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; as the number of flips increases, the proportion of heads approaches 0.5.

Probability Models

A probability model assigns probabilities to all possible outcomes of a random experiment.

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

• Invalid Probabilities: Any probability less than 0 or greater than 1 is not valid.

• Example: For a die, probabilities assigned to each face must add up to 1.

Types of Probability

• Empirical Probability: Based on observed data (relative frequency).

• Classical Probability: Based on equally likely outcomes. $P(E) = \frac{N(E)}{N(S)}$

• Subjective Probability: Based on personal judgment or experience.

• Example: Rolling a die (classical), observing 3 heads in 10 coin tosses (empirical), estimating the chance of rain (subjective).

Special Probabilities

• Certain Event: Probability = 1

• Impossible Event: Probability = 0

• Unusual Event: Probability less than 0.05

Calculating Probabilities

• Empirical Approach: $P(E) = \frac{\text{frequency of } E}{\text{number of trials}}$

• Classical Approach: $P(E) = \frac{N(E)}{N(S)}$

Disjoint (Mutually Exclusive) Events

• Definition: Two events are disjoint 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

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

• General Addition Rule (Not Disjoint): $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$

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

Contingency (Two-Way) Tables

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

• Example: Calculating the probability of being in a certain row or column, or both.

Complement Rule

• Formula: $P(A^c) = 1 - P(A)$

• Application: Useful for "at least" or "at most" probability questions.

Independence vs. Disjoint Events

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

• Disjoint Events: Cannot occur together; if one happens, the other cannot.

• Example: Flipping two coins (independent); drawing a red or black card (disjoint).

Multiplication Rule for Independent Events

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

"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.

Conditional Probability

• Definition: Probability of event B given that event A has occurred.

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

• Application: Reduces the sample space based on given information.

Multiplication Rule of Counting

• Definition: If a task consists of a sequence of choices, multiply the number of ways each choice can be made.

• Example: Choosing a sandwich (3 breads, 4 meats, 2 cheeses): $3 \times 4 \times 2 = 24$ combinations.

Permutations and Combinations

• 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)

Counting vs. Probability

• Total Outcomes: Counting the number of possible arrangements.

• Probability: Often calculated as $\frac{1}{\text{total number of possible outcomes}}$ for equally likely outcomes.

Chapter 6: Discrete Probability Distributions

Random Variables

A random variable assigns a numerical value to each outcome of a random experiment.

• Discrete Random Variable: Takes on countable values (e.g., number of heads in 10 coin tosses).

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

Probability Distributions

• Definition: Lists each possible value of a random variable and its probability.

• Requirements:

  • Each probability is between 0 and 1.

  • The sum of all probabilities is 1.

Probability Histograms

• Definition: A bar graph representing the probability distribution of a discrete random variable.

• Application: Used to visualize the distribution and compare probabilities.

Mean (Expected Value) of a Discrete Probability Distribution

• Formula: $\mu = \sum x \cdot P(x)$

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

• Example: For X with values 1, 2, 3 and probabilities 0.2, 0.5, 0.3: $\mu = 1 \times 0.2 + 2 \times 0.5 + 3 \times 0.3 = 2.1$

Expected Value Problems

• Definition: The long-run average value of repetitions of the experiment.

• Steps:

  1. Identify all possible outcomes and their values.

  2. Multiply each outcome value by its probability.

  3. Sum the results to find the expected value.

• Example: Calculating expected winnings in a game of chance.

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)

• 68%: Within 1 standard deviation of the mean.

• 95%: Within 2 standard deviations.

• 99.7%: Within 3 standard deviations.

• Note: Use only when specifically instructed.

Z-Scores

• Definition: The number of standard deviations a value is from the mean.

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

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

Areas Under the Normal Curve

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

• Finding 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, $P(X = x) = 0$

Applications to Normal Distributions

• Given X ~ N($\mu$, $\sigma$):

  • To find $P(X < x)$, convert $x$ to $z$ and use the standard normal table.

  • To find $P(X > x)$, use the complement rule.

  • To find $P(x_1 < X < x_2)$, find the area between the corresponding z-scores.

• Example: Heights, test scores, etc.

Percentiles and Reverse Lookup

• Given Area (Probability, Proportion, or Percentile): Find the corresponding z-score from the table.

• Find X from Z: $x = \mu + z\sigma$

• Example: Finding the test score corresponding to the 90th percentile.

Summary Table: Key Probability Rules and Formulas

Concept	Formula	Description
Classical Probability	$P(E) = \frac{N(E)}{N(S)}$	Equally likely outcomes
Empirical Probability	$P(E) = \frac{\text{frequency of } E}{\text{number of trials}}$	Based on experiment/observation
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
Conditional Probability	$P(B|A) = \frac{P(A \text{ and } B)}{P(A)}$	Probability of B given A
Complement Rule	$P(A^c) = 1 - P(A)$	Probability event does not occur
Mean of Discrete Distribution	$\mu = \sum x \cdot P(x)$	Expected value
Z-Score	$z = \frac{x - \mu}{\sigma}$	Standardized value
Find X from Z	$x = \mu + z\sigma$	Reverse lookup
Permutations	$P(n, r) = \frac{n!}{(n-r)!}$	Order matters
Combinations	$C(n, r) = \frac{n!}{r!(n-r)!}$	Order does not matter