Addition Rules for Probability

Properties of Probability

• Range of Probability:

  • The probability of an event is always a number between 0 and 1.

  • Notation: 0 ≤ P(event) ≤ 1

  • The value of probability can only be 0 (impossible event) or 1 (certain event), or any value in between.

• Sample Space:

  • The total probability of a sample space (all possible outcomes) equals 1.

  • Sum of probabilities for all outcomes = 100%.

• Null Set:

  • The probability of an empty set (null set) is equal to 0.

  • Example: If nothing happens, the probability of any event occurring is 0.

Complement of an Event

• Definition:

  • The complement of an event (E) includes all outcomes not in that event.

  • Example: For even numbers (2, 4, 6), the complement includes odd numbers (1, 3, 5).

• Calculation:

  • P(complement) = 1 - P(event)

  • For even numbers: P(even) = 1 - P(odd)

Addition Rule for Probability

• General Rule:

  • For two events E and F, the probability that either E or F occurs is:

  • P(E or F) = P(E) + P(F) - P(E and F)

• Example Using a Deck of Cards:

  • Events: Drawing a heart (E) or a queen (F).

  • Hearts in a deck: 13

  • Queens in a deck: 4 (including Queen of Hearts)

  • Overlap: Queen of Hearts counted twice.

• Calculation Example:

  • P(Heart) = 13/52

  • P(Queen) = 4/52

  • P(Heart and Queen) = 1/52

  • By applying the addition rule:

    • P(Heart or Queen) = P(Heart) + P(Queen) - P(Heart and Queen)

    • = (13/52) + (4/52) - (1/52)

    • = 16/52

    • = 0.3077 or 30.77%.

Conclusion

• The probability of drawing either a heart or a queen in a standard deck of cards is approximately 30.77%.