3.3 The Adddition Rule

Learning Objectives

  • Determine whether two events are mutually exclusive.

  • Use the addition rule to find the probability of two events.

Mutually Exclusive Events

Definition

  • Two events, A and B, are mutually exclusive if they cannot occur at the same time.

  • Essentially, A and B have no outcomes in common.

Visual Representation

  • Venn Diagrams:

    • Example 1:

      • Event A and Event B represented without any overlap, indicating they are mutually exclusive.

    • Example 2:

      • Event A and Event B with an overlap indicates they are not mutually exclusive.

Example Analysis

  • Example 1:

    • Event 1: Selecting a Jack from a deck of cards.

    • Event 2: Selecting a face card from the same deck.

    • Analysis: A Jack is a face card, so these events are not mutually exclusive.

  • Example 2:

    • Event 1: Selecting a vehicle that is a Ford.

    • Event 2: Selecting a vehicle that is a Toyota.

    • Analysis: A vehicle cannot be both a Ford and a Toyota at the same time, so these events are mutually exclusive.

Addition Rule for Probability

Definition

  • The probability of either event A or event B occurring is denoted as P(A or B).

  • The addition rule states:P(A or B) = P(A) + P(B) - P(A and B)

    • If A and B are mutually exclusive, then P(A and B) = 0, thus: P(A or B) = P(A) + P(B)

Example Problems

  • Example 3:

    • Situation: Roll a die to find the probability of rolling a 6 or an odd number.

    • Analysis: Cannot roll a 6 and an odd number together, so they are mutually exclusive.

      • P(rolling a 6) = 1/6

      • P(rolling an odd number) = 3/6

      • P(6 or odd) = 1/6 + 3/6 = 4/6 = 0.667

  • Example 4:

    • Situation: Select a card from a deck, find the probability of being a face card or a heart.

    • Analysis: Possible to be both (jack, queen, king of hearts), so not mutually exclusive.

      • P(face card) = 12/52

      • P(heart) = 13/52

      • P(face card and heart) = 3/52

      • P(face card or heart) = 12/52 + 13/52 - 3/52 = 22/52 = 0.423

Blood Type Probability Analysis

  • Situation: Blood bank example with donors classified by blood type and Rh factor.

  • Example 5:

    • Find the probability of a donor having type B or type AB blood.

    • Analysis: Type B and Type AB are mutually exclusive.

      • Count: Type B = 45, Type AB = 16

      • P(B or AB) = 45/409 + 16/409 = 61/409 = 0.149

  • Example 6:

    • Determine probability that donor does not have type O or A blood.

    • Analysis: Equivalent to the probability of having type B or AB.

      • Same math as above yields: P(not O or A) = 0.149

  • Example 7:

    • Find probability of type O blood or Rh positive.

    • Analysis: O and Rh positive can occur simultaneously.

      • P(O) = 184/409

      • P(Rh positive) = 344/409

      • P(both O and Rh positive) = 156/409

      • Calculation: P(O or Rh positive) = 184/409 + 344/409 - 156/409 = 372/409 = 0.910

  • Example 8:

    • Determine probability of type A blood or Rh negative.

    • Analysis: P(A) = 164/409, P(Rh negative) = 65/409, and P(both) = 25/409.

      • Calculation: P(A or Rh negative) = 164/409 + 65/409 - 25/409 = 204/409 = 0.499.

Conclusion

  • The lesson detailed methods to identify mutually exclusive events and apply the addition rule for calculating probabilities.

  • Apply this knowledge in practice problems.