Probability: Addition and Multiplication Rules in Elementary Statistics

Probability

Basic Concepts of Probability

Probability is a fundamental concept in statistics, used to quantify the likelihood of events occurring. It forms the basis for statistical inference and decision-making.

• Probability is the measure of how likely an event is to occur, expressed as a number between 0 and 1.

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

Addition Rule

The addition rule is used to find the probability that either event A or event B occurs in a single trial. The word "or" in probability is associated with addition.

• Notation: $P(A \text{ or } B)$ is the probability that event A occurs, event B occurs, or both occur.

• 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 by the total number of outcomes in the sample space.

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

Disjoint (Mutually Exclusive) Events

Events A and B are disjoint (mutually exclusive) if they cannot occur at the same time. For disjoint events, $P(A \text{ and } B) = 0$.

• Example: Selecting a person who is either male or female for a clinical trial. These events are disjoint.

• Non-disjoint Example: Selecting a person who is taking a statistics course and who is female. These events can overlap.

Summary of Addition Rule

• Associate "or" with addition.

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

Complementary Events and the Addition Rule

Complementary events are pairs of events where one event occurs if and only if the other does not. The probability of an event and its complement always sum to 1.

• Notation: $\overline{A}$ denotes the event that A does not occur.

• Rule: $P(A) + P(\overline{A}) = 1$

• Complement Formula: $P(\overline{A}) = 1 - P(A)$

Example: Smartphone Ownership

If the probability of randomly selecting a household with a smartphone is 0.87, then the probability of selecting one without a smartphone is:

• $P(\text{not smartphone}) = 1 - 0.87 = 0.13$

Multiplication Rule

The multiplication rule is used to find the probability that both event A and event B occur. The word "and" in probability is associated with multiplication.

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

• Conditional Probability: $P(B | A)$ is the probability of event B occurring given that event A has already occurred.

• Formal Multiplication Rule: $P(A \text{ and } B) = P(A) \cdot P(B | A)$

Independence and Dependence

• Independent Events: The occurrence of one event does not affect the probability of the other.

• Dependent Events: The occurrence of one event affects the probability of the other.

Example: Drug Screening

Suppose 50 subjects are tested for drugs: 45 positive, 5 negative.

• With Replacement: Probability first is positive and second is negative: $\frac{45}{50} \cdot \frac{5}{50} = 0.0900$

• Without Replacement: Probability first is positive and second is negative: $\frac{45}{50} \cdot \frac{5}{49} = 0.0918$

Sampling Methods

• Sampling with Replacement: Selections are independent events.

• Sampling without Replacement: Selections are dependent events.

• 5% Guideline: If the sample size is no more than 5% of the population, treat selections as independent.

Example: Drug Screening and the 5% Guideline

Three employees are randomly selected from 130,639,273. Probability all test positive:

• $P(\text{all positive}) = (0.042)^3 = 0.0000741$

Redundancy: Application of the Multiplication Rule

Redundancy increases reliability by using multiple components. If one fails, others can compensate.

• Example: Seagate hard drive failure rate is 2.89% ($P(\text{failure}) = 0.0289$).

• Probability at least one of two independent hard drives does not fail: $1 - (0.0289)^2 = 0.999$

• Using two hard drives reduces risk of failure dramatically.

Summary of Addition and Multiplication Rules

• Addition Rule: Use for "or" events, ensuring no double counting.

• Multiplication Rule: Use for "and" events, considering independence or dependence.