Making Sense of Probabilities: Union, Tables, and Conditional Thinking

When working with data, it’s critical to understand how probabilities interact — especially when events overlap or depend on one another. In this post, you’ll learn how to calculate unions, read contingency tables, and understand conditional probabilities using intuitive visuals and real-life examples.

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📚 This post is part of the "Intro to Statistics" series

🔙 Previously:How Random Is Random? Understanding Probability and Events

🔜 Next: Understanding Bayez-Rule<

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🔗 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

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📊 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: \( P(\text{Owns Dog}) = 45 / 100 = 0.45 \)

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🔄 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.

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📌 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

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🧪 Problem 1 — Using the Table

Q: What’s the probability that a person owns a cat given they own a dog?

\[ P(\text{Cat} \mid \text{Dog}) = \frac{P(\text{Cat and Dog})}{P(\text{Dog})} = \frac{0.20}{0.45} \approx 0.444 \]

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🧪 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.

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🤖 Why This Matters to Machine Learning

Understanding probability rules is foundational to many ML algorithms:

• Naive Bayes classifiers rely on conditional probability.
• Confusion matrices and precision/recall relate to joint and marginal probabilities.
• Probabilistic reasoning underlies Bayesian networks, Hidden Markov Models, and likelihood estimation.

By mastering unions, intersections, and conditional logic, you’re building the intuition needed for more complex ML reasoning.

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📌 Try It Yourself: Union & Conditional Thinking

Q1: If events A and B are disjoint, what is P(A ∪ B)?

💡 Show Answer

P(A ∪ B) = P(A) + P(B) — because disjoint events can’t happen together, so there’s no overlap to subtract.

Q2: In a contingency table, how do you calculate a joint probability?

💡 Show Answer

Divide the frequency in a single cell by the total sample size.
P(A ∩ B) = count in cell / grand total

Q3: What are marginal probabilities and where do you find them?

💡 Show Answer

They’re totals across rows or columns in a table, divided by the grand total.
They show the overall probability of a single variable.

Q4: What’s the formula for conditional probability?

💡 Show Answer

P(A | B) = P(A ∩ B) / P(B) — it’s the probability of A given B has occurred.

Q5: Why can’t marginal probabilities alone determine joint probabilities?

💡 Show Answer

Because they don’t show how two variables interact — they only describe each variable separately.
You need joint or conditional data to understand overlap.

Q6: You know that 90% of cat owners like coffee. What kind of probability is that?

💡 Show Answer

It’s a conditional probability: the chance of liking coffee given that someone owns a cat.

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🧠 Level Up: Know When to Use Conditional Probabilities

Conditional probability isn’t just for math exams — it powers real-world decisions:

• 🩺 In medical testing, it tells us the chance a patient actually has a condition, given a positive test.
• 📈 In finance, it helps estimate risks based on market behavior.
• 🧠 In machine learning, it's the backbone of models like Naive Bayes and Bayesian networks.

Mastering it helps you move from counting events to thinking conditionally — just like algorithms do.

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✅ Best Practices for Probability Thinking

• Use **Venn diagrams** or tables to clarify overlap and independence.
• Double-check whether events are **mutually exclusive** or **independent** — they’re not the same!
• Normalize tables to probabilities for clearer analysis.
• Label everything clearly: A, B, A ∩ B, A ∪ B, etc.

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⚠️ Common Pitfalls

• ❌ Assuming disjoint events are independent.
• ❌ Forgetting to subtract the intersection in union calculations.
• ❌ Using marginal totals to infer conditional relationships directly.

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🧠 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

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💬 Got a question or suggestion?

Leave a comment below — I’d love to hear your thoughts or help if something was unclear.

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✅ Up Next

We’ll now explore probability rules in depth, including:

• Complement rules
• Bayes’ Theorem
• Advanced conditional reasoning

See you there!