The Addition Rule and Complements in Probability

Section 5.2: The Addition Rule and Complements

Learning Objectives

• Use the Addition Rule for Disjoint Events

• Use the General Addition Rule

• Compute the Probability of an Event Using the Complement Rule

Addition Rule for Disjoint (Mutually Exclusive) Events

Definition of Disjoint Events

Two events are disjoint (or mutually exclusive) if they have no outcomes in common. If one event occurs, the other cannot occur.

• Example: Drawing a king or a queen from a deck of cards in a single draw—these events cannot happen simultaneously.

Venn Diagram Representation

Events are often represented as circles within a rectangle (the sample space) in a Venn diagram. Disjoint events are shown as non-overlapping circles.

Addition Rule for Disjoint Events

If E and F are disjoint events, the probability that E or F occurs is the sum of their probabilities:

$P(E \text{ or } F) = P(E) + P(F)$

• This rule can be extended to more than two disjoint events:

$P(E \text{ or } F \text{ or } G \text{ or } \ldots) = P(E) + P(F) + P(G) + \ldots$

Example: Probability with Digits

Suppose a bag contains chips labeled 0 through 9. Let E, F, and G be disjoint events representing the selection of specific digits. If $P(E) = 0.3$, $P(F) = 0.2$, and $P(G) = 0.2$, then:

$P(E \text{ or } F \text{ or } G) = 0.3 + 0.2 + 0.2 = 0.7$

Benford's Law Example

Benford's Law describes the frequency of the first digit in many real-life sources of data. The probabilities for the first digit (1 through 9) are not equal, as shown in the table below:

Digit	Probability
1	0.301
2	0.176
3	0.125
4	0.097
5	0.079
6	0.067
7	0.058
8	0.051
9	0.046

• Example: Probability that the first digit is 1 or 2:

$P(1 \text{ or } 2) = P(1) + P(2) = 0.301 + 0.176 = 0.477$

• Example: Probability that the first digit is at least 6:

$P(6 \text{ or } 7 \text{ or } 8 \text{ or } 9) = 0.067 + 0.058 + 0.051 + 0.046 = 0.222$

Example: Deck of Cards

• Probability of drawing a king: $P(\text{king}) = \frac{4}{52} = \frac{1}{13}$

• Probability of drawing a king, queen, or jack: $P(\text{king or queen or jack}) = \frac{4}{52} + \frac{4}{52} + \frac{4}{52} = \frac{12}{52} = \frac{3}{13}$

The General Addition Rule

Definition and Formula

The General Addition Rule applies to any two events, whether or not they are disjoint:

$P(E \text{ or } F) = P(E) + P(F) - P(E \text{ and } F)$

• This formula corrects for double-counting the outcomes that are in both E and F.

Example: Cards That Are Not Disjoint

• Probability of drawing a king or a diamond:

$P(\text{king}) = \frac{4}{52}$ $P(\text{diamond}) = \frac{13}{52}$ $P(\text{king and diamond}) = \frac{1}{52}$ $P(\text{king or diamond}) = \frac{4}{52} + \frac{13}{52} - \frac{1}{52} = \frac{16}{52} = \frac{4}{13}$

Contingency Tables (Two-Way Tables)

A contingency table displays the frequency distribution of variables. It is useful for calculating probabilities involving two categorical variables.

	Males (in millions)	Females (in millions)
Never married	46.0	40.1
Married	63.3	62.0
Widowed	3.3	11.9
Divorced	12.1	16.1
Separated	2.2	3.1

• Probability that a randomly selected resident is male: $P(\text{male}) = \frac{126.9}{260.1} = 0.488$

• Probability that a resident is widowed: $P(\text{widowed}) = \frac{15.2}{260.1} = 0.058$

• Probability that a resident is widowed or divorced (disjoint): $P(\text{widowed or divorced}) = \frac{15.2 + 28.2}{260.1} = 0.167$

• Probability that a resident is male or widowed (not disjoint): $P(\text{male or widowed}) = \frac{126.9 + 15.2 - 3.3}{260.1} = 0.534$

The Complement Rule

Definition of Complement

The complement of an event E, denoted $E^C$, consists of all outcomes in the sample space that are not in E.

Complement Rule Formula

The probability of the complement of E is:

$P(E^C) = 1 - P(E)$

Examples Using the Complement Rule

• Lottery Example: If 52% of Americans have played the lottery, the probability that a randomly selected American has not played is $1 - 0.52 = 0.48$.

• Income Example: Probability that a household earned $200,000 or more in 2017:$P = \frac{8284}{120,063} = 0.069$

• Probability that a household earned less than $200,000: $1 - 0.069 = 0.931$

• Probability that a household earned at least $10,000: $1 - \frac{7804}{120,063} = 0.935$