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Data AnalysisCore

Probability and Conditional Probability

Pull probabilities out of two-way tables — especially the conditional ones where the words quietly shrink your denominator.

⏱ 7 min📍 Learn first: Two-Variable Data and Scatterplots
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Learn Probability and Conditional Probability in the format that works for you

Start with a quick audio review, take the deeper podcast lesson, or move directly into the visual guide and knowledge check below.

13 free resources
00:00 / 01:37

Shortcuts for Hard DSAT Probability Questions

A short spoken refresher on this concept.

00:00 / 22:59

Digital SAT Probability Logic and Shortcuts

The full lesson, worked through in depth.

Concept Snapshot

Definition
Probability = favorable outcomes / total outcomes. Conditional probability restricts the total to a given group first.
What the SAT tests
Reading two-way frequency tables; P(A given B) questions where the condition changes the denominator.
Key rule
"Given X" or "of the X" → the denominator becomes the X row or column total, not the grand total.
Most common mistake
Using the grand total as the denominator when the question restricts to one group.
Prerequisite
Two-Variable Data and Scatterplots

Visual Explanation

What to picture: "Given X" or "of the X" → the denominator becomes the X row or column total, not the grand total.

A diagram for this concept is in production. Until it is ready, the worked example below shows the same relationship step by step.

Concept Mind Map

DefinitionProbability = favorable outcomes / total outcomes. Conditional probability restricts the total to a given group first.
Key rule"Given X" or "of the X" → the denominator becomes the X row or column total, not the grand total.

Probability and Conditional Probability

Data Analysis

Table ReadStraight probability from a two-way table cell over a total.
ConditionalThe condition names the denominator group.
Build the TableFill missing cells from row/column totals before computing.

How the SAT Tests It

Pattern

Table Read

Straight probability from a two-way table cell over a total.

Clue: "selected at random"
Pattern

Conditional

The condition names the denominator group.

Clue: "given that", "of the students who…"
Pattern

Build the Table

Fill missing cells from row/column totals before computing.

Clue: A table with blanks

Worked Example

A survey table: 120 juniors (70 play sports), 80 seniors (60 play sports). If a selected student plays sports, what is the probability they are a senior?

"If…plays sports" is a condition — we only look inside the sports players.
Sports players: 70 + 60 = 130. The 200 total students is now irrelevant.
Seniors within that group: 60.
P(senior | plays sports) = 60/130 = 6/13. Compare: P(senior) overall = 80/200 = 2/5 — the condition changed the answer.

Try It Yourself

A table shows 250 plants: 90 tomato (36 diseased), 160 pepper (24 diseased). What is the probability that a randomly chosen DISEASED plant is a tomato plant?

Common Traps

What students do

Confusing P(A|B) with P(B|A).

Why it fails

Both use the same two cells — only the denominator differs, and the wording decides which.

Do this instead

Circle the phrase after "given that" or "of the" — that group IS the denominator. Everything else in the table goes dark.

Interactive Flashcards

Review deckCard 1 of 8

Four-Question Knowledge Check

4 quick questions drawn from this concept's own snapshot, patterns and traps.

Question 1 of 4. Which statement is the key rule for this concept?

Audio Lesson

Digital SAT Probability Logic and Shortcuts

Conditional probability without formulas — just table discipline.

00:00 / 22:59

22:59 · the full lesson

What to notice

"Given X" or "of the X" → the denominator becomes the X row or column total, not the grand total.

What the lesson covers

  • 0:00 Table anatomy
  • 1:20 The condition rule
  • 2:50 Reversed conditionals
  • 4:10 Practice

Study Guide and Transcript

One-page study guide

The snapshot, the key rule, the patterns and the traps, on one page.

Source-Grounded AI Tutor

Ask the Probability and Conditional Probability notebook

A tutor grounded in this concept's own sources — ask for a simpler explanation, another example, or why a particular answer is wrong. It is in development and will appear here once its sources pass academic review.

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