Chapter 4: Introduction to Probabilities – Business Statistics Study Notes

Introduction to Probabilities

What is Probability?

Probability is a fundamental concept in statistics that quantifies the likelihood of an event occurring. It is expressed as a numerical value between 0 and 1, where 0 indicates impossibility and 1 indicates certainty.

• Probability measures the chance or likelihood of a specific event.

• If there is no chance of the event occurring, the probability is 0.

• If the event is certain to occur, the probability is 1.

Key Definitions

Experiment

An experiment is the process of measuring or observing an activity for the purpose of collecting data. For example, rolling a single six-sided die is an experiment.

Sample Space

The sample space is the set of all possible outcomes of an experiment. For a single die roll, the sample space is {1, 2, 3, 4, 5, 6}.

Event

An event is one or more outcomes of an experiment. It is a subset of the sample space. For example, rolling an even number with a die is an event.

Simple Event

A simple event is an event with a single outcome in its most basic form, which cannot be further simplified. For example, rolling a five with a die.

Sample Space Examples

Experiment	Sample Space
Flip a coin	{heads, tails}
Answer a multiple-choice question	{a, b, c, d, e}
Inspect a product	{defective, not defective}
Draw a card from a standard deck	{52 cards in the deck}

Methods of Assigning Probability

There are three primary methods for assigning probabilities to events:

• Classical Probability

• Empirical Probability

• Subjective Probability

Classical Probability

Used when the number of possible outcomes is known and each outcome is equally likely.

• Formula: $P(A) = \frac{\text{Number of possible outcomes that constitute Event A}}{\text{Total number of possible outcomes in the sample space}}$

• Example: Rolling a die once. The probability of rolling a five is $P(5) = \frac{1}{6} \approx 0.167$.

Empirical Probability

Based on observations or experiments, not theoretical reasoning.

• Formula: $P(A) = \frac{\text{Frequency in which Event A occurs}}{\text{Total number of observations}}$

• Example: If 102 out of 400 graduates have $5,000-$9,999 in student loans,$P(\$5,000-\$9,999) = \frac{102}{400} = 0.255$.

Subjective Probability

Based on intuition, experience, or judgment when classical and empirical probabilities are not available.

• Example: Estimating the probability that inflation will exceed 4% next year.

Law of Large Numbers

The law of large numbers states that as an experiment is repeated many times, the empirical probability of an event will converge to its theoretical (classical) probability.

• Example: Flipping a coin many times will result in the proportion of heads approaching 0.5.

Basic Properties (Rules) of Probability

• Rule 1: If $P(A) = 1$, event A is certain to occur.

• Rule 2: If $P(A) = 0$, event A will not occur.

• Rule 3: $0 \leq P(A) \leq 1$ for any event A.

• Rule 4: The sum of probabilities for all simple events in the sample space is 1.

• Rule 5 (Complement Rule): The probability of the complement of event A is $P(A') = 1 - P(A)$.

Probability Rules for More Than One Event

Contingency Tables

Contingency tables display the frequency or probability of events classified by two categorical variables. They are useful for calculating joint, marginal, and conditional probabilities.

Intersection of Events (Joint Probability)

The intersection of events A and B (denoted as $A \cap B$ or "A and B") is the event that both A and B occur.

• Formula: $P(A \cap B)$

• Example: Probability that a student is both female and a freshman.

Union of Events (Addition Rule)

The union of events A and B (denoted as $A \cup B$ or "A or B") is the event that at least one of A or B occurs.

• For mutually exclusive events: $P(A \text{ or } B) = P(A) + P(B)$

• For non-mutually exclusive events: $P(A \text{ or } B) = P(A) + P(B) - P(A \cap B)$

• Example: Probability that a student is female or a freshman.

Mutually Exclusive Events

Events are mutually exclusive if they cannot occur at the same time. For such events, $P(A \cap B) = 0$.

• Example: A customer cannot be both satisfied and unsatisfied simultaneously.

Independent and Dependent Events

• Independent events: The occurrence of one event does not affect the probability of the other. $P(A|B) = P(A)$

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

• Events cannot be both independent and mutually exclusive.

Conditional Probability

Conditional probability is the probability of event A occurring given that event B has occurred, denoted as $P(A|B)$.

• Formula: $P(A|B) = \frac{P(A \cap B)}{P(B)}$

• Example: Probability that a student scored 601-800 on the SAT given they completed a prep class.

Multiplication Rule

The multiplication rule is used to find the probability of the intersection of two events.

• For dependent events: $P(A \cap B) = P(A)P(B|A) = P(B)P(A|B)$

• For independent events: $P(A \cap B) = P(A)P(B)$

• For multiple independent events: $P(A_1 \cap A_2 \cap ... \cap A_n) = P(A_1)P(A_2)...P(A_n)$

Counting Principles

Fundamental Counting Principle

If there are $n_1$ ways to do the first event, $n_2$ ways to do the second, ..., $n_k$ ways to do the k-th event, then the total number of possible outcomes is $n_1 \times n_2 \times ... \times n_k$.

• Example: Choosing a model number with one letter (A-D) and two digits (0-9): $4 \times 10 \times 10 = 400$ possible model numbers.

Permutations

Permutations count the number of ways to arrange objects where order matters.

• Formula: $P(n, x) = \frac{n!}{(n-x)!}$

• Example: Number of ways to arrange 3 numbers (1, 2, 3): $3! = 6$

• Excel function: =PERMUT(n, x)

Combinations

Combinations count the number of ways to select objects where order does not matter.

• Formula: $C(n, x) = \frac{n!}{x!(n-x)!}$

• Example: Number of ways to choose 2 letters from A, B, C, D: $C(4, 2) = 6$

• Excel function: =COMBIN(n, x)

Summary Table: Probability Methods

Method	When Used	Formula
Classical	Known, equally likely outcomes	$P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}}$
Empirical	Based on observed data	$P(A) = \frac{\text{Frequency of A}}{\text{Total observations}}$
Subjective	Based on judgment/experience	N/A

Example Applications

• Business: Estimating the probability of a customer defaulting on a loan.

• Quality Control: Calculating the probability that a product is defective.

• Marketing: Determining the likelihood that a campaign will increase sales.