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Video Lectures


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This suite of videos, arranged in twelve tableaux, invites the curious viewer into the magical world of probability. Read on for a quick summary of what the viewer can expect to see as she winds her way through the suite and what she can hope to learn. These video lectures originally appeared as a top-rated MOOC on Coursera in 2015. They appear in full in this sequence. In total these lectures cover material equivalent to a semester course on undergraduate probability at the University of Pennsylvania where this academic has taught since 1986.


Probability is, with the possible exception of Euclidean geometry, the most intuitive of the mathematical sciences. Games of chance have an ancient and honoured history and, in the modern day, chance and its language pervade our common experience. Today there is scarce an area of investigation that is left untouched by probabilistic considerations. The renowned mathematical physicist Pierre Simon, Marquis de Laplace wrote with more than a soupçon of prescience in his opus on probability in 1812 that “the most important questions of life are, for the most part, really only problems in probability”. His words ring particularly true today in this the century of “big data”.


This suite of video lectures takes us through the development of a modern, axiomatic theory of probability. But, unusually for a mathematical subject, the material is presented in its lush and glorious historical context, the mathematical theory buttressed and made vivid by rich and beautiful applications drawn from the world around us. The student will see surprises in election-day counting of ballots, a historical wager the sun will rise tomorrow, the folly of gambling, the sad news about lethal genes, the curiously persistent illusion of the hot hand in sports, the unreasonable efficacy of polls and its implications to medical testing, and a host of other beguiling settings. A curious individual taking this as a stand-alone course will emerge with a nuanced understanding of the chance processes that surround us and an appreciation of the colourful history and traditions of the subject. And for the student who wishes to study the subject further, this course provides a sound mathematical foundation for courses at the advanced undergraduate or graduate levels.

Video Organisation and Ancillary Documents


This suite of videos comprising a course on probability is organised into 12 tableaux. Each tableau explores one facet of the theory of probability.  Tableaux are typically broken up into two or three sub-tableaux, each comprising a cohesive group of lectures.  The lecture numbering convention follows this organisational schema: for example, Lecture 3.1: b refers to the second lecture (b) of the first sub-tableau of Tableau 3; Lecture 8.2: h refers to the eighth lecture (h) in the second sub-tableau of Tableau 8 and Lecture 11.3: f refers to the sixth lecture (f) of the third sub-tableau of Tableau 11.  A lecture-by-lecture breakdown by title can be found in the Syllabus in the probability folder on the Courses page.  Additional ancillary documents in this folder flesh out the subject: these include problem sets for solution and detailed solutions.

For the viewer who prefers all her videos in one place they are also available in a single Playlist on YouTube: Probability | Santosh S. Venkatesh, University of Pennsylvania [FULL COURSE].


These lectures are intended to be absorbed sequentially, much as you would in a live college class, with each tableau appearing in order as you scroll down this page.   The viewer new to this sequence should begin by casting an eye over the introductory videos in Tableau 1 which outline the organisation of videos in the entire suite, the philosophy behind these lectures, what a viewer may hope to learn from them, and the notational conventions in force. The suite is intended to be watched sequentially from Tableau 1 through Tableau 12 and for someone new to the subject that is likely to be the best strategy. There are delicious bon mots scattered here and there for the expert as well, however, and any such may, of course, wish to flit here and there through the suite as interest and time allow.

Recommended Background


With all its historical context and diverse modern applications, these lectures fundamentally deal with mathematical probability. To fully appreciate them, as background a student should have a solid exposure to at least one semester of calculus.


Further Reading


While the lectures are designed to be self-contained, for the student who likes to have a reference book handy, the material itself is drawn primarily from the first half of my book The Theory of Probability: Explorations and Applications (Cambridge University Press, 2013). This expansive book contains an eclectic mixture of undergraduate and graduate material, all presented in the narrative spirit of this course. It is likely to be most useful for a student who has a solid calculus background and who wishes to make a deeper study of the theory and see an even richer cast of applications by self-study or by taking further courses in the subject. If used in conjunction with the course the student should look at the suggestions on how to read the book on page xxi and, especially, the suggested chapters for undergraduates, before launching into it.



The production of these lectures involved a labour of love of very many chefs working tirelessly behind the scenes. I am indebted to our magnificent teaching staff, Behnaz Arzani, Mahyar Fazlyab, Natalie Miller, Amin Rahimian, and Saraswathi Venkatesh who vetted the content, created and worked out problem sets, took care of a myriad details in production, and saved me from making grave mistakes; our video editor Chris Cook who did a yeoman’s job in producing these beautiful videos from the rather unpromising material in front of his cameras; and Anshul Tripathi who alternatively cajoled and encouraged, and took it upon himself to port all these videos into this platform — a labour of love indeed! My thanks also go to my many teaching assistants over the years who have contributed so richly to my teaching and the many long-suffering generations of my students at the University of Pennsylvania whom it has been my privilege to teach.


Winding Roads

Tableau 1
Prelude to a theory of chance

Tableau 1

Tableau 1 consists of 9 videos in total arranged in three sub-tableaux.  Thumbnails can be previewed by clicking on the menu bar at the bottom of the video and clicking on the right and left arrows to scroll through the collection.


Sub-Tableau 1.0 consists of four video lectures labelled 1.0: a – d. These videos provide essential background on the arrangement of videos in this collection, explanations of the conventions used, and how best to get the most out of the lectures in this suite. The expenditure of time is modest and the viewer who begins here will have saved herself some aggravation in decoding the organisation and conventions in use downstream.


Sub-Tableau 1.1 is a standalone video providing a glimpse of the manifold manifestations of chance around us in diverse settings.


The four videos in sub-Tableau 1.2: a – d complete the introduction. In these are explored unexpected ramifications of the simplest chance experiment: from the humble toss of a coin to a problem in sports psychology. Never fear. The tantalising questions posed are answered later in the sequence.

Tableau 1
Watch Now

Tableau 2

Beautiful Landscape

Tableau 2
Combinatorial elements