Table Truth Slides THE TABLE TRUTH
Interpreting Two-Way Frequency Tables
PROBABILITY UNIT // LESSON 01
The Core Question
If a medical test comes back positive, what is the probability that you actually have the disease?
"The answer isn't a single percentage. It's a relationship between groups of people, and to see it, we need a table."
Two-Way Frequency Tables
Also known as Contingency Tables, these organize data by two categorical variables.
Infected Healthy Total Vaxed 12 488 500 Unvaxed 85 415 500 Total 97 903 1000
The Power of the Denominator
Joint Probability
Probability of both things happening together.
\( P(A \cap B) = \frac{\text{cell count}}{\text{GRAND TOTAL}} \)
Ex: Chance a random person is vaxed AND infected.
Conditional Probability
Probability given we ALREADY KNOW something is true.
\( P(A|B) = \frac{\text{cell count}}{\text{SUBSET TOTAL}} \)
Ex: Chance someone is infected GIVEN they are vaxed.
Why the Denominator Matters
In 2021, reports showed that "50% of hospitalized patients were vaccinated."
Does this mean the vaccine doesn't work?
Without the denominator (how many people total were vaccinated in the population), that percentage is mathematically meaningless for risk assessment.
Data Detective Worksheet Lab Analysis: Case 101
Subject: Conditional Probability & Contingency Tables
Technician:
Date:
Case Briefing: The "Dax-9" Virus
In a controlled study of 2,000 residents in the city of Oakhaven, health officials tracked the efficacy of a new antiviral treatment against the "Dax-9" virus. The following raw data counts were collected over a six-month period.
Treatment Group Infected (Dax+) Not Infected (Dax-) TOTAL Received Treatment 45 ? 1,100 Placebo Control ? 780 ? TOTAL ? ? 2,000
Part 1: Data Reconciliation
Complete the missing values in the table above using the grand total of 2,000.
Part 2: Probability Analysis
Show your fractional setup before calculating the decimal/percentage.
1. Joint Probability:
What is the probability that a randomly selected resident was in the Placebo group AND became infected?
2. Marginal Probability:
What is the probability that a randomly selected resident in the study was infected, regardless of treatment?
3. Conditional Probability (The Risk Assessment):
Given that a resident received the treatment, what is the probability they became infected? \( P(\text{Infected} | \text{Treatment}) \)
4. Conditional Probability (The Flip):
Given that a resident was infected, what is the probability they were in the Placebo group? \( P(\text{Placebo} | \text{Infected}) \)
Part 3: Critical Interpretation
Compare your answers from Question 3 and Question 4. Explain why the denominators are different even though both questions involve infection and treatment/placebo.
A local news outlet reports: "Over 25% of all infected people were already taking the new treatment! Is the treatment useless?" Based on your calculations in Question 4, evaluate the accuracy and fairness of this headline.
Table Truth Teacher Guide Teacher Guide: The Table Truth
Lesson 01: Interpreting Two-Way Frequency Tables
Instructional Goals
Identify joint vs. marginal frequencies
Calculate P(A|B) vs. P(B|A)
Interpret data in real-world news contexts
Understand the "denominator effect"
Completed Data Table (Data Detective)
Group Infected (Dax+) Healthy (Dax-) Total Treatment 45 1,055 1,100 Placebo 120 780 900 TOTAL 165 1,835 2,000
Probability Analysis Key
1. Joint Probability (Placebo AND Infected):
120 / 2,000 = 0.06 or 6%
2. Marginal Probability (Infected):
165 / 2,000 = 0.0825 or 8.25%
3. Conditional (Infected | Treatment):
45 / 1,100 ≈ 0.0409 or 4.09%
4. Conditional (Placebo | Infected):
120 / 165 ≈ 0.7272 or 72.7%
Discussion Facilitation
The Misleading Headline
The news reports "27% of infected people were treated" (45/165). While factually true, it ignores that the treatment group was much larger than the placebo group. The actual risk of infection dropped from 13.3% (Placebo) to 4.1% (Treatment). This is a classic example of "The Base Rate Fallacy."
