Investigation Guide Teacher Guide Luck Lab Investigation
Teacher Facilitation Guide
Tier 2 Intervention
Standard: CO HS.S-CP.A.2
Learning Objective
Students will determine if two events are independent by comparing the product of their individual probabilities to the probability of both events occurring together.
Small Group Setup
Group Size: 3-5 studentsMaterials: Pennies (1 per student), sets of 10 marbles (4 Red, 6 Blue), "Luck Lab" Worksheets.Duration: 30-40 minutes
Key Concept
"Does knowing the outcome of Event A change the chance of Event B?"
\[P(A \cap B) = P(A) \cdot P(B)\]
Instructional Routine
1. Concrete Exploration (10 mins)
The Hook: Give each student a penny.
"If I flip a Head first, does that change the chance of flipping a Head the second time? No. The coin doesn't have a memory. These are independent."
Have students flip their coin twice and record the data. Emphasize that the first result didn't 'force' the second result.
2. Bridging to Math (15 mins)
The Experiment: Use the marble bag (4 Red, 6 Blue).
Scaffolded Questioning:
"What is the probability of picking Red? (4/10)"
"If I put it back (with replacement), does the bag look different for the second pick?"
"What if I KEEP it (without replacement)? Now what is the probability of picking Red?"
3. The Independence Formula (10 mins)
Show the product check. This is where most Tier 2 students struggle: Multiplying fractions.
Intervention Tip:
Use a visual grid or "area model" to show why we multiply. If half the coins are Heads, and half of those are Heads again, that's half of a half (a quarter).
Error Analysis & Scaffolding
Common Mistake
Students add probabilities instead of multiplying for "A and B".
REMEDY:
Ask: "Is it MORE likely or LESS likely that two things happen together?" Addition makes the number bigger; multiplication of fractions makes it smaller.
Common Mistake
Confusing 'independent' with 'disjoint/mutually exclusive'.
REMEDY:
Explain that disjoint means they CAN'T happen together. Independent means they CAN happen together, but they don't affect each other.
Luck Lab Slides Statistics Unit 4
LUCK LAB
Mastering Independence
The Big Question
"Does knowing the result of Event A change the probability of Event B?"
Yes
The events are
No
The events are
Classic Independence
Trial 1
Flip a coin. It lands on HEADS.
Trial 2
What is the probability of flipping HEADS again?
1 / 2
The coin does not have a "memory." The first flip had ZERO impact on the second.
The Math Check
Two events A and B are independent IF:
\[P(A \cap B) = P(A) \cdot P(B)\]
To test for independence, multiply the individual probabilities. If they equal the joint probability, the events are independent.
The Marble Test
Scenario 1: With Replacement
You draw a marble, record the color, and PUT IT BACK.
INDEPENDENT
Scenario 2: No Replacement
You draw a marble, and you KEEP IT OUT.
DEPENDENT
TIME TO INVESTIGATE
Use your "Luck Lab" worksheet to complete the marble experiments. Prove independence using the math rule!
Luck Lab Worksheet Luck Lab Investigation
PROBABILITY EXPERIMENT 01
Researcher:
Date:
Experiment 1: The Penny Proof
Flip a penny two times. Does the first flip affect the second? Let's check the math.
Your Trials:
Flip 1:
Heads / Tails
Flip 2:
Heads / Tails
The Theory:
The probability of Heads is \(P(H) = \frac{1}{2}\).
Math Check
\(P(H \text{ and } H) = P(H) \cdot P(H)\)
\( \frac{1}{4} = \frac{1}{2} \cdot \frac{1}{2} \)
Conclusion:
Did your coin "remember" what happened on the first flip? Why or why not?
Experiment 2: The Marble Mix
4 Red
6 Blue
10 Total
Scenario A: With Replacement
Pick 1, put it back, pick 2.
1. P(Red) first pick:
2. P(Red) second pick:
Is the bag the same for pick #2?
Yes
No
Scenario B: No Replacement
Pick 1, KEEP IT, pick 2.
1. P(Red) first pick:
2. P(Red) second pick:
Is the bag the same for pick #2?
Yes
No
The Ultimate Check
Two events are independent if \(P(A \text{ and } B) = P(A) \cdot P(B)\).
If \(P(A) = 0.5\) and \(P(B) = 0.4\), and we know \(P(A \cap B) = 0.2\):
0.5 \(\times\) 0.4 =
Are these events independent? Why?
Luck Lab Exit Ticket Luck Lab Exit Ticket
End of Session 01 Check-In
Researcher ID:
1
The Intuition Test
A bag contains 5 green cards and 5 orange cards. You pick a card, keep it in your hand, and then pick a second card. Are these events independent? Why?
Yes, they are independent.
No, they are dependent.
Explain your reasoning:
2
The Math Check
In a different experiment, the probability of Event A is \(0.6\) and the probability of Event B is \(0.3\). The probability of both occurring together is \(P(A \cap B) = 0.18\).
Is \(P(A \cap B) = P(A) \cdot P(B)\)?
Product Check
0.6 \(\times\) 0.3 =
Verdict: Are they independent?
INDEPENDENT
DEPENDENT
Self-Reflection
How confident do you feel explaining the difference between independent and dependent events?
Need Help
Getting There
I've Got It
Luck Lab Answer Key Luck Lab Master Key
Teacher Resource
Luck Lab Worksheet Answers
Exp 1: Penny Proof
Conclusion Answer:
The coin did not "remember." The probability of the second flip remained 1/2 regardless of the first outcome. Multiplying 1/2 * 1/2 = 1/4 matches the observed probability of getting two heads in a row.
Exp 2: Marble Mix
Scenario A (Replacement):
P(Red 1) = 4/10; P(Red 2) = 4/10. Independence: YES.
Scenario B (No Replacement):
P(Red 1) = 4/10; P(Red 2) = 3/9. Independence: NO.
Luck Lab Exit Ticket Answers
1
Intuition Test
Answer: No, they are dependent.
Reasoning: Because you kept the card, the total number of cards changed (from 10 to 9) and the number of that specific color changed. The "bag" is now different for the second pick.
2
Math Check
Calculation:
0.6 \(\times\) 0.3 = 0.18
Verdict:
INDEPENDENT
Reasoning: Since the product of the individual probabilities (0.18) is exactly equal to the joint probability provided (0.18), the mathematical rule proves independence.
Intervention Strategy
Scaffold: For students struggling with 0.6 \(\times\) 0.3, remind them to multiply 6 \(\times\) 3 first, then count the decimal places (2 total).Concrete Check: If a student gets the Exit Ticket #1 wrong, physically hand them two cards, have them keep one, and show them how the denominator in their probability fraction must change.Terminology: Clarify that "Independent" does not mean "doesn't happen," it means "doesn't affect the other." 
