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:
    • $P(\text{no major flaw}) = 1 - P(\text{major flaw}) = 1 - 0.11 = 0.89$