Rare Disease Paradox Slides THE RARE DISEASE PARADOX
Tree Diagrams & Positive Predictive Value
PROBABILITY UNIT // LESSON 02
"99% Accuracy"
A new test for Lupus is 99% accurate.
You take the test and it comes back POSITIVE.
Only 0.5% of the population actually has Lupus.
What is the probability you have the disease?
A) 99% B) 50% C) Less than 50%
Visualizing the Population
Population
Has Disease (0.5%)
Healthy (99.5%)
Test Positive (99%)
Test Negative (1%)
Test Positive (1%)
Test Negative (99%)
Calculating PPV
Positive Predictive Value (PPV) is the probability that subjects with a positive screening test truly have the disease.
Total Positive Tests
49.5 + 99.5 = 149
(True Positives + False Positives)
Final Probability
49.5 / 149 ≈ 33.2%
P(Disease | Positive Test)
The Takeaway
When a condition is rare, even a highly accurate test will generate more False Positives than True Positives.
This is why doctors don't screen the entire population for rare diseases—it would cause unnecessary panic and further invasive testing for healthy people.
Screening Simulation Worksheet Clinical Model: PPV-02
SCREENING SIMULATION
Assigned Student
Simulation Objective
"Map a population of 20,000 individuals through a diagnostic tree for 'Phage-X,' a condition affecting 1 in 200 people. Determine the predictive power of a test with 98% sensitivity and 95% specificity."
1
Population Mapping (Tree Diagram)
Total 20,000
Affected (0.5%)
?
Healthy (99.5%)
?
Pos Result (98%)
?
Neg Result (2%)
?
Pos Result (5%)
?
Neg Result (95%)
?
2
Data Synthesis
Total individuals who tested POSITIVE:
Sum of True Positives + False Positives
Positive Predictive Value (PPV):
\( P(\text{Affected} | \text{Positive Test}) \)
Initial Hypothesis Check
Before doing the math, many people assume a test with 98% accuracy means a 98% chance of having the disease.
How does your calculated PPV compare to that 98% assumption? Why is it so different?
3
Policy Recommendation
Imagine you are a Public Health Official. A pharmaceutical company wants to mandate this test for every citizen in a country of 50 million people. Based on your simulation of 20,000 people:
Approximately how many healthy people would be told they are sick across the whole country?
Would you recommend this mandatory screening for the entire population? Support your answer with your mathematical findings regarding False Positives and PPV.
Rare Disease Teacher Guide Teacher Guide: The Rare Disease Paradox
Lesson 02: Tree Diagrams & PPV
Instructional Core
Students often struggle with the idea that an "accurate" test can be wrong. The core of this lesson is shifting their focus from the test's accuracy (sensitivity/specificity) to the prevalence of the condition. If the condition is rare, the sheer volume of healthy people means even a small 1-5% error rate will produce a massive number of false positives.
Simulation Calculations (20,000 People)
1. Initial Split
Affected (0.5%): 100 people
Healthy (99.5%): 19,900 people
2. Testing Outcomes
True Positives: 100 × 0.98 = 98
False Positives: 19,900 × 0.05 = 995
Total Positives: 98 + 995 = 1,093
Final Probability (PPV)
98 / 1,093 ≈ 0.0896 (8.96%)
Conclusion: Even with a 98% accurate test, a positive result only means a ~9% chance of actually being sick. 91% of people told they were sick are actually healthy.
Facilitation Notes
Q1
"Why do we see so many False Positives?"
Q2
"What is the cost of a False Positive?"
Justice by the Numbers Slides Justice by the Numbers
Forensic Probability & The Prosecutor's Fallacy
Case File // LESSON 03
"One in a Million"
A prosecutor says: "The DNA found at the crime scene matches the defendant. The probability that a random person has this DNA profile is 1 in 1 million."
