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Probability Theory - Calculus-Based Statistics - Online Course for Academic Credit

Often referred to as the "higher Probability & Statistics course", or even "Calculus-based Statistics", our Probability Theory course is actually an introduction to the study of statistics and probability, but based upon the usage of Calculus to study both discrete and continuous aspects of the subject. Accordingly, there is no prerequisite of a previous study of statistics, but rather a prerequisite of having completed Calculus II (having completed or concurrently enrolled in Multivariable Calculus is recommended). The Probability Theory course is an essential part of the mathematical training for those wishing to study Data Science.

Course
DMAT 311 - Computational Probability Theory
Credits
3 Semester Credit Hours
Delivery
Fully Online, Asynchronous, Self-Paced

Completion of DMAT 311 - Computational Probability Theory earns 3 academic credit semester hours with an official academic transcript from Roger Williams University, in Providence, Rhode Island, USA, which is regionally accredited by the New England Commission of Higher Education (NECHE), facilitating transfer of credits nationwide to other colleges and universities.



Probability Theory Introductory Videos


Probability Theory Course Introduction

The curriculum for the course, Prob/Stat&Mathematica by Carpenter/Davis/Raschke/Uhl, is a thorough and advanced investigation of the subject matter, fascinating and challenging at the same time. The usage of the powerful computer algebra and graphing system Mathematicaâ„¢ allows for a unique exploration of distributions - both discrete and continuous - and their application to the cornerstone of the subject - the data set from a real-world situation.

Probability Theory differs from the "lower" Statistics course significantly in both approach and difficulty level. Compare the prerequisites:

Course
Statistics
Probability Theory
Prerequisite
Algebra II from high school
Calculus II (rec: Multivariable Calculus)
Intended for Majors
Humanities, Social Studies, Biological Sciences
Math, Engineering, Physics, Economics, etc.

RWU Course Catalog - DMAT 311 • Computational Probability Theory
Course
DMAT 311
Course Title
Computational Probability Theory
Transcript Title
Probability Theory
Credits
3 Semester Credit Hours
Description
A first course in calculus-based statistics and probability with emphasis on computational software techniques, geometrical analysis, and numerical processes applications. Topics include Monte-Carlo method, measurements of probability, distributions, applications to data analysis, discrete and random variables, conditional and joint probability concepts, Central Limit Theorem, correlation and regression, confidence intervals.
Prerequisite
Successful completion (C- or higher) of Calculus II or equivalent, or consent of instructor.
E-Textbook
Probability, Statistics & Mathematica by Davis/Porta/Uhl
Software
Mathematica

DMAT 311 - Learning Outcomes

  1. To understand the core concepts of Probability, Sampling, Distributions, and Density
  2. To understand and compute Monte-Carlo method for integration
  3. To understand and compute Expected Value, Variance, Mean, Mode, Median
  4. To understand and compute Probability and Conditional Probability
  5. To understand and compute Markov's and Chebyshev's Theorems
  6. To understand and compute normally and exponentially distributions
  7. To understand and compute calculus-based formulas and relationships between Cumulative Distribution Functions and Probability Density Functions
  8. To understand and compute the Central Limit Theorem
  9. To understand the core concepts of discrete and random variables
  10. To understand and compute Joint Distributions, Correlations, and Covariance
  11. To understand the Law of Total Probability
  12. To understand and compute the classic statistics measures of Confidence Intervals and Hypothesis testing.

DMAT 311 - Syllabus of Topics

1. Getting Started
1.1. Email and Chat
1.2. Learning About the Course
1.3. Required Hardware
1.4. Software Fundamentals
2. Simulations
2.1. Uniform Distributions
2.2. Monte Carlo Methods
2.3. Random Walks
2.4. Shooting craps; Iterated Fractals.
3. Data Analysis
3.1. Frequency
3.2. Expected Value
3.3. Cumulative Distributions
3.4. Variance
3.5. Histograms
3.6. Related formulas for Expected Values and Variance
4. Probabilities
4.1. Calculating Probability
4.2. Union and Intersection and Probability
4.3. Conditional Probability Formula
4.4. Independence
4.5. Indicator functions
5. More Data Analysis
5.1. Markov’s Inequality
5.2. Chebyshev’s Theorem
5.3. Laws of Large Numbers
5.4. One-Sided Chebyshev Theorem
6. Normal and Exponential Distributions
6.1. Approximately Normally Distributed Sets
6.2. Normal Distribution
6.3. Approximately Exponentially Distributed Sets
6.4. Exponential Distribution
6.5. Memoryless Property of Exponential Distributions
7. Random Variables
7.1. “Random Variables”
7.2. Discrete Random Variables
7.3. Continuous Random Variables
7.4. Probability Density Functions
7.5. Cumulative Distribution Functions
7.6. Expected Values and Variance
7.7. Markov, Chebyshev, and Law of Large Numbers Revisited
7.8. Mean, Median, and Mode
8. Joint Distributions
8.1. Joint Probability Calculations
8.2. Discrete & Continuous
8.3. Expected Values, Covariance, and Correlation.
8.4. Conditional Probability Calculations
8.5. Conditional Expectations
8.6. The Law of Total Probability
9. Central Limit Theorem
9.1. Generating Functions for Discrete Random Variables
9.2. Generating Functions for Continuous Random Variables
9.3. Generating Functions and Independence
9.4. Central Limit Theorem
9.5. Chi-squared and Gamma random variables
10. Classic Statistics
10.1 Sampling
10.2. Confidence Intervals? 10.3. Hypothesis testing

