Probability Concepts

Key Terms and Definitions

• Independent Events: Two events are independent if the occurrence of one does not affect the probability of the other.

• Dependent Events: The probability of one event is affected by the occurrence of another.

• Disjoint (Mutually Exclusive) Events: Events that cannot occur at the same time.

• Complement: The complement of event X, denoted as X̅, is the event that X does not occur.

• Significantly High/Low Values: Values that are unusually high or low, often determined by statistical rules (e.g., Range Rule of Thumb or 5% Rule).

Probability Basics

• Probability: A measure of how likely an event is to occur, ranging from 0 (impossible) to 1 (certain).

• Smallest Possible Value: $0$

• Largest Possible Value: $1$

• Inequality:

Three Approaches to Probability

• Classical (Theoretical): Based on equally likely outcomes.

• Empirical (Experimental): Based on observed data.

• Subjective: Based on personal judgment or experience.

Probability Notation and Calculations

• Probability of an event:

• Complement:

• Union (or):

• Intersection (and):

• Conditional Probability:

• With Replacement: Probability remains unchanged after each selection.

• Without Replacement: Probability changes after each selection.

• At Least One:

Odds vs. Probability

• Odds Against: Ratio of unfavorable to favorable outcomes.

• Probability: Proportion of favorable outcomes to total outcomes.

• Formula for Odds Against:

Counting Rules

Fundamental Counting Rule

If there are ways to do one thing and ways to do another, there are ways to do both.

Factorial Rule

• Definition: is the product of all positive integers up to .

• Example:

Permutations and Combinations

• Permutations (All Different):

• Permutations (Some Identical):

• Combinations (Order Does Not Matter):

Discrete Probability Distributions (Chapter 5)

Expected Value

• Definition: The mean of a probability distribution.

• Formula:

• Example: If can be 1, 2, 3 with probabilities 0.2, 0.5, 0.3, then

Requirements for a Valid Probability Distribution

• Each probability must be between 0 and 1.

• The sum of all probabilities must be 1:

Binomial Distribution

• Requirements:

  • Fixed number of trials ()

  • Each trial is independent

  • Each trial has two outcomes (success/failure)

  • Probability of success () is constant

• Probability Formula:

• Mean:

• Standard Deviation:

Poisson Distribution

• Requirements:

  • Counts number of events in a fixed interval

  • Events occur independently

  • Mean rate () is constant

• Probability Formula:

• Mean:

• Standard Deviation:

Identifying Distribution Types

• Binomial: Fixed number of trials, two outcomes per trial.

• Poisson: Counts events in a fixed interval, no fixed number of trials.

Calculator Functions

• BINOMPDF: Computes binomial probabilities.

• POISSONPDF: Computes Poisson probabilities.

• Note: Always write down calculator inputs for partial credit.

Parameters

• p: Probability of success

• n: Number of trials

• x: Number of successes/events

Normal Probability Distributions (Chapter 6)

Normal Distribution

• Definition: A continuous, symmetric, bell-shaped distribution characterized by mean () and standard deviation ().

• Standard Normal Distribution: Normal distribution with and .

• Difference: Standard normal is a special case; normal can have any mean and standard deviation.

Uniform Distribution

• Definition: All outcomes are equally likely within a given interval.

• Probability Formula: for

• Visualization: The graph is a rectangle.

Significantly High or Low Values

• Range Rule of Thumb: Values outside are considered significant.

• 5% Rule: Values in the lowest or highest 5% are significant.

Area Under the Normal Curve

• Probability: Area under the curve represents probability.

• Use: Sketching helps visualize probability regions.

Z-score Calculations

• Z-score:

• Finding x:

• Application: Used to find probabilities and percentiles.

Central Limit Theorem (CLT)

• Definition: The sampling distribution of the sample mean approaches a normal distribution as sample size increases.

• Formula for Standard Error:

• Application: Used when working with sample means.

Unbiased Estimators

• Definition: Statistics whose expected value equals the population parameter.

• Examples: Sample mean (), sample proportion (), sample variance ().

• Target: Population mean, proportion, variance.

Summary Table: Probability Distributions

Additional info:

• Always round answers to three significant digits or a reasonable fraction.

• Show your work, as calculator and table answers may differ slightly.

• Table A-2 (Standard Normal Table) can be used for calculations.