Probability: Addition and Multiplication Rules in Statistics

Chapter 4: Probability

Overview

This chapter introduces the fundamental concepts of probability, focusing on the addition and multiplication rules, the use of complements, and the application of Venn diagrams. These concepts are essential for understanding how to calculate the likelihood of events in statistics.

Basic Concepts of Probability

Definition of Probability

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

• The sample space is the set of all possible outcomes of a random experiment.

• An event is any subset of the sample space.

Addition Rule and Multiplication Rule

Addition Rule

The addition rule is used to find the probability that either of two events occurs in a single trial.

• The word "or" in probability is associated with the addition of probabilities.

• Compound event: Any event combining two or more simple events.

Intuitive Addition Rule

• To find $P(A \text{ or } B)$, add the number of ways event A can occur and the number of ways event B can occur, but avoid double counting outcomes that are common to both events.

• The probability is then the sum divided by the total number of outcomes in the sample space.

Formal Addition Rule

• The formal rule for any two events A and B is:

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

• $P(A \text{ and } B)$ is the probability that both A and B occur at the same time.

Disjoint (Mutually Exclusive) Events

• Events A and B are disjoint (or mutually exclusive) if they cannot occur at the same time.

• For disjoint events, $P(A \text{ and } B) = 0$, so $P(A \text{ or } B) = P(A) + P(B)$.

Examples

• Disjoint events: Selecting someone older than 65 and someone younger than 40 in a clinical trial (the same person cannot be both).

• Not disjoint: Selecting someone taking a statistics course and someone who is a freshman (the same person could satisfy both conditions).

Venn Diagrams

Visualizing Events

• A Venn diagram is a graphical representation of the sample space and events as areas within a rectangle.

• Disjoint (mutually exclusive) events do not overlap in a Venn diagram.

• Events that are not disjoint overlap, representing outcomes common to both events.

Complements and the Addition Rule

Complementary Events

• The complement of event A, denoted as $A'$, is the event that A does not occur.

• The sum of the probabilities of an event and its complement is 1:

$P(A) + P(A') = 1$

• Therefore, $P(A') = 1 - P(A)$

Example: Probability of Not Having a Smartphone

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

Multiplication Rule

Definition

• The multiplication rule is used to find the probability that two events both occur (in sequence).

• The word "and" in probability is associated with the multiplication of probabilities.

Formal Multiplication Rule

$P(A \text{ and } B) = P(A) \times P(B|A)$

• $P(B|A)$ is the probability that event B occurs given that event A has already occurred.

• Alternatively, $P(A \text{ and } B) = P(B) \times P(A|B)$

Independence and Dependence

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

• If not independent, the events are dependent.

Sampling

• Sampling with replacement: Selections are independent events.

• Sampling without replacement: Selections are dependent events.

5% Guideline for Cumbersome Calculations

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

Example: Drug Screening

Test Result	Number of Subjects
Positive	45
Negative	5
Total	50

• Probability (with replacement) that first is positive and second is negative:

$P(\text{positive first}) = \frac{45}{50}$ $P(\text{negative second}) = \frac{5}{50}$ $P(\text{positive first and negative second}) = \frac{45}{50} \times \frac{5}{50} = 0.09$

• Without replacement, the second probability is adjusted:

$P(\text{negative second}) = \frac{5}{49}$ $P(\text{positive first and negative second}) = \frac{45}{50} \times \frac{5}{49} = 0.091$

Probability of "At Least One"

Definition and Calculation

• "At least one" means one or more occurrences of an event.

• The complement is "none" of the event occurring.

• Probability of at least one occurrence:

$P(\text{at least one}) = 1 - P(\text{none})$

Example: Manufacturing Defects

• If the defect rate is 15% and a customer buys 12 products, the probability that at least one is defective is:

$P(\text{all 12 are good}) = 0.85^{12} = 0.142$ $P(\text{at least one defective}) = 1 - 0.142 = 0.858$

Example: Hard Drive Reliability

• Failure rate for one hard drive is 2.89% ($P(\text{failure}) = 0.0289$).

• Probability that at least one of two independent hard drives works:

$P(\text{at least one works}) = 1 - (0.0289)^2 = 0.999$

• Redundancy (using two drives) greatly increases reliability.

Summary Table: Addition and Multiplication Rules

Rule	Key Word	Formula	When to Use
Addition Rule	or	$P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$	Finding probability of either event A or B occurring
Multiplication Rule	and	$P(A \text{ and } B) = P(A) \times P(B|A)$	Finding probability of both events A and B occurring in sequence