Does this mean there is only a 1 in 1 million chance that the defendant is innocent?
The Conditional Flip
Evidence Probability
P(Match | Innocent)
"The chance of a random match if you didn't do it."
Guilt Probability
P(Innocent | Match)
"The chance you are innocent despite the match."
The Fallacy
Assuming that these two probabilities are the same is called the Prosecutor's Fallacy.
They are almost NEVER equal. One measures the rarity of evidence; the other measures the probability of guilt.
A Tragedy of Math
In 1999, Sally Clark was convicted of murdering her two sons who died of SIDS.
A "medical expert" testified that the chance of two SIDS deaths in one family was 1 in 73 million.
The jury interpreted this as: "There is only a 1 in 73 million chance she is innocent."
The Error
The expert ignored that SIDS deaths are dependent events (genetics/environment).
They committed the Prosecutor's Fallacy by equating the rarity of the event with the likelihood of guilt.
The Royal Statistical Society later intervened to help free her.
Doing the Math: Random Matches
If a DNA match is "1 in 1 million" and you live in a country of 300 million people...
300
Potential Matches
1/300
Chance of being the one
"The DNA gets you in the room; the other evidence has to prove you did it."
Verdict Analysis Worksheet Legal Analysis: Case 492-B
Documenting the Prosecutor's Fallacy
Confidential
Part 1: Defining the Variables
In legal statistics, the order of the condition changes the entire meaning of the statement. Define the following in plain English using the terms "Innocent," "Evidence Match," and "DNA Found."
P(Match | Innocent)
Plain English definition...
P(Innocent | Match)
Plain English definition...
Part 2: The "City of Metro" Case
Case Facts:
A robbery occurred in Metro City (Population: 1,000,000).
A partial fingerprint was found. The probability of a random person matching this print is 1 in 20,000.
A suspect is arrested because they matched the print. They have no previous criminal record and no other evidence connects them to the crime.
1. How many people in Metro City would be expected to match this partial fingerprint, including the actual perpetrator?
2. If the only evidence against the suspect is the fingerprint match, what is the probability that the suspect is actually innocent? \( P(\text{Innocent} | \text{Match}) \)
Critical Reasoning:
The prosecutor argues: "The chance of a random match is 1 in 20,000, so there is a 99.995% chance he is guilty." Identify the logical error in this statement based on your answer to Question 2.
Part 3: Closing Argument
"DNA and fingerprints don't lie, but statistics can be made to say anything. As a juror, how would you explain to the rest of the jury that a 'one in a million' match doesn't automatically mean the defendant is the one in a million?"
Justice Teacher Guide Teacher Guide: Justice by the Numbers
Lesson 03: Forensic Probability & The Prosecutor's Fallacy
Instructional Focus
The core of this lesson is transposing the conditional. Students must realize that \( P(A|B) \neq P(B|A) \). In a legal context, \( P(\text{Match} | \text{Innocent}) \) is very small (rarity of DNA), but \( P(\text{Innocent} | \text{Match}) \) can be quite large if the population is big and there is no other evidence.
Metro City Analysis Key
Q1: Expected Matches
1,000,000 / 20,000 = 50 people
There are 50 people in the city who would have this fingerprint profile.
Q2: Probability of Innocence P(I|M)
49 / 50 = 0.98 or 98%
Of the 50 people who match, only 1 is guilty. If the suspect is just a random match, there is a 98% chance they are innocent.
The Fallacy Point:
The prosecutor confuses "the chance of a random person matching" (0.005%) with "the chance the suspect is innocent" (98%). This is the Prosecutor's Fallacy.
Common Misconceptions
"1 in a million means he's guilty."
Explain that "one in a million" just defines the size of the group of potential suspects. It doesn't pinpoint the individual unless the group size is 1.
"Independence of events."
