanation.
The table summarizes the 360 purchases made at a store one day, classified by payment method and by purchase size. One of these purchases is selected at random. Given that the purchase was not paid for with cash, what is the probability that it was a large purchase?
Show the answer and explanationThe answer: $\frac{144}{290}$
The table shows the 200 books in a classroom library, classified by subject and by binding. If one of the fiction or biography books is selected at random, what is the probability that it is a hardcover book?
Show the answer and explanationThe answer: $\frac{71}{160}$
Questions students ask
How do you find conditional probability from a two-way table?
Restrict yourself to the row or column named in the condition, divide the matching cell by that row's or column's total, not by the grand total.
What does "given that" mean in an SAT probability question?
It tells you which group to look inside. That group's total becomes the denominator.
Practise this type until it is automatic
Five more two-way-table probability questions, checked instantly, with every wrong choice explained. Free, no card.