AP Statistics – Unit 4 Lesson 2: Basic Probability Rules

Topic 4.2 — Complement Rule & Addition Rule

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1. Warm-Up Questions

1. If the chance of rain today is 30%, what is the chance it will not rain? Explain your thinking.

2. A bag contains red, yellow, and green candies. If the chance of drawing a red candy is 0.4, what does that mean in real life?

These questions preview complement thinking and the interpretation of probabilities in context.

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2. Core Vocabulary & Concepts 

Event

An event is something that can happen in a random process.
Example: “Rolling a 4 on a die” is an event.

Probability of an event P(A)

A number from 0 to 1 describing how likely A is to occur.

Complement of an event (not A)

All outcomes where event A does not happen.

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3. The Complement Rule (UNC-2.B.1)

Rule:

P(not A)=1−P(A)

Why this works:

All possible outcomes add up to 1 (or 100%).
So whatever is not part of event A must make up the rest.

Example:

Probability it snows tomorrow:

(snow)=0.25

Then the probability it does not snow:

P(not snow)=1−0.25=0.75

Real-world meaning:

There’s a 75% chance the weather will be something other than snow.

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4. Mutually Exclusive Events + The Addition Rule (UNC-2.B.2)

 

Mutually Exclusive Events

Two events that cannot happen at the same time.

Examples:

• Rolling a die: “rolling a 2” and “rolling a 5”

• Drawing a card: “ace of hearts” and “ace of clubs”

• Choosing a school lunch: “pizza” or “chicken rice” (you pick only one)

Addition Rule for Mutually Exclusive Events:

P(A or B)=P(A)+P(B)

Example:

A die is rolled.

• P(rolling a 1) = 1/6

• P(rolling a 6) = 1/6

They can’t happen together → mutually exclusive.

P(A or B)=P(A)+P(B)

🎲 Example: Rolling a die

• P(rolling a 1)=61​

• P(rolling a 6)=61​

These cannot happen together, so they are mutually exclusive.

P(1 or 6)=1/6+1/6​
=2/6​
=1/3​​

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5. Visual Tools: Tables & Venn Diagrams (Skill 3.A).

Example: Candy Bag Table

Color	Probability
Red	0.4
Yellow	0.3
Green	0.3

• P(not red) = 1 – 0.4 = 0.6

• P(red or yellow) = 0.4 + 0.3 = 0.7 (mutually exclusive)

Venn Example (Non-overlapping Sets)

If A = “students in drama club” and B = “students in chess club” and the school rules say a student cannot join both, then A and B do not overlap → mutually exclusive.

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6. Main Activity – “Weather Center: Build the Forecast”

Scenario:

You are the statistician for a city weather center. You will calculate probabilities for events based on a weather model.

Weather Type	Probability
Sunny	0.55
Rainy	0.30
Snowy	0.10
Storm	0.05

Activity Tasks:

1. Calculate:
   a. P(not sunny)
   b. P(rainy or storm)
   c. P(not snow)
   d. P(sunny or snowy)
   e. Are “Rainy” and “Storm” mutually exclusive? Explain.

2. Interpret each result in context, e.g.:
   “P(not sunny) = ___ means there is a ___ chance the weather will be anything except sunny.”

3. Create your own two new weather events:
   Example:

• Event A = “precipitation” (rain, snow, storm)

• Event B = “calm weather” (sunny)

Then calculate P(A) and P(B) showing the addition rule.

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7. Real-World Discussion (Skill 4.B)

Connect probability rules to real examples:

• “If a doctor says your chance of side effects is 8%, what is the chance of no side effects?”

• “If two sports outcomes cannot happen together (win vs. lose), how do we find P(win or lose)?”

• “Why do weather forecasts always add up to 100%?”

Briefly explain in writing.

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8. Discussion Question

Is it possible for two events to both happen at the same time and still be mutually exclusive? Explain why or why not.

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Homework Exercise Set

📝 Exercise 1 – Complement Rule Practice

Use the complement rule:

P(not A)=1−P(A)

1. A website has a 0.82 probability of loading successfully.
   Find P(failure).

2. A basketball player makes free throws with probability 0.64.
   Find the probability the player misses.

3. A certain type of light bulb has a 12% chance of burning out in the first month.
   Find the probability the bulb does not burn out.

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📝 Exercise 2 – Addition Rule for Mutually Exclusive Events

Use:

P(A or B)=P(A)+P(B)

4. A drawer contains:

   • 30% pens

   • 25% pencils

   • 45% markers
     If you only take one item, what is P(pen or pencil)?

5. At a restaurant, 20% order sushi, 15% order ramen, and 65% order rice bowls.
   Find P(sushi or ramen).

6. A random Spotify playlist has:

   • 40% pop

   • 10% country

   • 50% rap
     What is the probability the next song is pop or rap?

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📝 Exercise 3 – Reading a Probability Table

A survey asked students about their favorite pet.