AP Statistics 4.4 Mutually Exclusive Events Study Notes
AP Statistics 4.4 Mutually Exclusive Events Study Notes- New syllabus
AP Statistics 4.4 Mutually Exclusive Events Study Notes -As per latest AP Statistics Syllabus.
LEARNING OBJECTIVE
- The likelihood of a random event can be quantified.
Key Concepts:
- Mutually Exclusive Events
Mutually Exclusive Events
Mutually Exclusive Events
In probability, some events cannot happen at the same time. These are called mutually exclusive events, also known as disjoint events.
Two events are mutually exclusive if they cannot both occur in the same trial.
These outcomes are mutually exclusive.
- Formally: Events A and B are mutually exclusive if \( P(A \text{ and } B) = 0 \).
- If one event occurs, it rules out the possibility of the other.
- They have no outcomes in common in the sample space.
- Example: Rolling a single die → Event A = “roll an even number,” Event B = “roll a 3.” These are mutually exclusive.
Probability Rule for Mutually Exclusive Events
- If events A and B are mutually exclusive, then \( P(A \text{ or } B) = P(A) + P(B) \).
- No subtraction of overlap is needed, because overlap is zero.
- This rule only applies if events cannot happen together.
Mutually Exclusive vs Independent Events
- Mutually Exclusive: Events cannot occur together, so \( P(A \text{ and } B) = 0 \).
- Independent: The occurrence of one does not affect the probability of the other. They may still occur together.
- Important distinction: Mutually exclusive events are not independent (except in trivial cases).
Comparison Table
| Concept | Definition | Probability Rule | Example |
|---|---|---|---|
| Mutually Exclusive | Events cannot occur together | \( P(A \text{ or } B) = P(A) + P(B) \) | Roll a die: A = “2,” B = “5” |
| Independent | One event does not affect the other | \( P(A \text{ and } B) = P(A) \times P(B) \) | Flip a coin & roll a die |
Example
A single card is drawn from a deck. Let A = “drawing a heart” and B = “drawing a spade.” Are A and B mutually exclusive? Find \( P(A \text{ or } B) \).
▶️ Answer / Explanation
Step 1: A and B cannot happen together (a card cannot be both heart and spade).
Step 2: \( P(A) = \dfrac{13}{52} \), \( P(B) = \dfrac{13}{52} \).
Step 3: \( P(A \text{ or } B) = P(A) + P(B) = \dfrac{13}{52} + \dfrac{13}{52} = \dfrac{26}{52} = \dfrac{1}{2} \).
Conclusion: Yes, A and B are mutually exclusive. Probability = 0.5.
Example
A die is rolled. Let A = “rolling an even number” and B = “rolling a 3.” Are A and B mutually exclusive? Find \( P(A \text{ or } B) \).
▶️ Answer / Explanation
Step 1: Event A = {2,4,6}, Event B = {3}. They have no overlap → mutually exclusive.
Step 2: \( P(A) = \dfrac{3}{6} = \dfrac{1}{2}, \; P(B) = \dfrac{1}{6} \).
Step 3: \( P(A \text{ or } B) = \dfrac{1}{2} + \dfrac{1}{6} = \dfrac{4}{6} = \dfrac{2}{3} \).
Conclusion: Yes, they are mutually exclusive. Probability = 2/3.
Example
A coin is flipped twice. Let A = “first flip is heads” and B = “second flip is heads.” Are A and B mutually exclusive?
▶️ Answer / Explanation
Step 1: A = {HH, HT}, B = {HH, TH}.
Step 2: Intersection = {HH}, so both can occur together.
Step 3: Therefore, A and B are not mutually exclusive.
Conclusion: They can happen together (both flips heads).