Probability, Discrete and Continuous Probability Distributions, and the Normal Distribution: Study Guide

Chapter 5: Probability

Law of Large Numbers

The Law of Large Numbers states that as the number of repetitions of a probability experiment increases, the observed proportion of a particular outcome approaches the theoretical probability of that outcome.

• Key Point: The more trials conducted, the closer the experimental probability gets to the actual probability.

• Example: Flipping a fair coin many times; the proportion of heads approaches 0.5 as the number of flips increases.

Probability Models

A probability model is a mathematical description of a random phenomenon, consisting of a sample space and a way of assigning probabilities to events.

• Valid Probability Model: All probabilities are between 0 and 1, and the sum of all probabilities is 1.

• Invalid Probabilities: Negative probabilities or probabilities greater than 1 are not allowed.

• Example: For a die, probabilities assigned to each face must be between 0 and 1 and sum to 1.

Types of Probability

• Empirical Probability: Based on observations from experiments or historical data. $P(E) = \frac{\text{Number of times } E \text{ occurs}}{\text{Total number of trials}}$

• Classical Probability: Based on equally likely outcomes. $P(E) = \frac{N(E)}{N(S)}$ where $N(E)$ is the number of outcomes in event $E$ and $N(S)$ is the total number of outcomes in the sample space.

• Subjective Probability: Based on personal judgment or experience, not formal calculations.

• Example: Rolling a die (classical), observing weather patterns (empirical), estimating the chance of a sports team winning (subjective).

Probability of Events

• Certain Event: Probability is 1.

• Impossible Event: Probability is 0.

• Unusual Event: Probability less than 0.05 (5%).

Disjoint (Mutually Exclusive) Events

Two events are disjoint or mutually exclusive if they cannot occur at the same time (no common outcomes).

• Example: Drawing a card that is both a heart and a club is impossible; these events are disjoint.

Addition Rules for Probability

• Addition Rule for Disjoint Events: $P(A \text{ or } B) = P(A) + P(B)$ if $A$ and $B$ are disjoint.

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

• Example: Probability of drawing a heart or a king from a deck of cards.

Complement Rule

• Complement of Event A: $P(A^c) = 1 - P(A)$

• Example: Probability of not rolling a 6 on a die: $1 - \frac{1}{6} = \frac{5}{6}$

Independence vs. Disjoint Events

• Independent Events: The occurrence of one event does not affect the probability of the other. $P(A \text{ and } B) = P(A) \times P(B)$

• Disjoint Events: Cannot occur together; $P(A \text{ and } B) = 0$

• Example: Flipping a coin and rolling a die are independent; drawing a red and a black card simultaneously is disjoint.

Multiplication Rule for Independent Events

• Formula: $P(A \text{ and } B) = P(A) \times P(B)$ if $A$ and $B$ are independent.

Conditional Probability

• Definition: Probability of event $A$ given that event $B$ has occurred: $P(A|B) = \frac{P(A \text{ and } B)}{P(B)}$

• Reduces Sample Space: Only consider outcomes where $B$ has occurred.

• Example: Probability a randomly chosen student is female, given that the student is left-handed.

Contingency (Two-Way) Tables

Used to organize data for two categorical variables and calculate probabilities for combinations of events.

• Example: Table showing counts of students by gender and major.

"At Least" Probabilities

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

• Example: Probability of getting at least one head in three coin tosses: $1 - P(\text{no heads}) = 1 - (0.5)^3 = 0.875$

Counting Principles

• Multiplication Rule of Counting: If a task consists of a sequence of choices, multiply the number of ways each can be made.

• Permutations: Arrangements where order matters: $P(n, r) = \frac{n!}{(n-r)!}$

• Combinations: Selections where order does not matter: $C(n, r) = \frac{n!}{r!(n-r)!}$

• Arrangements of n Distinct Items: $n!$

• Arrangements of n Non-Distinct Items: $\frac{n!}{n_1! n_2! \ldots n_k!}$ where $n_1, n_2, \ldots, n_k$ are counts of identical items.

• Example: Arranging the letters in "BALLOON" (with repeating letters).

Chapter 6: Discrete Probability Distributions

Random Variables

• Discrete Random Variable: Takes on countable values (e.g., number of heads in coin tosses).

• Continuous Random Variable: Takes on any value in an interval (e.g., height, weight).

Probability Distributions

• Valid Probability Distribution: Each probability is between 0 and 1, and the sum of all probabilities is 1.

• Probability Histogram: Graphical representation of a probability distribution for a discrete random variable.

Mean (Expected Value) of a Discrete Probability Distribution

• Formula: $\mu = E(X) = \sum [x \cdot P(x)]$

• Note: Do not divide by the number of outcomes; multiply each value by its probability and sum.

• Example: Expected value of winnings in a game.

Expected Value Problems

• Set Up: Identify all possible outcomes and their values, multiply each by its probability, and sum to find the expected value.

• Application: Used in games of chance, insurance, and economics to determine long-term averages.

Chapter 7: The Normal Probability Distribution

Probability Density Function (PDF)

• Definition: The probability distribution for a continuous random variable is called a probability density function (PDF).

• Properties: The total area under the curve is 1; the curve never dips below the horizontal axis.

Normal Distribution Properties

• Symmetry: The normal curve is symmetric about the mean $\mu$.

• Inflection Points: Occur at $\mu \pm \sigma$.

• Area: Half the area is to the left of the mean, half to the right.

Empirical Rule (68-95-99.7 Rule)

• Approximate Probabilities: 68% of data within 1 standard deviation, 95% within 2, 99.7% within 3.

• Note: Use only when specifically instructed.

Z-Scores

• Formula: $z = \frac{x - \mu}{\sigma}$

• Interpretation: Number of standard deviations a value $x$ is from the mean.

• Example: If $x = 85$, $\mu = 80$, $\sigma = 5$, then $z = 1$.

Areas Under the Normal Curve

• Area to the Left: Use Table V to find $P(Z < a)$.

• Area to the Right: $P(Z > a) = 1 - P(Z < a)$

• Area Between Two Z-Scores: $P(a < Z < b) = P(Z < b) - P(Z < a)$

• Probability of a Single Value: For continuous variables, $P(X = x) = 0$

Finding Probabilities and Percentiles

• Given X: Convert to z-score, then use the standard normal table to find area/probability.

• Given Area/Percentile: Find the corresponding z-score, then solve for $x$ using $x = \mu + z\sigma$

• Example: Find the probability that a randomly selected person has a height less than 68 inches, given $\mu = 65$, $\sigma = 3$.

Summary Table: Key Probability Rules

Rule	Formula	When to Use
Addition Rule (Disjoint)	$P(A \text{ or } B) = P(A) + P(B)$	Events cannot occur together
Addition Rule (General)	$P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$	Events may overlap
Multiplication Rule (Independent)	$P(A \text{ and } B) = P(A) \times P(B)$	Events do not affect each other
Complement Rule	$P(A^c) = 1 - P(A)$	Probability of event not occurring
Conditional Probability	$P(A|B) = \frac{P(A \text{ and } B)}{P(B)}$	Given another event has occurred
Expected Value	$E(X) = \sum [x \cdot P(x)]$	Mean of a probability distribution
Z-Score	$z = \frac{x - \mu}{\sigma}$	Standardize a value

Additional info: This guide expands on the review points by providing definitions, formulas, and examples for each concept, ensuring a self-contained and comprehensive study resource for exam preparation.