Remind students that in the Sally Clark case, the 1 in 73 million figure was wrong because it assumed SIDS deaths were independent, ignoring genetic and environmental factors that make them dependent.
Note: For a deep dive, mention the "People v. Collins" case (1968) where a prosecutor multiplied random physical characteristics to claim guilt, which was later overturned due to mathematical errors.
Formulaic Precision Slides Formulaic Precision
Sensitivity, Specificity & The Formal Formula
PROBABILITY UNIT // LESSON 04
The Algebra of Context
We have used tables and trees. Now, we use the Definition of Conditional Probability:
\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]
Numerator
Intersection: P(A and B)
Denominator
The Condition: P(B)
Medical Vocabulary
Sensitivity
The "True Positive Rate"
P(Positive | Disease)
How good is the test at finding the disease when it's there?
Specificity
The "True Negative Rate"
P(Negative | Healthy)
How good is the test at identifying healthy people correctly?
The Engineer's Dilemma
Airport Security Scanners
They have High Sensitivity. They want to catch every threat, even if it means lots of false alarms (keys, belt buckles).
Court of Law
The system aims for High Specificity. "Better that ten guilty persons escape than that one innocent suffer."
You can't always have both. Where you set the threshold depends on the cost of being wrong.
Calculate the Risk
If a test has 90% Sensitivity and 90% Specificity, and the disease is present in 1% of the population...
P(Disease | Positive) = ?
Time to work in the Lab
Formula Lab Worksheet Formal Analysis: Lab 04
SENSITIVITY & SPECIFICITY WORKSHOP
Formula Reference
\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]
Sensitivity (True +)
Probability of a positive test given the person has the disease.
P(Pos | Disease)
Specificity (True -)
Probability of a negative test given the person is healthy.
P(Neg | Healthy)
1 The Mammogram Study
A specific mammogram test has a sensitivity of 85% and a specificity of 90%. In a population of women, the prevalence of breast cancer is 1%.
A) Calculate the Joint Probability:
What is the probability that a random woman has cancer AND tests positive? \( P(C \cap \text{Pos}) \)
B) Calculate the Total Probability of Positive:
What is the probability that a random woman tests positive? \( P(\text{Pos}) \)
C) Final Conditional Probability:
Given a positive test, what is the probability she has cancer? \( P(C | \text{Pos}) \)
2 Engineering Thresholds
"A smoke detector is designed to have 99.9% sensitivity. However, this high sensitivity causes it to have only 80% specificity (triggering for burnt toast or steam)."
If 95% of alarms are false positives, why might engineers STILL choose this design over one with 99% specificity but only 80% sensitivity?
Advanced Workshop: Deriving from Counts
Test Outcome Disease (+) Disease (-) Positive Test 190 1,000 Negative Test 10 8,800
Calculate the Sensitivity of this test:
Calculate the Specificity of this test:
Formula Teacher Guide Teacher Guide: Formulaic Precision
Lesson 04: Sensitivity, Specificity & Formal Models
Instructional Objective
This lesson transitions students from intuitive models (trees/tables) to the formal algebraic definition. The goal is for students to recognize that the Sensitivity is just a specific type of conditional probability: \( P(\text{Pos} | \text{Disease}) \).
Mammogram Study Solutions
A) \( P(C \cap \text{Pos}) \)
\( P(C) \times P(\text{Pos}|C) = 0.01 \times 0.85 = \mathbf{0.0085} \)
B) \( P(\text{Pos}) \)
\( (C \cap \text{Pos}) + (H \cap \text{Pos}) = 0.0085 + (0.99 \times 0.10) = 0.0085 + 0.099 = \mathbf{0.1075} \)
C) \( P(C|\text{Pos}) \)
\( 0.0085 / 0.1075 \approx \mathbf{0.079} \) or 7.9%
Engineering Tradeoffs (Smoke Detector)
"In life-safety systems (fire alarms, emergency brakes), the cost of a False Negative (a fire with no alarm) is death. The cost of a False Positive (burnt toast alarm) is just an annoyance. Therefore, we deliberately design for maximum Sensitivity at the expense of Specificity."
