AP Statistics 4.5 Conditional Probability Study Notes
AP Statistics Link Study Notes- New syllabus
AP Statistics Link Study Notes -As per latest AP Statistics Syllabus.
LEARNING OBJECTIVE
- The likelihood of a random event can be quantified.
Key Concepts:
- Conditional Probability
Conditional Probability
Conditional Probability
Conditional probability measures the probability of one event occurring given that another event has already occurred.
Definition
- The probability of A given B is written as \( P(A \mid B) \).
- Formula: \( P(A \mid B) = \dfrac{P(A \text{ and } B)}{P(B)} \), provided \( P(B) > 0 \).
- This tells us how likely A is, assuming we already know B happened.
- It narrows the sample space down to event B.
Key Rules
- Multiplication Rule: \( P(A \text{ and } B) = P(A \mid B) \cdot P(B) = P(B \mid A) \cdot P(A) \).
- Independent Events: If A and B are independent, then \( P(A \mid B) = P(A) \) and \( P(B \mid A) = P(B) \).
- General Addition Rule: \( P(A \text{ or } B) = P(A) + P(B) – P(A \text{ and } B) \).
Comparison: Independence vs Dependence
| Concept | Definition | Probability Rule | Example |
|---|---|---|---|
| Independent Events | Occurrence of one does not affect the other | \( P(A \mid B) = P(A) \) | Flip coin & roll die |
| Dependent Events | Occurrence of one changes probability of the other | \( P(A \mid B) \neq P(A) \) | Drawing cards without replacement |
Example
A card is drawn from a standard deck. Let A = “drawing a king” and B = “drawing a face card.” Find \( P(A \mid B) \).
▶️ Answer / Explanation
Step 1: Face cards = 12 total (J, Q, K in each suit).
Step 2: Kings = 4, and all are face cards → overlap = 4.
Step 3: \( P(A \mid B) = \dfrac{P(A \text{ and } B)}{P(B)} = \dfrac{4/52}{12/52} = \dfrac{4}{12} = \dfrac{1}{3} \).
Conclusion: If you know the card is a face card, probability it is a king is 1/3.
Example
A bag contains 5 red balls and 3 blue balls. One ball is drawn without replacement. Find \( P(\text{blue on 2nd draw} \mid \text{red on 1st draw}) \).
▶️ Answer / Explanation
Step 1: After removing 1 red, bag has 4 red and 3 blue (7 total).
Step 2: Probability of blue on 2nd draw = \( \dfrac{3}{7} \).
Step 3: This differs from \( \dfrac{3}{8} \) if draws were independent.
Conclusion: Events are dependent. \( P(B_2 \mid R_1) = 3/7 \).
Example
A fair die is rolled twice. Find \( P(\text{sum = 7} \mid \text{first roll = 3}) \).
▶️ Answer / Explanation
Step 1: First roll is fixed at 3 → need 4 on second roll to make 7.
Step 2: Probability = \( \dfrac{1}{6} \).
Conclusion: \( P(\text{sum=7} \mid \text{first=3}) = 1/6 \).