Basic Probability: The Multiplication Rule

Introduction to the Multiplication Rule

  • Assumes familiarity with conditional probability and independent events.

  • Suggests a review of these topics if needed.

Conditional Probability

  • Formula: P(A|B) = P(A ∩ B) / P(B)

    • P(A|B): Probability of A given B is true.

    • Intersection: P(A ∩ B) is the probability both events occur.

    • P(B) must be greater than zero for the formula to be valid.

  • Independence:

    • If A and B are independent, P(A|B) = P(A).

    • The roles of A and B can be switched arbitrarily.

The Multiplication Rule for Two Events

  • Formula:

    • P(A ∩ B) = P(A) * P(B|A)

    • Alternatively: P(A ∩ B) = P(B) * P(A|B).

  • Intuition: Multiply the probability of one event by the conditional probability of the other given the first occurs.

  • Independence case simplifies to:

    • P(A ∩ B) = P(A) * P(B) (only true if A and B are independent).

Example: Probability of Drawing Two Red Balls

  • Problem: An urn contains 3 blue and 5 red balls; draw 2 balls without replacement.

  • Events defined:

    • A = First ball is red,

    • B = Second ball is red.

  • Steps to Find P(A ∩ B):

    1. Calculate P(A): Drawing a red ball first

      • P(A) = 5/8.

    2. Calculate P(B|A): Probability the second ball is red given the first is red

      • P(B|A) = 4/7 (1 red ball removed, leaving 4 red out of 7 total).

    3. Multiply: P(A ∩ B) = (5/8) * (4/7) = 20/56 = 5/14 ≈ 0.357.

Example: Inability to Calculate Intersection

  • Given: P(A) = 0.4, P(B) = 0.3.

  • Issue: Cannot find P(A ∩ B) due to insufficient information.

  • Attempted but not useful:

    • Trying to use P(B|A) and P(A|B) but lacking necessary probabilities.

  • Conclusion: Need extra information (like independence) or union probabilities to solve accurately.

Multiplication Rule for Three or More Events

  • Formula:

    • P(A ∩ B ∩ C) = P(A) * P(B|A) * P(C|A ∩ B).

  • Generalization: Continues to work for any number of events by multiplying the conditional probabilities progressively.

Example: Probability of a Randomly Selected Canadian

  • Given Information:

    • Probability of being male (M) = 0.5

    • Probability of living in Ontario (O) = 0.39

    • Probability of being 65 or older given living in Ontario (E|O) = 0.18

    • Probability of being male given 65+ and living in Ontario (M|E ∩ O) = 0.45.

  • To find the intersection: P(M ∩ O ∩ E).

  • Calculation:

    • Use the provided information in the order:

      • P(M ∩ O ∩ E) = P(O) * P(E|O) * P(M|E ∩ O)= 0.39 * 0.18 * 0.45 ≈ 0.032.

  • Conclusion: Approximately 3% of Canadians are senior males living in Ontario.

Summary

  • Multiplication rule allows for calculating probabilities involving intersections of events based on conditional probabilities.

  • Understanding independence is crucial to simplifying calculations.

  • Useful to recognize when sufficient information is not available to apply these rules.