AP Statistics – Unit 4 Lesson 3: Addition Rule for Non-Mutually Exclusive Events

Topic 4.3 — Events That Can Overlap

 

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1. Warm-Up Questions

Write short answers.

1. Can two events happen at the same time? Give an example from real life.

2. Why might simply adding two probabilities sometimes give an answer that is too large?

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2. Key Vocabulary & Concepts

Event A and Event B

Two things that might happen in a random situation.

Overlap (“A and B”)

“A and B” means both events happen at the same time.

Examples of overlaps:

• A student plays soccer and basketball.

• A day has rain and wind.

• A customer buys fries and a drink.

These are NOT mutually exclusive because both can occur together.

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3. The General Addition Rule (UNC-2.B.3)

When events can overlap, we must subtract the overlap to avoid double counting.

Rule:

P(A or B) = P(A) + P(B) - P(A and  B)

Why subtract?

Because the overlap is counted twice when you add P(A) and P(B).
Subtracting removes the extra count.

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4. Example With Explanation

Example Scenario: School Clubs

At a school:

• 40% of students are in Drama (A)

• 30% are in Art (B)

• 12% are in both Drama and Art (A and B)

We want:

P(A or  B)

Solution

P(A or  B) = 0.40 + 0.30 - 0.12 = 0.58

Interpretation

58% of students are in at least one of the two clubs.

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5. Visual Tools: Venn Diagram + Two-Way Table (UNC-2.B.4)

Venn Diagram Example

         _________A_________
        /                   \
       /   A and B (12%)     \
      /                       \
     /___________     _________\
                 \   /
            B (30%) \ /

Two-Way Table Example

	In Art	Not in Art	Total
In Drama	12%	28%	40%
Not in Drama	18%	42%	60%
Total	30%	70%	100%

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6. Think–Pair–Share Activity: “Rain or Wind?”

Students work individually, then discuss with a partner, then share with the class.

Scenario:

A weather model predicts:

• P(rain) = 0.45

• P(wind) = 0.55

• P(rain and wind) = 0.35

These can overlap because weather conditions often occur together.

Tasks:

1. Calculate P(rain or wind).

2. Explain why we subtract the overlap.

3. Write an interpretation in a full sentence.

4. Discuss with a partner:
   “Does a higher probability of wind mean the day will be windy the entire time?”
   Explain what probability does and does not tell us.

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7. Main Activity – “Café Orders: What Do Customers Choose?”

A café tracks customer orders:

• 60% order coffee (Event C)

• 50% order pastries (Event P)

• 25% order both

Use the general addition rule to answer the following.

Activity Tasks

1. Calculate P(coffee or pastries).
Show the formula and every step.

2. Fill in a Venn diagram with the three percentages.

3. Create a two-way table showing all four regions:

• Coffee only

• Pastries only

• Both coffee and pastries

• Neither

4. Interpret P(coffee or pastries) using a clear context sentence.

5. Invent one additional question based on the café data and solve it.

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8. Real-World Interpretation Practice (Skill 4.B)

Answer each in one or two sentences.

A. A patient could have symptom A (fever), symptom B (cough), or both.
If (P(A) = 0.62), (P(B) = 0.55), and (P(A \text{ and } B) = 0.40):

• What does P(A or B) represent in medical terms?

B. A music app recommends songs to you.
Event A = “song is pop,” Event B = “song is upbeat.”
Why might P(pop or upbeat) be much larger than either event alone?

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9. Exit Question

Why can’t we add P(A) + P(B) when the events overlap? Explain in your own words.