AP Statistics 4.6 Independent Events and Unions of Events Study Notes
AP Statistics 4.6 Independent Events and Unions of Events Study Notes- New syllabus
AP Statistics 4.6 Independent Events and Unions of Events Study Notes -As per latest AP Statistics Syllabus.
LEARNING OBJECTIVE
- The likelihood of a random event can be quantified.
Key Concepts:
- Independent Events and Unions of Events
Independent Events and Unions of Events
Independent Events
Two events A and B are independent if knowing one occurs does not change the probability of the other.
- Rule: \( P(A \mid B) = P(A) \) and \( P(B \mid A) = P(B) \).
- Multiplication Rule: \( P(A \text{ and } B) = P(A) \cdot P(B) \).
- Example: Flipping a coin and rolling a die are independent events.
Dependent Events
Two events are dependent if knowing one occurs changes the probability of the other.
- For dependent events: \( P(A \text{ and } B) = P(A) \cdot P(B \mid A) \).
- Example: Drawing cards without replacement makes events dependent.
Union of Events
The union of A and B (written \( A \cup B \)) means “A or B or both.”
- General Addition Rule: \( P(A \cup B) = P(A) + P(B) – P(A \text{ and } B) \).
- If A and B are mutually exclusive: \( P(A \cup B) = P(A) + P(B) \).
Comparison Table
| Concept | Key Idea | Formula | Example |
|---|---|---|---|
| Independent Events | One does not affect the other | \( P(A \text{ and } B) = P(A) \cdot P(B) \) | Coin flip & die roll |
| Dependent Events | One affects the other | \( P(A \text{ and } B) = P(A) \cdot P(B \mid A) \) | Cards without replacement |
| Union of Events | Either A, or B, or both | \( P(A \cup B) = P(A) + P(B) – P(A \text{ and } B) \) | Rolling 2 or even number |
Example
A coin is flipped and a die is rolled. Find \( P(\text{Heads and 4}) \).
▶️ Answer / Explanation
Step 1: Events are independent.
Step 2: \( P(H) = \dfrac{1}{2}, \; P(4) = \dfrac{1}{6} \).
Step 3: \( P(H \text{ and } 4) = \dfrac{1}{2} \cdot \dfrac{1}{6} = \dfrac{1}{12} \).
Example
A card is drawn from a deck. Let A = “drawing a heart” and B = “drawing a face card.” Find \( P(A \cup B) \).
▶️ Answer / Explanation
Step 1: \( P(A) = \dfrac{13}{52}, \; P(B) = \dfrac{12}{52} \).
Step 2: Overlap: 3 face cards are hearts → \( P(A \text{ and } B) = \dfrac{3}{52} \).
Step 3: \( P(A \cup B) = \dfrac{13}{52} + \dfrac{12}{52} – \dfrac{3}{52} = \dfrac{22}{52} = \dfrac{11}{26} \).
Example
A bag contains 3 red and 2 blue balls. Two are drawn without replacement. Find \( P(\text{first red and second blue}) \).
▶️ Answer / Explanation
Step 1: Dependent events.
Step 2: \( P(R_1) = \dfrac{3}{5} \).
Step 3: After removing 1 red, \( P(B_2 \mid R_1) = \dfrac{2}{4} = \dfrac{1}{2} \).
Step 4: Multiply: \( P(R_1 \text{ and } B_2) = \dfrac{3}{5} \cdot \dfrac{1}{2} = \dfrac{3}{10} \).