Reality Check Slides Reality Check
Empirical Probability & Expected Value
Tier 2 Intervention Small Group Study
The Random Variable (X)
A random variable is a numerical description of the outcome of an experiment.
Examples:
Number of pets in a home
People in a waiting room
Goals scored in a game
X
The "What" we are measuring
Empirical Distribution
Empirical probabilities are based on actual data rather than theoretical math.
1
Count
How many times each value occurred.
2
Calculate
Divide count by the total total sample size.
3
Table
List values of \(X\) and \(P(X)\).
Formula: \(P(X) = \frac{\text{Frequency}}{\text{Total Count}}\)
Expected Value \(E(X)\)
The "long-run average" or what we expect to happen over time.
\[E(X) = \sum X \cdot P(X)\]
Multiply each value by its probability, then sum them up.
X P(X) X \(\cdot\) P(X)
0 0.40 0.00
1 0.60 0.60
Total: 1.00 EV = 0.60
Case Study: The Neighborhood Survey
We surveyed 100 households in our district to see how many people live in each home. We will use this data to calculate the Expected Occupancy .
Household Data Challenge Worksheet Household Data Challenge
Empirical Probability & Expected Value
Name:
Date:
The Scenario: The Green Valley Survey
Researchers surveyed 100 households in Green Valley to count the number of functional vehicles owned by each household. This is our Random Variable (X). Use the raw data below to build a probability distribution and calculate the expected value.
Phase 1: Build the Distribution
Raw Survey Results
Vehicles (X) Frequency 0 15 households 1 35 households 2 40 households 3 10 households Total 100 households
Your Task:
Calculate the probability \(P(X)\) for each value of \(X\). Remember: \[P(X) = \frac{\text{Frequency}}{\text{Total (100)}}\]
X P(X) (decimal) 0 1 2 3 Total Sum
Phase 2: Calculate the Expected Value \(E(X)\)
To find the expected value, multiply each \(X\) by its probability \(P(X)\), then add the results.
Value (X)
Probability P(X)
Multiply (X · P(X))
Subtotal
0
×
Write Prob Here
=
1
×
=
2
×
=
3
×
=
Add them all up:
Total Sum = E(X)
Phase 3: Reality Check Analysis
1. Interpret the number: Based on your calculation, what is the "average" number of cars in a Green Valley household?
2. Comparison: Is it possible for a single household to have exactly your expected value number of cars? Why or why not?
Small Group Challenge
If a new housing development adds 500 more homes to Green Valley, how many total vehicles would we expect there to be in the whole neighborhood? (Hint: Multiply your E(X) by 500!)
Probability Pulse Check Assessment Probability Pulse Check
Progress Monitoring Exit Ticket
Score:
Name:
Date:
The Challenge
A survey asked 50 students how many siblings they have. The results are shown in the frequency table. Construct the probability distribution and find the Expected Number of Siblings .
Raw Data
Siblings (X) Frequency 0 10 students 1 25 students 2 10 students 3 5 students Total 50 students
1. Distribution Table
Note: Probabilities should add to 1.00
2. Calculate Expected Value \(E(X)\)
Show your work below. Multiply \(X\) values by their \(P(X)\) and sum the results.
Expected Value =
3. Concept Check
If the Expected Value is 1.2, what does this tell us about the number of siblings a "typical" student has in this group?
Masters Concept
Progressing
Needs Support
Intervention Facilitation Guide Intervention Facilitation Guide
HS.S-MD.A.4 Tier 2 Small Group 30-45 Minutes
Step-by-Step Facilitation
1. Hook & Random Variable (5 mins)
Use the Reality Check Slides to define the random variable \(X\). Ask: "If we survey this group for 'number of siblings', what are the possible values for \(X\)?" Emphasize that \(X\) must be a number.
2. Empirical Data vs. Theoretical (5 mins)
Explain that we aren't "guessing" probabilities (like with a fair die). We are looking at real history. If 15/100 homes have 0 cars, the probability is 0.15.
3. Guided Construction (15 mins)
Distribute the Household Data Challenge. Walk through Phase 1 together.
4. Expected Value Modeling (10 mins)
Model the multiplication for \(X=0\) and \(X=1\). Have students finish the table. Discuss: "If the EV is 1.45, does it mean a house has 1.45 cars? No! It's the balance point of the data."
Answer Key: Household Data Challenge
Phase 1: Probabilities
\(P(0) = 0.15\)
\(P(1) = 0.35\)
\(P(2) = 0.40\)
\(P(3) = 0.10\)
Phase 2: Expected Value
\(0 \cdot 0.15 = 0.00\)
\(1 \cdot 0.35 = 0.35\)
\(2 \cdot 0.40 = 0.80\)
\(3 \cdot 0.10 = 0.30\)
Sum: 1.45 vehicles
Common Pitfalls
Adding frequencies instead of probabilities. Remind students: \(P(X)\) is always \(\le 1\).
Forgetting \(X=0\). Emphasize that \(0 \cdot P(X)\) is \(0\), but it must still be accounted for in the table.
Confusion about non-integer results. Explain that Expected Value is a parameter, not a possible outcome.
Targeted Questions
"If we doubled the sample size to 200, would our probabilities change? (No, because the proportions likely stay the same.)"
"Why don't we just add the number of cars and divide by 4? (Because each count has a different frequency/weight.)"
"Look at your E(X). Is it closer to the value with the highest frequency?"
Pulse Check Key
P(X): 0.2, 0.5, 0.2, 0.1
E(X): \(0(0.2) + 1(0.5) + 2(0.2) + 3(0.1)\)
Result: \(0 + 0.5 + 0.4 + 0.3 = \mathbf{1.2}\)
