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Official Data Lab

Public data sources for examples, charts, and R practice.

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STATLABAcademyStatistics through data and R practice

All STATLAB Academy lessons, quizzes, datasets, and notes are original or properly licensed. Textbooks may be used only as curriculum references.

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STATLAB Academy | Learn Statistics with R and Real-World Data

Lesson 4 of 16

Addition Rules and Event Structure

Home/Modules/Module 5/Addition Rules and Event Structure

Use general and special addition rules while distinguishing exclusivity from independence.

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38 minutes25% path

Learning objectivesBy the end of this lesson, students will be able to:

1Define the events and probability quantities used in addition rules and event structure.
2Select and calculate an appropriate probability rule with a visible denominator or assumption.
3Interpret the result for a business decision without overstating certainty or causation.

Theory

Core probability reasoning

Use general and special addition rules while distinguishing exclusivity from independence.

Explanation 1

Overlap: The intersection counted in both marginal probabilities.

Explanation 2

Mutually exclusive: Events that cannot occur together.

Explanation 3

Collectively exhaustive: Events that cover the whole sample space.

Terminology

Concept map

Overlap

The intersection counted in both marginal probabilities.

A loyalty member with a high-value cart.

Mutually exclusive

Events that cannot occur together.

One order assigned exactly one priority.

Collectively exhaustive

Events that cover the whole sample space.

Low, medium, or high cart band.

Notation and formulas

General addition rule

Key relationship

P(A or B) = P(A) + P(B) - P(A and B)

Define every event before substituting values, then validate the result and denominator.

Step 1Add both marginal probabilities.
Step 2Subtract the overlap counted twice.
Step 3Check that the result lies between 0 and 1.

Interactive Mission

addition rule calculator

P(A)P(B)P(A and B)

P(A or B)

0.7

P(A) + P(B) - P(A and B)

The overlap is subtracted once because it was counted in both marginal probabilities.

Worked example

Review workload

Scenario

A review is triggered by a high cart or an unusual payment route.

Subtracting the overlap avoids double counting.

R connection

R note: Complements and Addition Rules

Run and edit the code, inspect intermediate output, and use the movable R Code Guide while practising.

Live R Lab

Complements and Addition Rules

Calculate complements, unions, and intersections from original checkout events.

Editable R code

Results

Ready to run

Ready to run

Ready to run. Results, errors, tables, plots, and fallback messages will appear here.

Define A and B.

Calculate intersection.

Verify the general addition rule.

Dataset provenance

Original synthetic data generated for STATLAB Academy. No textbook data used.

Interpretation safeguard

Mutually exclusive events are generally not independent when both have positive probability.

Reflection

Pause and write

Where could an unclear event definition or assumption change the conclusion in addition rules and event structure?

Name the event, denominator, probability interpretation, and one operational limitation.

Assistant

Ask your probability coach

Help me reason through Addition Rules and Event Structure by asking questions about my event definitions, denominator, formula choice, assumptions, and interpretation. Do not give a quiz answer before I submit.

Exit check

Lesson checkpoint

Use a hint before submitting if needed. Full calculations and interpretations appear after submission.

Checkpoint

Module 5 Lesson 4 Checkpoint

Question 1 of 2. Answered 0/2. Passing score: 70%.

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Question 1: Numeric answer

mediumCalculate and responsibly interpret probability for a defined event.

P(A)=0.45, P(B)=0.35, and P(A and B)=0.10. Find P(A or B).

Resource

Recommended reading

Explore additional probability explanations and exercises from properly licensed or publisher resources.

Resource

Module 5 learning material

Download an original worksheet or guide for this lesson.

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