Statistics and Probability · 35 minutes
Suggested grades 10–12 · Unit 3: Independence and expected value · Lesson 1 of 2
Independence and an Inclusive “Or”
Use a complete sample space to distinguish independent from mutually exclusive events.
Mr. S Teaches · Original starter lesson · HS-STAT-05 · Grade range is planning guidance.Watch, notice, discuss
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Addition rule for probability | Probability and Statistics | Khan Academy
By Khan Academy
Use a Venn diagram to explain inclusive union and prevent double-counting.
Open on YouTube: Addition rule for probability | Probability and Statistics | Khan Academy (opens a new tab) ↗General multiplication rule example: independent events | Probability & combinatorics
By Khan Academy
Compare the product rule for independent events with the lesson’s numerical independence test.
Open on YouTube: General multiplication rule example: independent events | Probability & combinatorics (opens a new tab) ↗How do you test independence, and why must the overlap be subtracted for an inclusive “or”?
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Learning objective
Test independence with a product and calculate an inclusive union probability.
Before you begin
- Simple probability
- Fractions
Gather your materials
- Paper and pencil
- Graph paper
- Calculator for checking results
Language to know
- independent events
- Events whose joint probability equals the product of their probabilities.
- mutually exclusive
- Events that cannot happen together.
- union
- The event that at least one of the named events occurs.
Understand the concept
Consider equally likely labeled cards R1,R2,R3,B1,B2,B3. Let A mean red and B mean number 1. They overlap at R1. Independence is about probability relationships, not about events having nothing to do with one another linguistically. Mutually exclusive positive-probability events cannot be independent because their joint probability is zero.
Teach it step by step
- List the six cards and mark A,B and their overlap.
- Compute P(A)=1/2,P(B)=1/3 and P(A and B)=1/6; test the product criterion.
- Count A or B inclusively and derive the addition rule by subtracting the double-counted overlap.
- Compare red versus blue, which are mutually exclusive but not independent in this draw.
Make the thinking visible
Worked example or model response
P(A)P(B)=1/6 equals P(A and B), so red and number 1 are independent. P(A or B)=1/2+1/3−1/6=2/3. Red and blue have joint probability 0 rather than 1/4.
Guided practice
On a fair die, test independence of “even” and “greater than 4.” Then calculate their union.
Independent practice
1. If P(C)=0.4,P(D)=0.5 and P(C and D)=0.1, are C and D independent? 2. Find P(C or D).
Quick assessment
Why does ordinary “either/or” wording need clarification in probability?
Answer key & assessment criteria
Answers and teaching guidance
- Guided: P(even)=1/2,P(>4)=1/3,joint={6}=1/6, so independent. Union={2,4,5,6}, probability 2/3.
- Independent 1: No; the product is 0.2, not 0.1. 2:0.4+0.5−0.1=0.8.
- Quick check: Mathematical “or” is usually inclusive, allowing both events; everyday wording may be read exclusively.
Look for: Require the learner to explain the interpretation and check, not only state a result. Distinguish an exact value from a rounded approximation and attend to the stated domain.
Make it work for your learner
More support
Use a Venn diagram with one card label per sample-space outcome.
Extension challenge
Find two overlapping but nonindependent events on the same six-card set.
Verified references · Selected contributions
Standards connections
Subject-content standards
These records identify the part of a subject standard practiced here. Course placement is instructional guidance; the source’s category or band remains authoritative.
Contributes to Common Core standard
HSS.CP.A.2
Common Core · High school · Statistics and Probability
Regular high school mathematics standard
Test independence using the product of event probabilities.
Read the official standard ↗This lesson contributes selected skills; broader applications and full standard coverage require further instruction.
Verified 2026-11-04. Selected contribution, not full attainment.
Contributes to Common Core standard
HSS.CP.B.7
Common Core · High school · Statistics and Probability
Regular high school mathematics standard
Use the addition rule for an inclusive union and interpret its overlap.
Read the official standard ↗This lesson contributes selected skills; broader applications and full standard coverage require further instruction.
Verified 2026-11-04. Selected contribution, not full attainment.