3.2 Conditional Probabillity and the Multiplication Rule

Learning Objectives

  • Probability of an Event Given Another Event

  • Distinguishing Between Independent and Dependent Events

  • Using the Multiplication Rule for Probability

Understanding Conditional Probability

  • Definition: The probability of an event occurring given that another event has already occurred.

  • Notation: Denoted as P(B|A), read as "the probability of B given A."

    • Example: Survey of 970 adults on whether they have ridden in a self-driving vehicle.

      • Age breakdown:

        • Ages 18-64: 202 yes, 549 no.

        • Ages 65 and older: 23 yes, 196 no.

      • Conditional Probability Calculation:

        • Probability of an adult being 65 or older given that they have not ridden in a self-driving vehicle:

        • P(65+|No) = 196 / (196 + 549) = 196 / 745 = 0.2631.

Independent vs. Dependent Events

  • Independent Events: The occurrence of one event does not impact the occurrence of the other.

    • Condition: P(B|A) = P(B) or P(A|B) = P(A).

    • Example 1: Smoking cigarettes (A) and developing emphysema (B) - Dependent.

    • Example 2: Tossing a coin (head - A) and then tossing again (tail - B) - Independent.

Multiplication Rule of Probability

  • Definition: For two events A and B occurring in sequence, P(A and B) = P(A) x P(B|A).

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

  • Examples:

    1. Successful Salmon Swimming Probability:

    • Probability of success P(A) = 0.85.

    • For two salmons: P(A and B) = 0.85 x 0.85 = 0.7225 (72.25%).

    1. Card Drawing without Replacement:

    • Probability both cards are hearts:

      • P(first heart) = 13/52.

      • P(second heart given first heart) = 12/51.

      • P(A and B) = (13/52) x (12/51) = 0.0588.

    1. Rotator Cuff Surgeries:

    • Probability all three successful (assuming independence): P(success) = 0.9.

      • P(success for all three) = 0.9 x 0.9 x 0.9 = 0.729.

    • P(none successful) = 0.1 x 0.1 x 0.1 = 0.001.

    • P(at least one successful) = 1 - P(none) = 0.999.

Additional Example Problems

  1. Jury Selection Probability:

    • Probability a juror is female = 0.65.

    • Probability female works in health field = 1/4.

    • P(female and in health field) = P(female) x P(health|female) = 0.65 x 0.25 = 0.1625 (16.25%).

  2. Female Not in Health Field:

    • P(female and not in health field) = P(female) x P(not health|female) = 0.65 x 0.75 = 0.4875 (48.75%).

Conclusion

  • Understanding probabilities of given events is crucial for forecasting outcomes in various situations.

  • Practice with independent and dependent events using the multiplication rule will solidify concepts learned.