Conditional Probability

SAT Problem-Solving and Data Analysis

In this lesson:

• Define : probability of given
• Read conditional probability from two-way tables
• Distinguish from

Your Learning Goals for This Lesson

By the end of this lesson, you will be able to:

1. Define and interpret as "probability of given "
2. Calculate conditional probability from a two-way table
3. Read conditional language in SAT problem statements
4. Distinguish from and explain the difference

Knowing Something Changes the Question

Imagine asking: "What is the probability that this person brought an umbrella today?"

• Without any information: you use all people as the denominator
• If you know it rained today: you restrict to people who experienced rain

The sample space shrinks when we have additional information.

Notation: Means "Given B"

The vertical bar is read as "given that".

Formula:

With counts from a table:

Reading Conditional Probability from Tables

Rule: "Given that..." → find that group's total → use as denominator.

Example 1:

Given "full-time" → denominator = 180

Example 2:

Same table — now the condition is "part-time."

• "Given part-time" → denominator = 120 (part-time column total)
• "Not satisfied and part-time" → numerator = 70

Quick Check: Switching the Condition

Using the same table:

Find . Think before advancing…

vs : Different Denominators

The same two numbers (120, 300) — but different denominators.

Comparing P(A|B) and P(B|A) Side-by-Side

Same numerator. Different denominators. Different answers.

These are different questions — read the condition carefully.

SAT Language Cues for Conditional Probability

These phrases signal a conditional question:

• "given that..." → classic conditional notation
• "among those who..." → restricts to subgroup
• "of the [group]..." → e.g., "of the part-time workers"
• "if a [group member] is selected..." → restricts

Strategy: Underline the phrase. That group = denominator.

SAT-Style Problem: Parse the Language

200 students; 80 play sports. Of sports players, 60 are Grade 10.

P(Grade 10 | plays sports)?

• Condition: "plays sports" → denominator = 80
• "Grade 10 and sports" = 60

Quick Check: Identify the Given Group

"A worker is selected at random from the part-time employees. What is the probability they are satisfied?"

What is the denominator?

Think before advancing…

Common Trap: Using the Grand Total

"Given full-time" → correct denominator = 180, not 300.

Test yourself: Is there a "given" phrase? If yes, use that group's total.

Fill in Missing Table Values Before Computing

Complete the table first — then apply the conditional rule.

Practice: Conditional Probability from a Table

Check Your Work: Practice Solutions

1. Given Grade 11 (total=90):

2. Given English (total=80):

3. Given Math (total=100):

Notice: .

Key Takeaways: Conditional Probability Rules

• ✓ : given B → denominator = B's total
• ✓ "Given that," "among," "of the" = conditional signal
• ✓ — order matters

Never use grand total for conditional P

Underline the given phrase to find the denominator

What You Will Learn Next

Future topics build on today's conditional probability skills:

• Independence: when does ?
• Bayes' theorem: connecting and
• Combined probability: multi-step problems

Conditional probability is the foundation for all advanced probability.