Lecture on the Multiplication Rule

Introduction to the Multiplication Rule

• Multiplication Rule: A fundamental rule used to calculate joint probabilities.
• Joint Probability: Probability of two or more events occurring in the same experiment.
  • Example: Probability of drawing a card that is both a jack and a spade.
  • Joint probability involves an "AND" condition (events happening together).
  • Often represented with a logical AND sign or intersection sign in set theory.

Calculating Joint Probabilities

• Basic Principle: Multiply the probabilities of individual events to find the joint probability.
• Example Problem: Flipping a coin three times and calculating the probability of getting three heads.
  • Probability of head on one flip: 1/2.
  • For three flips (three heads): (1/2) * (1/2) * (1/2) = 1/8.
  • Important to break down events:
    • Event A: Head on first flip.
    • Event B: Head on second flip.
    • Event C: Head on third flip.

Formal Definition of the Multiplication Rule

• Formula: Probability of A and B = Probability of A * Probability of B given A.
  • Introduces the concept of conditional probability.
• Example: Selecting a person over six feet tall and also taller than five feet.
  • Probability of being over six feet might be 10%.
  • Given over six feet, probability of being over five feet is 100%.*

Properties of the Multiplication Rule

• Order and Time: Order doesn’t matter; Probability of A and B is the same as B and A.
• General Rule: Always true for calculating joint probabilities.

Special Case: Independent Events

• Independence: Events A and B are independent if the probability of B given A equals the probability of B.
• Simplified Rule for Independent Events:
  • Probability of A and B = Probability of A * Probability of B.
  • Example: Coin flips are independent events.*

Applications and Practice Problems

• Example of dealing cards without replacement:
  • Event A: First person gets a diamond.
  • Event B: Second person gets a diamond.
  • Event C: Third person gets a diamond.
  • Event D: Third person gets a king.
• Problem 1: Probability that all cards are diamonds.
• Problem 2: Probability the first two cards are diamonds and the third is a king.
  • Leave answers in fraction form without simplifying.

Conclusion

• Emphasized understanding and applying the multiplication rule in various contexts.
• Encouraged students to apply the rule to practice problems.
• Advised on maintaining answers in their unsimplified form as required.