Addition Rule and Rule of Complements

Compound Events and the General Addition Rule

  • A compound event combines two or more events.
  • The event A or B occurs if A occurs, B occurs, or both occur.
  • The general addition rule computes probabilities of events in the form A or B.

General Addition Rule

  • For any two events A and B, the probability of A or B is:
    P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)

Example

  • 1,000 adults were asked about favoring a law for higher education support and their likelihood to vote.
  • Classified as "likely to vote" or "not likely to vote."
Data:
  • Likely to vote: 372
  • Favor the law: 151
  • Both likely to vote and favor the law: 20
  • Only likely to vote: 262
  • Only Favor the law: 87
Calculations:
  • Probability of likely to vote:
    • Total people likely to vote = 372 + 262 + 87 = 721
    • P(\text{likely to vote}) = \frac{721}{1000} = 0.721
  • Probability of favoring the law:
    • Total people who favor the law = 372 + 151 = 523
    • P(\text{favors the law}) = \frac{523}{1000} = 0.523
  • Probability of both likely to vote and favoring the law:
    • P(\text{likely to vote and favors the law}) = \frac{372}{1000} = 0.372
  • Using the general addition rule:
    • P(\text{likely to vote or favors the law}) = P(\text{likely to vote}) + P(\text{favors the law}) - P(\text{likely to vote and favors the law})
    • P(\text{likely to vote or favors the law}) = 0.721 + 0.523 - 0.372 = 0.872

Mutually Exclusive Events

  • Two events are mutually exclusive if it is impossible for both events to occur simultaneously.

Examples

  • Rolling a die:
    • Event A: Die comes up as 3.
    • Event B: Die comes up as an even number.
    • These are mutually exclusive because the die cannot be both 3 and an even number at the same time.
  • Tossing a coin twice:
    • Event A: One of the tosses is heads.
    • Event B: One of the tosses is tails.
    • These are NOT mutually exclusive because it's possible to get heads, tails or tails, heads.

Addition Rule for Mutually Exclusive Events

  • If events A and B are mutually exclusive, then P(A \text{ and } B) = 0
  • Simplified general addition rule:
    • P(A \text{ or } B) = P(A) + P(B)

Example

  • Olympics example with 10,735 athletes.
    • 530 from the US
    • 277 from Canada
    • 102 from Mexico
  • What is the probability that an athlete chosen at random represents the US or Canada?
Solution:
  • Events are mutually exclusive (cannot compete for both US and Canada).
    • P(\text{US or Canada}) = P(\text{US}) + P(\text{Canada})
    • P(\text{US or Canada}) = \frac{530}{10735} + \frac{277}{10735} = \frac{807}{10735} \approx 0.07517

Complements

  • The complement of an event A is the event that A does not occur, denoted as A^C
  • Example: If there is a 60% chance of rain, there is a 40% chance it will not rain.

Examples: Statistics Class

  • 200 students enrolled.
    • Event: Exactly 50 are business majors.
      • Complement: The number of business majors is not 50.
    • Event: More than 50 are business majors.
      • Complement: 50 or fewer are business majors.
    • Event: At least 50 of them are business majors.
      • Complement: Fewer than 50 are business majors.

Rule of Complements

  • P(A^C) = 1 - P(A)
  • Example: Wall Street Journal reports 40% of cars sold were small cars.
    • Probability that a randomly chosen car is not a small car:
      • P(\text{not a small car}) = 1 - P(\text{small car}) = 1 - 0.4 = 0.6 or 60%.

Application Example: Foundry Manufacturing

  • 500 cast aluminum parts manufactured.
    • Some with major flaws.
    • Some with minor flaws.
    • Some with both.

Table Results:

  • Major flaw only = 35
  • Minor flaw only = 75
  • Both major and minor flaws = 20
  • No flaw = 370

Probability Calculations:

  • A. Probability of a major flaw:
    • Total with major flaw = 20 + 35 = 55
    • P(\text{major flaw}) = \frac{55}{500} = 0.11
  • B. Probability of a minor flaw:
    • Total with minor flaw = 20 + 75 = 95
    • P(\text{minor flaw}) = \frac{95}{500} = 0.19
  • C. Probability of a major or minor flaw:
    • Using the general addition rule:
      • P(\text{major flaw or minor flaw}) = P(\text{major flaw}) + P(\text{minor flaw}) - P(\text{major flaw and minor flaw})
      • P(\text{major flaw and minor flaw}) = \frac{20}{500} = 0.04
      • P(\text{major flaw or minor flaw}) = 0.11 + 0.19 - 0.04 = 0.26
  • D. Probability of no major flaw:
    • Using the complement rule:
      • P(\text{no major flaw}) = 1 - P(\text{major flaw}) = 1 - 0.11 = 0.89