AP Statistics – Unit 4 Lesson 4: Multiplication Rule & Independence

Topic 4.4 — “A and B” for Dependent & Independent Events

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1. Warm-Up Questions

Write your answers clearly.

1. If you flip a coin and roll a die, can the coin outcome affect the die outcome? Explain.

2. If Event A becomes more or less likely after Event B happens, what does that tell you about the relationship between A and B?

These questions prepare you for thinking about independence and conditional probability.

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2. Key Concepts & Vocabulary

Event A and Event B

Two things that might occur in a random situation.

Intersection (“A and B”)

Means both A and B happen together.

Conditional Probability: P(B|A)

“How likely is B given that A has already happened?”

Example:

• P(student eats vegetables | student buys lunch from the cafeteria).

This is a key part of the multiplication rule.

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3. Multiplication Rule for “A and B” (UNC-2.C.1)

This rule works for all events—dependent or independent.

General Multiplication Rule:

P(A and B)=P(A)×P(B∣A)

Meaning:

• First: probability that A happens.

• Then: probability that B happens given A already happened.

• Multiply them to get the probability that they both occur.

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Example (Dependent Events): Drawing Cards Without Replacement

A deck has 52 cards.

Event A: first card drawn is a heart.

P(A)=13/52=0.25

Event B: second card is a heart given that the first was a heart.
Now only 12 hearts left out of 51 cards:

«math»«semantics»«mrow»«mi»P«/mi»«mo stretchy=¨false¨»(«/mo»«mi»B«/mi»«mi mathvariant=¨normal¨»§#8739;«/mi»«mi»A«/mi»«mo stretchy=¨false¨»)«/mo»«mo»=«/mo»«mn»12«/mn»«mi mathvariant=¨normal¨»/«/mi»«mn»51«/mn»«/mrow»«annotation encoding=¨application/x-tex¨»P(B|A) = 12/51«/annotation»«/semantics»«/math»P(B∣A)=12/51

Using the rule:

P(A and B)=5213​×5112​=2652156​

These events are not independent because the first heart changes the probability of the second.

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4. Independence (UNC-2.C.2)

Definition:

Two events A and B are independent if:

P(B∣A)=P(B)

This means A happening does not affect B.

Special Independence Rule:

If A and B are independent:

P(A and B)=P(A)×P(B)

No conditional probability needed.

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Example (Independent Events): Rolling Dice

Roll two dice:

• P(rolling a 6 on the first die) = 1/6

• P(rolling a 6 on the second die) = 1/6

Since rolling one die cannot affect the other:

P(6 on both)=61​×61​=361​

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5. Practice: Checking Independence (Skill 3.A)

Example Scenario

A survey of 500 students shows:

	Owns a dog	Does not own a dog	Total
Plays a sport	120	80	200
Does not play sport	100	200	300
Total	220	280	500

Is owning a dog independent of playing sports?

Students must check whether:

P(Dog | Sport)=P(Dog)

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6. Think–Pair–Share Activity: “Rain and Traffic”

Scenario:

A city report states:

• P(rain) = 0.30

• P(heavy traffic) = 0.40

• P(rain and heavy traffic) = 0.20

Tasks

1. Find

   P(heavy traffic | rain).

2. Use the multiplication rule to verify the value of

   P(rain and heavy traffic).

3. Decide if rain and traffic are independent.

4. With your partner, discuss why weather and traffic might realistically be dependent.

5. Share with the class how the context helps you decide.

This activity builds both computation and real-world interpretation skills.

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7. Main Activity – “Genetics Lab: Inheritance Patterns”

(Unique activity aligned to Skill 4.B)

You are analyzing genetic traits. Flower color is determined independently by two different genes.

Gene A:

• P(A = dominant allele) = 0.6

• P(A = recessive allele) = 0.4

Gene B:

• P(B = dominant allele) = 0.7

• P(B = recessive allele) = 0.3

Assume the genes are independent.

Activity Tasks

1. Calculate the probability that a plant receives dominant alleles for both genes:
   P(A-dominant AND B-dominant).

2. Calculate the probability that a plant has at least one recessive allele
   (hint: complement might be easier).

3. Explain why independence is a reasonable assumption in this genetic model.

4. In real biological data, genes can interact (dominance, linkage).
   Explain how interaction might break independence.

5. Create your own question about allele combinations and solve it.

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8. Real-World Interpretation Questions

Answer each in 1–2 sentences.

A. Why does independence matter when predicting outcomes like genetic traits or multiple medical test results?

B. If two events are not independent, what mistake might someone make if they multiply P(A) × P(B) without using conditional probability?

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9. Exit Question

State clearly how you can tell whether two events A and B are independent.