Table Derivation Key
Sensitivity:
190 / 200 = 95%
Specificity:
8,800 / 9,800 ≈ 89.8%
Crisis Command Slides Crisis Command
Synthesizing Probability for Public Safety
Final Assessment // LESSON 05
Mission Briefing
You are the Risk Assessment Team for the Department of Public Safety.
A shipment of potentially contaminated baby formula has been distributed nationwide. You have 20 minutes to analyze the probability data and decide whether to issue a Level 1 Nationwide Recall.
The Stakes:
"A false negative costs lives. A false positive costs $400 million and causes a massive supply shortage."
Your Command Report
Your team's 3-minute briefing must include:
The Math (PPV Calculation)
The Visual (Tree or Table)
The Recommendation
"Justify your risk."
Why did you prioritize specificity or sensitivity in this specific case?
Raw Intelligence
Prevalence
0.2%
Estimated percent of batches contaminated.
Test Sensitivity
99.5%
How many bad batches the test catches.
Test Specificity
92.0%
How many safe batches are correctly ID'd.
Crisis Command: LIVE
The press is waiting. The families are waiting. Use the math to make the call.
RECALL
MONITOR
Risk Dossier Document Incident Dossier: C-909
Status: Priority Assessment // Risk Division
Confidential Briefing
"The 'BrightGrow' infant formula line is suspected of salmonella contamination. 250,000 units have reached store shelves. Initial reports from the factory suggest a 0.2% probability of contamination per unit. Our rapid test is fast but carries risks: 99.5% Sensitivity, 92.0% Specificity."
The Danger:
Failure to recall contaminated formula results in severe illness/death (Potential Liability: \$5B+).
The Economic Impact:
Recall costs \$400M and creates a national formula shortage lasting 4 months.
Part 1: The Probability Model
Construct a two-way frequency table or tree diagram for a hypothetical sample of 100,000 units based on the dossier stats.
Modeling Space
Part 2: Quantitative Results
True Positives
False Positives
Calculated PPV
Part 3: Final Command Recommendation
Based on your calculated PPV, what percentage of the units being pulled from shelves in a recall are actually contaminated?
The Executive Decision:
Will you recommend the recall? Justify your decision by weighing the risk of a False Negative against the high rate of False Positives. Mention how your choice aligns with the mission of public safety.
Crisis Command Teacher Guide Facilitator's Rubric: Crisis Command
Lesson 05: Final Risk Assessment Project
Calculation Solution Key (Sample: 100,000 units)
1. Initial Split (0.2% prevalence)
Contaminated: 200
Safe: 99,800
2. Testing Results
True Positives: 200 × 0.995 = 199
False Positives: 99,800 × 0.08 = 7,984
Total Positives: 8,183
Final PPV Calculation:
199 / 8,183 ≈ 0.0243 (2.43%)
Only 2.4% of units identified as "bad" are actually contaminated. Over 97% of recalled formula would be safe.
Assessment Rubric
Criteria Proficient (4) Developing (2) Mathematical Modeling Accurate tree/table with correct application of sensitivity/specificity. Math errors in splitting population or confusing specificity for FPR. Conditional Reasoning Correctly calculates PPV and interprets it as the risk of "the recall being wrong." Cannot differentiate between P(Pos Ethical Justification Explicitly weighs the "cost of failure" (lives) against "economic cost" using math. Recommendation is purely emotional or lacks data-backed reasoning.
Facilitation Notes
The Moral Dilemma: Most students will instinctively say "Recall!" because of the babies. Push them on the "formula shortage" aspect. If 97% of the formula is safe, and we recall everything, will more babies starve from the shortage than get sick from the salmonella?
Peer Review: Have teams swap dossiers and check each other's PPV math before the presentations.
