Rules of Probability: Complement and Addition Rules

Rules of Probability

Complement Rule

The complement rule is a fundamental concept in probability, allowing us to determine the probability of an event not occurring by knowing the probability that it does occur. The complement of an event A, denoted as "not A" or A', consists of all outcomes in the sample space that are not in A.

• Definition: The probability of the complement of event A is given by:

$P(\text{not } A) = 1 - P(A)$

• Alternative Form: If you know the probability of "not A" and want to find the probability of A:

$P(A) = 1 - P(\text{not } A)$

• Application: This rule is especially useful when it is easier to calculate the probability of the complement rather than the event itself.

• Example: If the probability of rolling a 5 on a fair die is 0.1667, then the probability of not rolling a 5 is $1 - 0.1667 = 0.8333$.

• Example: If the probability of rain tomorrow is 20% (0.20), then the probability of no rain is $1 - 0.20 = 0.80$ or 80%.

Addition Rule for Probability

The addition rule helps us find the probability that at least one of several events occurs. The rule varies depending on whether the events are mutually exclusive (cannot happen at the same time) or not.

Special Addition Rule (Mutually Exclusive Events)

• Definition: Two events are mutually exclusive if they cannot occur simultaneously (i.e., they have no outcomes in common).

• Rule: For mutually exclusive events A and B:

$P(A \text{ or } B) = P(A) + P(B)$

• Example: In a spinner game with digits 0–9, the probability of winning by landing on 0 or a number greater than 7 (i.e., 8 or 9): $P(\text{win}) = P(0) + P(8) + P(9) = 1/10 + 1/10 + 1/10 = 0.3$

• Example: If 23% of flowers are white and 12% are yellow, the probability that a randomly picked flower is white or yellow is $0.23 + 0.12 = 0.35$.

General Addition Rule (Non-Mutually Exclusive Events)

• Definition: If two events are not mutually exclusive (i.e., they can both occur), simply adding their probabilities will overcount the outcomes they share.

• Rule: For any two events A and B:

$P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$

• Example: In the spinner game, the probability of landing on an odd number (1, 3, 5, 7, 9) or a number greater than 7 (8, 9): - Odd numbers: 5/10 - Numbers >7: 2/10 - Both (number 9): 1/10 $P(\text{odd or >7}) = 0.5 + 0.2 - 0.1 = 0.6$

• Note: The file states that the general addition rule is not emphasized in this course, but it is important to understand why the special rule only applies to mutually exclusive events.

Mutually Exclusive Events

• Definition: Events are mutually exclusive if:

  • They do not have any outcomes in common.

  • They do not overlap.

  • They cannot happen simultaneously.

• Example: "Spent the evening at home watching TV" and "Went out clubbing" are mutually exclusive events for the same evening.

• More than Two Events: For more than two events to be mutually exclusive, no two can happen at the same time. If any pair can occur together, the set is not mutually exclusive.

• Example: In a spinner game, events H (land on 3), G (land on a number >7), and E (land on an even number) are not mutually exclusive because G and E can both occur (e.g., landing on 8).

Special Addition Rule for More Than Two Mutually Exclusive Events

• Rule: For mutually exclusive events A, B, and C:

$P(A \text{ or } B \text{ or } C) = P(A) + P(B) + P(C)$

Application: Relative Frequency Distributions as Probabilities

Relative frequency distributions can be interpreted as probabilities when each outcome is equally likely and the experiment is random. Each relative frequency represents the probability of selecting a household with a given number of people.

Number in Household	Relative Frequency (Probability)
1	0.083
2	0.140
3	0.328
4	0.353
5	0.092
6	0.003
7 or more	0.001

• Example: The probability that a randomly selected household has 3 people is 0.328.

• Example: The probability that a household has 1 or 2 people is $0.083 + 0.140 = 0.223$.

• Example: The probability that a household has less than 4 people is $0.083 + 0.140 + 0.328 = 0.551$.

Summary Table: When to Use the Addition Rule

Situation	Rule	Formula
Events are mutually exclusive	Special Addition Rule	$P(A \text{ or } B) = P(A) + P(B)$
Events are not mutually exclusive	General Addition Rule	$P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$

Key Takeaways:

• The complement rule is a quick way to find the probability of an event not occurring.

• The addition rule allows you to find the probability of at least one of several events occurring, but you must check if the events are mutually exclusive.

• Relative frequencies from data tables can be interpreted as probabilities for random selection.