Decision Lab Workbook Decision Lab Workbook
Topic: Probability-Based Strategy Analysis (HS.S-MD.B.7)
Scientist:
Date:
The Lab Objective
In this lab, you will act as a data analyst. You will use Expected Values and Tree Diagrams to decide which strategy is mathematically superior in sports and medicine. Remember: A good decision is based on the data available at the time, not just the eventual outcome!
Case Study 1: The Goalie's Gamble
Strategy A: Keep Goalie In
• Chance to Tie: 5% (Score: 0)
• Chance to Lose: 95% (Score: -1)
Strategy B: Pull Goalie
• Chance to Tie: 15% (Score: 0)
• Chance to Lose by 1: 55% (Score: -1)
• Chance to Lose by 2: 30% (Score: -2)
1. Construct a tree diagram for Strategy B (Pull Goalie). Label the probabilities and outcomes.
2. Calculate the Expected Value (EV) for each strategy.
EV: Keep Goalie In
EV: Pull Goalie
3. Based on the data, which strategy is better? Justify your choice using the numbers calculated above.
Case Study 2: The Rare Disease Test
A rare disease affects 1 out of 1,000 people. A screening test is 99% accurate (meaning it correctly identifies both sick and healthy people 99% of the time).
4. Total Population Analysis: Assume you test 100,000 people. Fill in the table below.
Category Calculation Result Total with Disease 0.001 × 100,000 Total Healthy 0.999 × 100,000 True Positives (Sick & Test Positive) 0.99 × [Total with Disease] False Positives (Healthy & Test Positive) 0.01 × [Total Healthy]
5. Calculate the Probability of actually being sick given a positive test result:
\( P(\text{Sick} | \text{Positive}) = \)
6. Strategy Decision: If a patient tests positive, should they immediately start an intensive, high-risk treatment? Why or why not?
Decision Lab Notebook | Module 4.2
Decision Lab Slides Decision Lab
Decisive
Data
Analyzing Strategies with Probability
Targeted Intervention | HS.S-MD.B.7
Today's Mission
The Goal
Use probability to calculate the expected outcome of different choices in high-stakes situations.
The Scenarios
The Goalie's Gamble (Sports)
The Lab Result (Medicine)
P(A|B)
Decision Math
The Goalie's Gamble
Down by 1. 60 seconds left. What do you do?
Option A: Goalie In
5% chance to Tie
95% chance to Lose
Option B: Pull Goalie
15% chance to Tie
30% Lose by 2
55% Lose by 1
The Core Question
Does increasing your chance to tie justify the risk of losing by more?
Visualizing Decisions
Start
Strategy 1
Strategy 2
Outcome A (%)
Outcome B (%)
Outcome C (%)
Outcome D (%)
Expected Value: The sum of (Probability × Value)
The Lab Result
A test for a rare disease is 99% accurate.
You just tested positive.
Prevalence
1 in 1,000
Sensitivity
99%
Specificity
99%
Don't assume 99% positive = 99% sick!
Think in Raw Numbers
Let's model a population of 100,000 people:
Sick: 100
99 True Positives
1 False Negative
Healthy: 99,900
999 False Positives
98,901 True Negatives
The Reality Check
Total Positive Tests: 1,098
9.0%
Chance of being sick
P(Sick | Positive) = 99 / 1,098
Decisive Data Teacher Guide Teacher Facilitation Guide
Decisive Data Intervention
Lesson ID: decisive-data-intervention
Instructional Objective
Students will use probability concepts, including expected value and tree diagrams, to analyze and justify decisions in sports and medical contexts. This lesson addresses Colorado Standard HS.S-MD.B.7 .
Target Group
Tier 2 Intervention (Small Group 3-5 students)
Duration
45-60 Minutes
Key Vocabulary
Expected Value: The long-term average outcome of a random variable.Base Rate: The original probability of an event (e.g., disease prevalence).False Positive: A test result that incorrectly indicates presence.
Facilitation Steps
1
The Hook: Goalie Gamble (10 mins)
Present the Hockey Scenario. Ask: "If you're going to lose anyway, why pull the goalie?"
Discussion Prompt: "Many students think Strategy A is better because the EV is higher (-0.95 vs -1.15). Ask students if losing by 2 is actually 'twice as bad' as losing by 1 in a playoff game. Guide them to the realization that in hockey, only the Tie outcome matters."
2
The Math: Tree Diagrams (15 mins)
Guide students through the Strategy B tree diagram on the worksheet. Ensure they multiply along branches to get final probabilities.
Check for Understanding: Do the probabilities at each node sum to 1?
Key Concept: EV = Σ[Outcome × P(Outcome)]
3
The Trap: Base Rate Fallacy (20 mins)
Move to the medical case. Most students will guess a positive test means 99% sick.
Scaffolding Strategy:
Use the 100,000 population model to make the numbers concrete.
Help students see that there are 999 healthy people testing positive, but only 99 sick people testing positive.
Answer Key
Case Study 1: Hockey
EV: Keep Goalie
\( (0.05 \times 0) + (0.95 \times -1) = -0.95 \)
EV: Pull Goalie
\( (0.15 \times 0) + (0.55 \times -1) + (0.30 \times -2) = -1.15 \)
Justification: Strategy B is often preferred because it triples the chance of a tie (15% vs 5%). In sports, the loss margin often doesn't matter as much as the chance to stay in the game.
Case Study 2: Medicine
<table class="w-full text-sm border-collapse border border-slate-200"><tbody><tr class="bg-slate-50"><td class="border border-slate-200 p-2 font-bold">Total Disease</td><td class="border border-slate-200 p-2 font-mono">100</td></tr><tr><td class="border border-slate-200 p-2 font-bold">Total Healthy</td><td class="border border-slate-200 p-2 font-mono">99,900</td></tr><tr class="bg-slate-50"><td class="border border-slate-200 p-2 font-bold">True Positives</td><td class="border border-slate-200 p-2 font-mono">99</td></tr><tr><td class="border border-slate-200 p-2 font-bold">False Positives</td><td class="border border-slate-200 p-2 font-mono text-rose-600">999</td></tr></tbody></table>
Quality Control Exit Ticket Exit Ticket
Lab: Quality Control Analysis
Analyst:
Score:
Scenario: Product Testing
A factory produces high-end sensors. 2% of sensors are defective. A quality control robot correctly identifies 95% of defective sensors, but it also incorrectly flags 5% of good sensors as defective.
1. If the robot flags a sensor as "Defective," what is the probability it is actually healthy (a False Positive)?
Hint: Use a population of 1,000 sensors.
• Defective Sensors: _________
• True Positives: _________
• Good Sensors: _________
• False Positives: _________
Workspace
2. Strategic Decision: If a sensor is flagged as defective, should the company throw it away immediately or run a second manual test?
Confidence Check
Lost
Getting There
Data Master
