UW-Madison offers several distinct introductory-level probability courses as well as a proof-based probability course in which students can enroll without taking an introductory-level probability course.
Some of the named option plan for the major in mathematics require a course in probability. The standard major does not. It may be possible to use the proof-based probability course MATH 531 to complete the requirement of a named option plan. Students should consult the Guide for more information. Note that in all versions of the major, only one of STAT/MATH 309 and MATH/STAT 431 may be used to fulfill any of the requirements. (MATH 331 may be used in the certificate, but at most one of MATH 331, STAT/MATH 309, and MATH/STAT 431 may be used in the certificate.) Note that this page does not discuss STAT 311-312, which are not cross-listed to mathematics and do not count toward any requirements for the mathematics major or certificate.
The purpose of this page is to describe the essential differences between the introductory-level probability courses as well as the proof-based undergraduate probability course.
MATH 331 (Introductory Probability)
MATH 331 is a first course in probability for students who have not taken multivariable calculus (MATH 234 or MATH 375-376). Students who have completed two semesters of calculus (either MATH 221-222 or MATH 211-213) may enroll in this class, which will introduce the necessary tools from multivariate integration alongside the topics in probability. This course is taught by the Department of Mathematics in a lecture-only format. Students who have taken or plan to take multivariable calculus should enroll in one of the other probability courses after completing multivariable calculus, as the other courses listed here count toward more programs and as prerequisites for more advanced courses.
This course does count toward the requirements of the mathematics certificate, but it does not count toward the requirements of the mathematics major (standard major or any of the named option plans). A student may use at most one of MATH 331, 309, and 431 toward requirements for the mathematics certificate. Students should consult the Guide for information on programs outside of mathematics that will count MATH 331. This course may be a good option for students pursuing the statistics certificate or certain programs from the Wisconsin School of Business. Computer Science majors who do not plan to take multivariable calculus but would like an introduction to probability should also consider this course.
In summary, MATH 331…
- Serves as an introduction to probability for students who have not completed multivariable calculus;
- Counts in the math certificate but not the mathematics major;
- Covers material in MATH 309 and therefore credit for only one of MATH 331 or STAT/MATH 309 can be applied to the math certificate;
- Covers material in MATH/STAT 431 and therefore credit for only one of MATH 331 or MATH/STAT 431 can be applied to the math certificate;
- Is a good introduction to probability for students in the statistics certificate who do not plan to take multivariable calculus; and
- Is not, on its own, suitable preparation for further advanced study of probability.
Suggested further courses are…
- MATH 234, MATH 421, and MATH 531 for students who wish to see an advanced, proof-based treatment of probability.
STAT/MATH 309 (Introduction to Probability and Mathematical Statistics I)
STAT/MATH 309 is a first course in probability for students who have taken multivariable calculus (MATH 234 or MATH 375-376), including concurrent enrollment in MATH 234 or MATH 376. This course is taught by the Department of Statistics in a lecture-discussion format. It is a mathematically-rigorous introduction to probability that is interchangeable with MATH/STAT 431 in program requirements and requisites. Because this course is taught by statisticians, applications introduced may be more related to statistical settings than in MATH/STAT 431.
This course does counts toward the requirements of the mathematics certificate and mathematics major. A student may use at most one of STAT/MATH 309 and MATH/STAT 431 toward requirements for the mathematics major.
In summary, STAT/MATH 309…
- Serves as an introduction to probability for students who have completed or are concurrently enrolled in multivariable calculus;
- Is taught by the Department of Statistics in a lecture-discussion format;
- Counts in both the mathematics certificate and the mathematics major;
- Covers material in MATH 331 and therefore credit for only one of MATH 331 or STAT/MATH 309 can be applied to the math certificate;
- Covers material in MATH/STAT 431 and therefore credit for only one of STAT/MATH 309 and MATH/STAT 431 can be applied to the math major; and
- Is good preparation for additional coursework in probability and mathematical statistics.
Suggested further courses are…
- STAT/MATH 310 for an introduction to the mathematical foundations of statistical inference (after also completing an introductory statistics course).
- MATH 531 for a proof-based approach to probability (after completing an appropriate introduction to proofs course).
- MATH 632 for an introduction to stochastic processes, which is a study of randomness with a time component (after completing additional prerequisites such as linear algebra or an appropriate introduction to proof).
MATH/STAT 431 (Introduction to the Theory of Probability)
MATH/STAT 431 is a first course in probability for students who have taken multivariable calculus (MATH 234 or MATH 375-376). Concurrent enrollment in MATH 234 or MATH 376 is not allowed. This course is taught by the Department of Mathematics in a lecture-only format. It is a mathematically-rigorous introduction to probability that is interchangeable with STAT/MATH 309 in program requirements and requisites. Because this course is taught by mathematicians, there may be less focus on potential statistical applications of probability than students will encounter in STAT/MATH 309.
This course does counts toward the requirements of the mathematics certificate and mathematics major. A student may use at most one of STAT/MATH 309 and MATH/STAT 431 toward requirements for the mathematics major.
In summary, MATH/STAT 431…
- Serves as an introduction to probability for students who have completed multivariable calculus;
- Is taught by the Department of Mathematics in a lecture-only format;
- Counts in both the mathematics certificate and the mathematics major;
- Covers material in MATH 331 and therefore credit for only one of MATH 331 or MATH/STAT 431 can be applied to the math certificate;
- Covers material in STAT/MATH 309 and therefore credit for only one of STAT/MATH 309 and MATH/STAT 431 can be applied to the math major; and
- Is good preparation for additional coursework in probability and mathematical statistics.
Suggested further courses are…
- STAT/MATH 310 for an introduction to the mathematical foundations of statistical inference (after also completing an introductory statistics course).
- MATH 531 for a proof-based approach to probability (after completing an appropriate introduction to proofs course).
- MATH 632 for an introduction to stochastic processes, which is a study of randomness with a time component (after completing additional prerequisites such as linear algebra or an appropriate introduction to proof).
MATH 531 (Probability Theory)
MATH 531 is a proof-based undergraduate course in probability. As such, students must have completed a suitable introduction to proofs (MATH 376, 421, or 521). Prior study of probability is not required. This course is taught by the Department of Mathematics in a lecture-only format. This course counts toward requirements in the mathematics major and certificate, and students may count both STAT/MATH 309 or MATH/STAT 431 along with MATH 531 toward the requirements. In some named option plans within the mathematics major, MATH 531 can be used to satisfy a probability requirement.
In summary, MATH 531…
- Is proof-based;
- Is taught by the Department of Mathematics in a lecture-only format;
- Counts in both the mathematics certificate and the mathematics major; and
- Is good preparation for additional coursework or graduate study in probability and mathematical statistics.
Suggested further courses are…
- STAT/MATH 310 for an introduction to the mathematical foundations of statistical inference (after also completing an introductory statistics course).
- MATH 632 for an introduction to stochastic processes, which is a study of randomness with a time component.