Legacy Course - DMAT 315 • Probability Theory

In 2023, Distance Calculus introduced a new catalog of courses. New DMAT 311 = Old DMAT 315 = Old MATH 315

Description
Emphasizes probability, probability density functions, distributions, statistical inferences and estimation, correlation, and regression. (3 credits)
Common Completion Timelines for DMAT 311 - Probability Theory
Hours/Week
Math Skills
Dedication
Completion
Advisory
5-10 hrs/wk
Weaker
1-2 hrs/day
12 weeks
Reasonable
7-12 hrs/wk
Modest
2-3 hrs/day
9 weeks
Reasonable
10-15 hrs/wk
Stronger
3-4 hrs/day
7 weeks
Reasonable
15-20 hrs/wk
Strong
5-6 hrs/day
5 weeks
Stretched
20-25 hrs/wk
Strong
5-7 hrs/day
3 weeks
Stretched
25-35 hrs/wk
Strong
6-8 hrs/day
12 days
World's Record

Time commitments are important for success in an online Probability Theory course for college credit from Distance Calculus. There are no fixed due dates in the Distance Calculus online courses, so it is important that students instead set their schedules for a dedicated amount of time towards the coursework.

It is also very important to consider that going faster through a course is DIRECTLY DEPENDENT upon your math skill level, and your successful engagement of the course. We require that you complete the course in a Mastery Learning format. If you are struggling with the course content, or trying to go too fast where the quality of your submitted work is suffering, then the instructors will force a slow-down of your progress through the course, even if you have fixed deadlines.

Probability Theory Examples of the Curriculum

Below are some PDF "print outs" of a few of the Mathematicaâ„¢ notebooks from Prob/Stat&Mathematica by Carpenter/Davis/Raschke/Uhl. Included as well is an example homework notebook completed by a student in the course, demonstrating how the homework notebooks become the "common blackboards" that the students and instructor both write on in their "conversation" about the notebook.



Distance Calculus Referenced Colleges & Universities (29 Years - 393+ Institutions)

Distance Calculus students have transferred course credits to these colleges and universities:

