SAT Math — two-way tables and conditional probability

Probability and conditional probability is one of the SAT Math skills in Problem-Solving and Data Analysis. College Board describes it as using one-way and two-way tables, area models and other representations to find relative frequencies, probabilities and conditional probabilities, and, less often, working backward from a probability to an unknown number in a table. Almost every question of this kind turns on a single decision: which total goes in the denominator. This material drills that decision from every side.

Three questions share one table of 400 commuters sorted by age group and by how they get to work. They ask, in turn, for a joint probability (under 35 and taking transit, out of everyone), a conditional probability given the row (transit, among commuters under 35) and the reversed condition given the column (under 35, among transit users). The numbers in the cells never change; only the group you select from does, and the wrong answers are exactly the other denominators.

A coffee shop's orders by temperature and size ask for a marginal relative frequency and for a fair comparison between two groups of different sizes, where raw counts mislead and proportions within each row decide. A partially completed table of voters by district has to be filled in from its totals before the question can be answered. A table given in percents of all respondents asks for a conditional percent, and a grouped bar graph of survey answers in three neighborhoods asks for a probability within one neighborhood.

The remaining questions leave the table behind but keep its logic. A probability of 0.2 in a club of 45 is turned into a count, a sample proportion is used to predict a count in a larger group, and two situations reverse a condition with percents: online orders and express delivery, and a screening test whose positive results come from two very different groups, so that most positives turn out to be false alarms. Each of these can be solved by imagining 100 or 1,000 cases and building the table yourself, which is the method the explanations show.

The tables and the bar graph are drawn in the questions, with all their values also given in words. Explanations name the denominator every wrong answer used, so a miss shows which total you picked. The flashcards define joint, marginal and conditional relative frequencies, compare P(A given B) with P(B given A), and summarize the techniques for completing tables, reading percent tables and handling two-stage percents.

The material offers a quiz and flashcards. It is independent practice with invented data; no question is taken from any published test.

  • Compute joint, marginal and conditional relative frequencies from a two-way table
  • Choose the correct denominator for a conditional probability and distinguish P(A given B) from P(B given A)
  • Complete a partially filled two-way table from its totals
  • Read conditional percents from a relative frequency table
  • Compare a category across groups of different sizes using proportions
  • Turn a probability into a count and use a sample proportion to predict a count
  • Solve two-stage percent and screening-test problems by building a table of 100 or 1,000 cases

Practice material written by Zestly, based on the SAT Math skill "Probability and conditional probability" in the Problem-Solving and Data Analysis domain, as described in College Board's Assessment Framework for the Digital SAT Suite (version 3.01, August 2024, Appendix B). All data sets and situations are invented. Zestly is not affiliated with College Board.

Sample question

Table of 200 drink orders at a coffee shop by temperature and size. Hot: small 48, medium 60, large 12, total 120. Iced: small 22, medium 40, large 18, total 80. Totals: small 70, medium 100, large 30, all orders 200.

The table summarizes 200 drink orders at a coffee shop by temperature and size. Based on the table, which statement is true?

See the answer

A randomly chosen iced order is more likely to be large than a randomly chosen hot order.

Compare the proportion of large drinks within each row. Among iced orders, $\frac{18}{80} = 0.225$ are large; among hot orders, $\frac{12}{120} = 0.1$. So an iced order is more likely to be large, and the reverse statement is false. Equal likelihood is false too: 0.225 and 0.1 differ, and being part of the same 30 large orders says nothing about each row's share. Finally, only 80 of the 200 orders were iced against 120 hot, so a randomly chosen order is more likely to be hot, not iced.

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