Probability: Addition Rules and Complements

Probability: Addition Rules and Complements

Disjoint (Mutually Exclusive) Events

In probability theory, understanding the relationship between events is crucial for calculating probabilities accurately. Two events are disjoint (or mutually exclusive) if they have no outcomes in common.

• Definition: Disjoint events cannot occur simultaneously. If event E and event F are disjoint, then E ∩ F = ∅ (the empty set).

• Venn Diagrams: These are graphical tools where events are represented as circles within a rectangle (the sample space). Disjoint events are shown as non-overlapping circles.

Example

Suppose we randomly select a chip from a bag labeled 0–9. Let E be "choose a number ≤ 2" and F be "choose a number ≥ 8". These events are disjoint.

• P(E): $P(E) = \frac{N(E)}{N(S)} = \frac{3}{10} = 0.3$

• P(F): $P(F) = \frac{2}{10} = 0.2$

• P(E \text{ or } F): $P(E \text{ or } F) = \frac{5}{10} = 0.5$

• Since E and F are disjoint: $P(E \text{ or } F) = P(E) + P(F) = 0.3 + 0.2 = 0.5$

Addition Rule for Disjoint Events

The Addition Rule for disjoint events allows us to find the probability that at least one of several mutually exclusive events occurs.

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

• Generalization: For pairwise disjoint events $E, F, G, \ldots$, $P(E \text{ or } F \text{ or } G \text{ or } \ldots) = P(E) + P(F) + P(G) + \ldots$

Benford’s Law

Benford’s Law describes the frequency distribution of leading digits in many real-life sets of numerical data. The first digit is not uniformly distributed; lower digits occur more frequently.

• Historical Context: Discovered by Simon Newcomb (1881) and Frank Benford (1938).

• Application: Used in fraud detection and data analysis.

Digit	Probability
1	0.30
2	0.18
3	0.12
4	0.10
5	0.08
6	0.07
7	0.06
8	0.05
9	0.04

Example Questions

1. Probability that X is 1 or 2: $P(1 \text{ or } 2) = 0.30 + 0.18 = 0.48$

2. Probability that X is even: $P(2) + P(4) + P(6) + P(8) = 0.18 + 0.10 + 0.07 + 0.05 = 0.40$

3. Probability that X is either 1 or even: $P(1) + P(2) + P(4) + P(6) + P(8) = 0.30 + 0.18 + 0.10 + 0.07 + 0.05 = 0.70$

Probabilities for Non-Disjoint Events

When events are not disjoint, they may overlap, and the simple addition rule does not apply because overlapping outcomes are counted twice.

• General Addition Rule: $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$

• Double-Counting: Subtract $P(A \text{ and } B)$ to correct for outcomes included in both events.

Example: Cards

Let A be "card is a diamond" and B be "card is a face card" in a standard deck.

• $P(A) = \frac{13}{52}$

• $P(B) = \frac{12}{52}$

• $P(A \text{ and } B) = \frac{3}{52}$ (three face cards are diamonds)

• $P(A \text{ or } B) = \frac{13}{52} + \frac{12}{52} - \frac{3}{52} = \frac{22}{52} = \frac{11}{26}$

Compound Events: Unions and Intersections

Unions

The union of events A and B, denoted A ∪ B, is the event that at least one of A or B occurs.

• Notation: $A \cup B$

• Interpretation: All sample points in A, B, or both.

Intersections

The intersection of events A and B, denoted A ∩ B, is the event that both A and B occur.

• Notation: $A \cap B$

• Interpretation: All sample points common to both A and B.

Mutually Exclusive Events

If A and B are mutually exclusive, $A \cap B = \emptyset$ (no overlap).

• Impossible Event: The intersection is the empty set, meaning both cannot occur together.

General Addition Rule

For any events A and B (not necessarily disjoint):

• Formula: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$

• Proof: $P(A \cup B) = \frac{n(A \cup B)}{n(S)} = \frac{n(A) + n(B) - n(A \cap B)}{n(S)} = P(A) + P(B) - P(A \cap B)$

Example: Cards

Suppose a single card is selected from a standard 52-card deck. Find the probability of drawing a king or a diamond.

• $P(\text{king}) = \frac{4}{52}$

• $P(\text{diamond}) = \frac{13}{52}$

• $P(\text{king of diamonds}) = \frac{1}{52}$

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

Example: Dice

Suppose a pair of dice are thrown. Let E be "first die is a two" and F be "sum of dice ≤ 5". Find $P(E \text{ or } F)$.

• $P(E) = \frac{6}{36}$

• $P(F) = \frac{10}{36}$

• $P(E \text{ and } F) = \frac{3}{36}$

• $P(E \text{ or } F) = \frac{6}{36} + \frac{10}{36} - \frac{3}{36} = \frac{13}{36}$

Complement of an Event

The complement of an event E, denoted Ec, consists of all outcomes in the sample space S that are not in E.

• Notation: $E^c$

• Interpretation: "Not E"

Complement Rule

The probability of the complement of E is given by:

• Formula: $P(E^c) = 1 - P(E)$

Example

According to a study, 52% of Americans have played state lotteries. What is the probability that a randomly selected American has not played?

• $P(\text{not played}) = 1 - 0.52 = 0.48$

Summary Table: Addition and Complement Rules

Rule	Formula	When to Use
Addition Rule (Disjoint)	$P(E \text{ or } F) = P(E) + P(F)$	Events are mutually exclusive
General Addition Rule	$P(A \cup B) = P(A) + P(B) - P(A \cap B)$	Events may overlap
Complement Rule	$P(E^c) = 1 - P(E)$	Probability of "not E"

Additional info: Venn diagrams are frequently used to visualize unions, intersections, and complements in probability. Benford’s Law is a real-world application of probability distributions and is used in statistical auditing and fraud detection.