Agnes Scott College • Aiken Technical College • Albany College of Pharmacy and Health Science • Alma College • American Graduate University • American Public University • American University • Andrews University • Arizona State University • Armstrong Atlantic State Univeristy • Athens State University • Auburn University • Auburn University MBA Program • Augusta State University • Austin Peay State University • Azusa Pacific University • Babson College • Baruch College • Baylor University • Belmont University • Beloit College • Bentley University • Berklee College of Music • Berry College • Bethany College • Binghamton University • Bloomsburg University • Bluefield State College • Bluegrass Community and Technical College • Borough of Manhattan Community College • Boston Conservatory • Boston University • Bryant University • Buena Vista University • California Lutheran University • California Polytechnic State University, San Luis Obispo • California state University • California State University Channel Islands • California State University, Dominguez Hills • California State University, Sacramento • Carleton College • Carnegie Mellon University • Cedarville University • Central Michigan University • Central Washington University • Champlain College • Chapman University • Charter Oak State College • Chicago State University • Clark University • Clarkson University • Clemson University • Cleveland State University • Coastal Carolina University • College of Santa Fe • College of William & Mary • Colorado Mesa University • Colorado State University • Columbia University • Columbia University School of Business • Cornell Univeristy • Cornell University • Covenant College • CUNY Medgar Evers College • Denison University • DePaul University • Drexel University • Duke University - Fuqua School of Business • Duke University School of Law • Duke University, Durham NC • Duke University, Fuqua School of Business, Law School, Graduate Programs • East Stroudsburg University • Eastern Illinois University • Eastern Kentucky University • Eastern Mennonite University • Eastern Nazarene College • Elon University • Embry Riddle Aeronautical University • Embry Riddle University • Endicott College • Evangel University • Excelsior College • Fairifield University • Fairleigh Dickenson University • Ferris State University • Florida A & M University • Florida Agricultural and Mechanical University • Florida Atlantic University • Florida Institute of Technology • Florida International University • Florida State College, Jacksonville • Florida State University • Fordham University • Fox Valley Technical College • Franklin University • Freed-Hardamen University • Fresno State University • Friends University • Gannon University • George Mason university • George Washington University • George Washington University School of Business • Georgetown University • Georgia Institute of Technology • Georgia State • Georgia State University • Georgia Tech • Gordon College • Governor's State University • Green Mountain College • Griffith University • Grinnell College • Grove City College • Hamline University • Hampshire College • Hampton University • Harvard University, Kennedy School of Government • Harvard University: Kennedy School of Government, Medical Schools • Hillsdale College • Hillsdale University • Hiram College • Hofstra University • Howard University • Huntingdon College • Illinois Institute for Technology • Illinois Institute of Technology • Indiana University • Iowa State University • Jacksonville State University • James Madison University • Jeff State Community College • Johns Hopkins Univerisity • Johns Hopkins University • Kalamazoo College • Kansas State University • Kaplan University • Kennesaw State University • Kentucky State University • Kettering University • Kings College, University of London • La Sierra University • Lebanon Valley College • Lee University • LeTourneau University • Liberty University • Lincoln University of 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Michigan University • Northwest Nazarene University • Northwestern University • Oberlin College • Occidental College • Oglethorpe University • Oklahoma Baptist University • Old Dominion University • Olympic College • Orange Coast College • Oregon State University • Pacific Lutheran University • Penn State University • Pennsylvania State University • Pepperdine University • Pomona College • Portland State University • Princeton University • Purdue University • Quinnipiac University • Randolph-Macon College • Regent University • Regis University • Rensselaer Polytechnic Institute • Rhode Island School of Design • Rice University • Robert Morris University • Rochester Institute of Technology • Roger Williams Univerity • Roger Williams University • Roosevelt University • Rowan University • Rutgers University • Saint Anselm College • Saint Joseph's University • Saint Louis University • Saint Michael's College • Salve Regina University • Samford University • San Diego State University • Santa Fe Community College • Shepherd University • Smith College • South Dakota School of Mines and Technology • Southern Adventist University • Southern Methodist University • St. Anselm College • St. John's College • St. Mary's College of Maryland • Stanford University • Stanford University, MBA • State University at Buffalo Law School • State University at Buffalo, Law School • State University of New York • Stevens Institute of Technology • Strayer University • SUNY Binghamton • Swarthmore College • Syracuse University • Texas A&M University • Texas A&M • Texas A&M University • Texas Tech University • The Art Institute of Atlanta • The Catholic University of America • The Citadel • The Citadel, Military College of South Carolina • The College of New Jersey • The College of St. Scholastica • The George Washington University • The Master's College • The New England Institute of Art • The Ohio State Universtity • The University of Alabama • The University of South Carolina • The University of Texas at Austin • The University of Virginia • Thomas Edison State College • Trinity University • TUI University • Tulane University • Union University • United States Air Force Academy • United States Military Academy • Univeristy of Puget Sound • University of Alabama, Huntsville • University of Arizona • University of Arkansas, Little Rock • University of Auckland, New Zealand • University of California, Berkeley • University of California, Los Angeles • University of California, Santa Barbara • University of California, Santa Cruz • University of Central Florida • University of Central Oklahoma • University of Central Texas • University of Chicago • University of Cincinnati • University of Colorado • University of Colorado, Boulder • University of Colorado,Colorado Springs • University of Connecticut • University of Dallas • University of Findlay • University of Florida • University of Georgia • University of Hartford • University of Hawai'i-Manoa • University of Illinois • University of Kentucky • University of La Verne • University of Maine • University of Maryland • University of Massachusetts • University of Massachusetts, Amherst • University of Memphis • University of Michigan • University of Michigan: MBA, Medical Schools, Graduate Programs • University of Minnesota • University of Minnesota, School of Public Health • University of Minnesota, Twin Cities • University of Minnesota-Twin Cities • University of Mississippi • University of Missouri • University of Missouri, Columbia • University of Montana • University Of Mount Union • University of Nebraska • University of Nevada • University of New Hampshire Law School • University of New Haven • University of New Orleans • University of North Carolina • University of North Carolina at Chapel Hill • University of North Carolina, Chapel Hill • University of North Carolina, MBA • University of North Dakota • University of North Texas • University of Northern Iowa • University of Notre Dame • University of Oklahoma • University of Otago • University of Pennsylvania • University of Pennsylvania Architectural School • University of Pennsylvania, Wharton School of Business • University of Phoenix • University of Pittsburgh • University of Portland • University of Redlands • University of Richmond • University of San Francisco • University of South Carolina • University of Southern California • University of Southern Indiana • University of Sussex • University of Tampa • University of Tennessee • University of Texas • University of Texas at Austin • University of Texas, Arlington • University of Texas, Austin • University of Texas, Brownsville • University of Texas, Dallas • University of Texas, Houston • University of Utah • University of Virginia • University of Warwick • University of West Alabama • University of West Florida • University of West Georgia • University of Wisconsin • University of Wisconsin, Madison • University of Wyoming • University West Florida • US Air 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Distance Calculus - Student Reviews

