Probability

Addition Rule and Multiplication Rule

This section introduces the fundamental rules for calculating probabilities involving compound events. Understanding these rules is essential for analyzing situations where multiple events may occur together or separately.

Compound Events

• Compound Event: Any event that combines two or more simple events.

Addition Rule

• Notation: P(A or B) is the probability that event A occurs, event B occurs, or both occur in a single trial.

• Intuitive Addition Rule: Add the number of ways event A can occur and the number of ways event B can occur, ensuring each outcome is counted only once. Divide this sum by the total number of outcomes in the sample space.

• Formal Addition Rule:

• Disjoint (Mutually Exclusive) Events: Events A and B are disjoint if they cannot occur at the same time (i.e., they do not overlap).

Example (Disjoint Events): Selecting a person who is male (A) and selecting a person who is female (B) are disjoint events, as one person cannot be both.

Example (Not Disjoint): Selecting a person taking a statistics course (A) and selecting a person who is female (B) are not disjoint, as a person can satisfy both conditions.

Summary of the Addition Rule

1. Associate the word "or" with addition.

2. Add the number of ways A and B can occur, avoiding double counting.

Complementary Events and the Addition Rule

• Complementary Events: The complement of event A (denoted as not A or \( \overline{A} \)) is the event that A does not occur.

• By definition, either A occurs or it does not, so:

Because A and \( \overline{A} \) are disjoint, the addition rule gives:

Therefore:

Example: If the probability that a randomly selected household has a smartphone is 0.87, then the probability it does not is .

Multiplication Rule

• Notation: P(A and B) is the probability that event A occurs in a first trial and event B occurs in a second trial.

• P(B | A): The probability that event B occurs given that event A has already occurred.

• Intuitive Multiplication Rule: Multiply the probability of event A by the probability of event B, assuming A has already occurred.

• Formal Multiplication Rule:

Independence and Dependence

• Independent Events: Two events A and B are independent if the occurrence of one does not affect the probability of the other.

• Dependent Events: If the occurrence of one event affects the probability of the other, the events are dependent.

Examples: Multiplication Rule in Practice

• Drug Screening Example (With Replacement): If 2 subjects are selected with replacement from 50 (45 positive, 5 negative), the probability the first is positive and the second is negative is:

• Drug Screening Example (Without Replacement): The probability changes because the first selection affects the second:

Sampling Methods and Independence

• Sampling with Replacement: Selections are independent events.

• Sampling without Replacement: Selections are dependent events.

5% Guideline for Cumbersome Calculations

• When sampling without replacement, if the sample size is no more than 5% of the population, treat selections as independent for simplicity.

Example: If 3 employees are selected without replacement from 130,639,273, and the probability of a positive drug test is 0.042, then:

Redundancy: Application of the Multiplication Rule

Redundancy increases system reliability by including backup components. For example, using two independent hard drives reduces the risk of total failure.

• Example: If a hard drive has a 2.89% failure rate, the probability it works for a year is .

• With two independent drives, the probability both fail is , so the probability at least one works is .

Summary Table: Addition and Multiplication Rules

Additional info: The above notes expand on the provided slides by including definitions, formulas, and practical examples for clarity and completeness.