Shrinking Sample Space

Teacher Lesson Guide • 9th Grade Algebra I

Duration

50 MIN

Objectives

• Distinguish between independent and dependent events.
• Calculate conditional probability using the formula \(P(A|B)\).
• Visualize the "narrowing" of a sample space when a condition is applied.

Materials

Shrinking Space Slides
Probability Funnel Worksheet
Final Filter Exit Ticket
Standard Decks of Cards (1 per pair)

Key Vocabulary

Conditional Probability

The probability of an event given that another has occurred.

Sample Space

The set of all possible outcomes.

Independent Events

One occurrence does not change the likelihood of the other.

Lesson Flow

1. The Swimsuit Paradox (5 min)

Start with the scenario: What is the probability a person is wearing a swimsuit right now? Then ask: What is the probability given they are at the pool?

"How did the 'condition' of being at the pool change our denominator? We aren't looking at all people anymore, only pool-goers."

2. Video-Guided Discovery (10 min)

Watch "Conditional Probability - Probability With Conditions". Pause to allow students to fill out the Probability Funnel Worksheet.

[2:40] PAUSE: Students copy the notation \(P(A|B)\).
[5:08] PAUSE: Give 1 min to solve the Dessert Tray problem.

3. Activity: Sample Space Shrink (20 min)

Students use physical card decks. They first calculate \(P(\text{King})\) with 52 cards. Then, introduce the condition: "The card is a Face Card (J, Q, K)."

The "Shrink" Step

Students must physically remove all non-face cards. They should be left with 12 cards. This visualizes the narrowing of the denominator.

4. The Gambler's Fallacy (10 min)

Discuss the coin flip example from the video. Ask: "If you flip 10 heads in a row, is the next flip more likely to be tails?" Explain that for Independent Events, the past condition does not shrink the current sample space.

5. Exit Ticket (5 min)

Distribute the Final Filter Exit Ticket. Students solve: \(P(6 \text{ on a die} | \text{even number})\).

Answer Key & Teacher Notes

Video Example 1: Dessert Tray

Condition: Only Donuts (5 sprinkle + 6 plain = 11 total)

Criteria: Sprinkles (5)

P(Sprinkles | Donut) = 5/11

Activity: Card Shrink

Simple P(King): 4/52 = 1/13

Condition: Face Cards only (12 total)

Criteria: Kings in that set (4)

P(King | Face Card) = 4/12 = 1/3

Exit Ticket: Die Roll

Problem: What is the probability of rolling a 6 on a die, given that the number is even?

Sample Space

{1, 2, 3, 4, 5, 6}

Filtered Space

{2, 4, 6}

Final Probability

1/3

*Note: If students answer 1/6, they did not apply the condition to the denominator.

Troubleshooting Common Misconceptions

Misconception: The "Past" is a Condition

Students often think a streak of tails makes a head more likely. Remind them: The coin has no memory. Unless the set of outcomes changes, it's independent.

Misconception: P(A|B) is the same as P(B|A)

P(Pool | Swimsuit) is NOT the same as P(Swimsuit | Pool). Use the "Funnel" visual: the first condition determines the mouth of the funnel.

Shrinking Sample Space

Conditional Command Teacher Guide

Shrinking Space Slides

Probability Funnel Worksheet

Final Filter Exit Ticket

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