Taylor C.★★★★★
Posted: May 27, 2025
Courses Completed: Multivariable Calculus
Im a visual learner and had a difficult time understanding multivariable calculus my first go around, and found this course offered by Roger Williams University, which uses real-time 3d graphing of the equations. I would see my work be manipulating the models, which gave a robust understanding of what each variable did. absolutely loved the course and class recordings.
Transferred Credits To: Califorina Baptist University
Coury Gaffney★★★★★
Posted: Jan 18, 2021
Courses Completed: Linear Algebra
The program used gives an amazing insight into everything that's happening, that you wouldn't get in a traditional course. All of the lessons are clear and clean, and the professor is very helpful along the way. I learned a lot and am happy with taking this course
Email: courygaffney@gmail.com
Transferred Credits To: Virginia Tech
I.★★★
Posted: May 5, 2025
Courses Completed: Calculus II
The course material was well explained through the videos and notebooks, and it was nice to receive feedback on each assignment and correct any mistakes. However, the software required for the course was ridiculously outdated, to the point where I had to reinstall a very old operating system on my computer to even be able to run it. It was annoying and tedious to use, and it sometimes crashed. The graders were also sometimes overly picky about the way that concepts had to be explained to be considered correct (specific wording, etc), but it wasn’t too bad. Having multiple final exams seemed a bit excessive but again was not too bad. Overall, the main strength of this course is that it is completely asynchronous and self-paced, so I do recommend it if that is the most important factor to you. You will learn all of the material thoroughly, but you must be willing to deal with the more annoying aspects of the course (primarily the outdated technology).
M M.★★★★★
Posted: Feb 8, 2026
Courses Completed: Precalculus, Calculus I
The courses were excellent. Very flexible and engaging and the platform offers a lot of upper-level courses. Dr. Curtis is an outstanding professor and very responsive. I would take again.
Transferred Credits To: None yet
Tanja B.★★★★★
Posted: Jan 28, 2026
Courses Completed: Calculus I
After two failed attempts at my university, this course helped me understand Calculus. The live maths tool along with Dr. Curtis were especially helpful, allowing me to visualize concepts and expand my understanding. The explanations were clear, the examples practical, and I could learn at my own pace, which built my confidence. Thank you.
Transferred Credits To: University of Namibia
Henry F.★★★★★
Posted: Dec 18, 2025
Courses Completed: Differential Equations
Transferred Credits To: Saint Joseph High School

Frequently Asked Questions

Do I Need To Take Statistics Before Probability Theory?

No. The actual topic coverage of Statistics and Probability are very close to one another. The Probability Theory course does everything with the machinery of Calculus, while the Statistics course stays away from Calculus and just concentrates on observing the patterns in the data.

Is Probability Theory The Same As Calculus-Based Statistics?

Yes. Probability Theory is exactly a first course in calculus-based Statistics.

Is Probability Theory Good for Data Science?

Yes. It is mandatory that any Data Science student will have taken Probability Theory (and more advanced courses after Probability Theory as well) prior to starting a Data Science certificate or degree.

Can I Take Probability Theory Without Calculus II?

No. Calculus II is very much a prerequisite for Probability Theory, and not a nominal prerequisite. Probability Theory relies very strongly on the mastery of the Calculus II content.

Is Probability Theory from Distance Calculus Accredited?

Yes, All Distance Calculus courses are offered through Roger Williams University in Providence, Rhode Island, USA, which is regionally accredited (the highest accreditation) through New England Commission of Higher Education (NECHE).

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