Lesson 87: Probability Rules (Addition & Multiplication)
Probability rules allow us to calculate the chance of combined events. The two most important are the Addition Rule and the Multiplication Rule.
Core Rules
- Addition Rule (OR):
For any two events A and B,
\( P(A \cup B) = P(A) + P(B) - P(A \cap B) \).
• If A and B are mutually exclusive (cannot both happen), then: \( P(A \cup B) = P(A) + P(B) \). - Multiplication Rule (AND):
For any two events A and B,
\( P(A \cap B) = P(A) \times P(B|A) \).
• If A and B are independent, then: \( P(A \cap B) = P(A) \times P(B) \).
Examples
Example 1 — Addition Rule:
A card is drawn from a standard deck. What is the probability it is a heart or a king? \( P(\text{Heart}) = 13/52 \), \( P(\text{King}) = 4/52 \), but the King of Hearts is counted twice. \( P = 13/52 + 4/52 - 1/52 = 16/52 = 4/13 \).
Example 2 — Multiplication Rule:
A coin is flipped twice. What is the probability of getting two heads? \( P(H \cap H) = (1/2) \times (1/2) = 1/4 \).
Example 3 — Dependent Events:
Two cards are drawn without replacement. What is the probability both are aces? \( P(\text{Ace first}) = 4/52 \). \( P(\text{Ace second | Ace first}) = 3/51 \). \( P = (4/52) \times (3/51) = 12/2652 = 1/221 \).
Practice Problems
- Roll one die. What is the probability of getting a 4 or an even number?
- Draw a single card. What is the probability it is a red card or a face card?
- Flip a coin 3 times. What is the probability of getting all heads?
- Draw 2 cards with replacement. What is the probability both are hearts?
- A jar has 5 red and 7 blue marbles. Two marbles are drawn without replacement. What is the probability both are red?