Making Sense of Probabilities: Union, Tables, and Conditional Thinking
How can we combine probabilities when events overlap?
What do those totals in a table mean?
Let’s break it all down — and build toward smarter probability thinking.
📚 This post is part of the "Intro to Statistics" series
🔙 Previously:How Random Is Random? Understanding Probability and Events
🔜 Next: Understanding Bayez-Rule<
🔗 Union of Events: The Addition Rule
When calculating the probability of A or B, we combine the probabilities — but subtract any overlap:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
This is called the Addition Rule.
✅ Special Case: Disjoint Events
If A and B are mutually exclusive (disjoint), then \( P(A \cap B) = 0 \), so:
\[ P(A \cup B) = P(A) + P(B) \]
📊 Visual Aid Placeholder
📊 Marginal and Joint Proportions with Tables
Let’s say we ask 100 people whether they own a dog or a cat:
| Owns Dog | No Dog | Total | |
|---|---|---|---|
| Owns Cat | 20 | 30 | 50 |
| No Cat | 25 | 25 | 50 |
| Total | 45 | 55 | 100 |
🧩 What Are Margins?
- The totals in the last row and column are called marginal totals.
- Their proportions (divided by total) are marginal proportions:
- Example: \( P(\text{Owns Dog}) = 45 / 100 = 0.45 \)
🔄 Joint Probability Table
To convert to probabilities, divide each cell by total:
| Owns Dog (D) | No Dog | Total | |
|---|---|---|---|
| Owns Cat (C) | 0.20 | 0.30 | 0.50 |
| No Cat | 0.25 | 0.25 | 0.50 |
| Total | 0.45 | 0.55 | 1.00 |
These are joint probabilities (each cell shows \( P(C \cap D) \), etc).
✅ Total of each row/column = 1
✅ This set of events is jointly exhaustive
💡 You can compute marginal probabilities by summing across rows or columns.
⚠️ But you can’t always reverse this — joint probability can’t be recovered from marginal alone.
📌 Conditional Probability: What If We Know B Happened?
Conditional probability is:
\[ P(A \mid B) = \frac{P(A \cap B)}{P(B)} \]
Read as “the probability of A given B.”
📊 Venn Diagram Placeholder
🧪 Problem 1 — Using the Table
Q: What’s the probability that a person owns a cat given they own a dog?
\[ P(\text{Cat} \mid \text{Dog}) = \frac{P(\text{Cat and Dog})}{P(\text{Dog})} = \frac{0.20}{0.45} \approx 0.444 \]
🧪 Problem 2 — Medical Testing Example
| Disease (+) | No Disease (−) | Total | |
|---|---|---|---|
| Test + | 40 | 10 | 50 |
| Test − | 10 | 40 | 50 |
| Total | 50 | 50 | 100 |
Let’s convert to probabilities:
\[ P(\text{Disease} \mid \text{Test +}) = \frac{40}{50} = 0.8 \]
🧠 The test is positive, and there’s an 80% chance the person actually has the disease.
🧠 Try It Yourself: Union & Conditional Thinking
Q1: If A and B are disjoint, what is \( P(A \cup B) \)?
💡 Show Answer
\( P(A) + P(B) \)
Q2: How do you compute a joint probability in a table?
💡 Show Answer
Divide each cell count by the grand total.
Q3: What is \( P(A \mid B) \)?
💡 Show Answer
\( P(A \cap B) / P(B) \)
📌 Try It Yourself: Union & Conditional Thinking
Q1: If A and B are disjoint, what is P(A ∪ B)?
💡 Show Answer
P(A) + P(B) — because they have no overlap.
Q2: In a contingency table, how do you compute a joint probability?
💡 Show Answer
Divide the count in a specific cell by the grand total.
Q3: What are marginal probabilities?
💡 Show Answer
The row or column totals divided by the overall total.
Q4: What is the equation for conditional probability?
💡 Show Answer
P(A | B) = P(A ∩ B) / P(B)
Q5: Why can’t you always recover joint probabilities from marginal probabilities?
💡 Show Answer
Because marginal probabilities don’t contain information about how the events overlap.
🧠 Summary
| Concept | Meaning |
|---|---|
| Union of A and B | Add both, subtract intersection |
| Disjoint events | No overlap: \( P(A \cup B) = P(A) + P(B) \) |
| Marginal proportion | Total rows/columns ÷ total cases |
| Joint probability | Cell ÷ total |
| Conditional \( P(A \mid B) \) | Probability of A given B |
✅ Up Next
We’ll now explore probability rules in depth, including:
- Complement rules
- Bayes’ Theorem
- Advanced conditional reasoning
See you there!