# 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.

Credits

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.

### Tableau

### Tableau 1

Prelude to a theory of chance

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 2

### Tableau 2

Combinatorial elements

### Tableau 2

The student of chance should be familiar with combinatorial methods, or the elementary theory of counting. These ideas have an honoured place in classical probability in the analysis of games of chance and continue to play an important rôle in the modern theory.

The lectures in Tableau 2, presented here, accordingly constitute a quick review of the key combinatorial principles. The twelve lectures comprising this tableau are grouped into two sub-tableaux, sub-Tableau 2.1 and sub-Tableau 2.2, each consisting of six lectures.

The six lectures in sub-Tableaux 2.1 are sequentially labelled 2.1: a – f and summarise the main ideas in sampling from finite sets.

The six lectures in sub-Tableau 2.2, also sequentially labelled 2.2: a – f, flesh out the key properties of binomial coefficients. The viewer who would like to quickly review these concepts should watch the lectures sequentially. The viewer who is familiar with combinatorial ideas can skip on ahead after a quick glance at Tableau 2.1: f and Tableau 2.2: f which contain brief summaries of the notation and the main results.

### Tableau 3

### Tableau 3

Chance in commonplace settings

Tableau 3, presented here, consists of three sub-tableaux. The 14 lectures in this tableau take the viewer on a passage through time, encountering chance problems in familiar and unfamiliar settings, with the goal of discovering fundamental features governing chance by exploration in commonplace settings.

Sub-Tableau 3.1 has six lectures labelled 3.1: a – f. In these video lectures the viewer is introduced to the classical metaphor of ball and urn problems in the setting of random sampling from finite sets. The illustrative problems are beguiling, beginning with a renaissance dicing paradox attributed to the Chevalier de Méré, and continuing with a startling birthday paradox.

Sub-Tableau 3.2 is an optional "dangerous bend" collection of five lectures 3.2: a – e which may be skipped by the viewer in a hurry to get to the res gestae. For the viewer who has a more leisurely progression in mind, this exotic digression leads her into urn models in statistical physics, the strange behaviour of particles in nature, and the unsettling realisation that elementary particles may not have common sense.

Sub-Tableau 3.3 is a return to the main progression of lectures. The three lectures 3.3: a – c in this sub-tableau build upon the ball and urn models of sub-Tableau 3.1 by taking the viewer beyond balls and urns and into a world of unequal probabilities and infinite spaces. The example problems in this sub-tableau include the casino game of craps and a return to an old familiar — the toss of a coin.

### Tableau 3

### Tableau 4

### Tableau 4

A little set theory

Tableau 4, presented here, consists of eight lectures labelled 4: a – h and is a self-contained review of the elements of set theory that are needed for this suite of lectures on probability. In this, the second of the two review tableaux, the viewer is introduced to the language of sets — terminology, notation, relations, basic operations, and properties.

The viewer who is comfortable with set theory may wish to jump forward to the summary lecture in Tableau 4: h to take on board the notational conventions that will be in force throughout these lectures before proceeding to Tableau 5. The viewer who is rusty with the conventions and ideas should absorb these lectures sequentially before proceeding.

### Tableau 4

### Tableau 5

### Tableau 5

The abstract probability space

Tableau 5, presented here, consists of 11 lectures labelled 5: a – k and comprises a descent into abstraction to establish the axiomatic foundations of a fully mathematical theory of probability. These lectures take the learner from the specification of an abstract sample space of idealised outcomes of a chance experiment, to the construction of an algebra of events, and rounds off the axiomatic foundations of the subject by articulating the axioms of probability measure, leading to a self-contained, consistent mathematical theory of chance.

### Tableau 5

### Tableau 6

Tableau 6 consists of two sub-tableaux. The 17 lectures in this tableau introduce the viewer to formal mechanisms which codify raw intuition in the simplest chance settings.

Sub-Tableau 6.1 has six lectures labelled 6.1: a – f. These video lectures summarise the lessons that may be accrued from the simplest chance problems and introduce the viewer to random choice in formal discrete spaces, the notions of atoms and mass functions, and the common distributions — combinatorial, binomial, Poisson, and geometric.

Sub-Tableau 6.2 is an optional "dangerous bend" collection of eleven lectures 6.2: a – k which may be skipped by the viewer in a hurry to get to the res gestae. For the viewer who has had exposure to integral calculus this collection of lectures segue from chance experiments in discrete domains to chance experiments in the continuum in one and more dimensions. The viewer will see mass functions segue to probability densities and encounter the basic densities — uniform, exponential, and normal.

### Tableau 6

### Tableau 7

Is the outcome of a coin toss really random?

Tableau 7 is an optional "dangerous bend" collection of five lectures 7.2: a – e which may be skipped, without loss of continuity, by the viewer in a hurry to get to the res gestae. For the viewer who has a more leisurely progression in mind, this digression has a philosophical cast as the viewer is asked to ponder whether randomness can exist in a physical world governed by natural law. This collection of five lectures begins with a piquant question: "Is the toss of a coin really random?" The succeeding lectures take the viewer on a journey of discovery culminating in Poincaré's delightful articulation of "The principle of arbitrary functions" with implications to the modelling of chance in physical systems and what that has to say about the casino game of roulette.

### Tableau 7

### Tableau 8

### Tableau 8

Conditional probability

Tableau 8 consists of two sub-tableaux. With the formal structure of a probability space having been formally introduced in Tableau 5 and built upon in Tableau 6, the 24 lectures in this tableau build a coherent structure of how a budding probabilist can fold in side-information about a chance experiment into a principled analysis. And, along the way, the viewer will discover a host of beguiling applications.

Sub-Tableau 8.1 consists of nine lectures labelled 8.1: a – i. In these video lectures the viewer will build intuition about the rôle of side-information in a chance experiment from the usual culprits, dice and those long-suffering families, leading to a formal notion of conditional probability. The setting is elementary but the consequences are subtle as the viewer will discover in a discussion of a fraught question from the gender wars: Is there gender bias in graduate admissions?

Sub-Tableau 8.2 is the largest sub-collection thus far in this suite of whimsical lectures. This sub-tableau consists of 15 lectures labelled 8.2: a – o. The fundamental pillar of additivity is blended back in this sequence of lectures and, in conjunction with conditioning based on side-information, leads to an extraordinarily rich domain of application spanning a quite incredible range: as a soupçon the viewer will encounter the Ballot Problem with echoes in the polemics of contested elections in the modern day, a historical wager that the sun will rise tomorrow leading to a doomsday principle and a whimsical take on humanity’s future, and an urn scheme which casts light on population growth and, in an era of pandemic, the spread of contagion.

### Tableau 8

### Tableau 9

### Tableau 9

Independence!

Tableau 9 consists of 36 lectures organised into three sub-tableaux. These lectures introduce the viewer to the principle of independence in probability. This fundamental principle is at the heart of the mathematical theory of chance and gives it much of its distinctive character. The lectures in this tableau begin with the development of the formal notion of independence followed by, as the viewer may have begun to expect, a host of tantalising applications.

Sub-Tableau 9.1 consists of six lectures labelled 9.1: a – f. In these video lectures the viewer will build intuition from multiplication tables and simple chance settings to arrive at a formal definition of independence.

Sub-Tableau 9.2 consists of 10 lectures labelled 9.2: a – j. These lectures build upon the definition of independence and flesh out a variety of subtle and important consequences. In these lectures the viewer will see how untrained intuition can play us false if we are not careful, and find a codification of the most natural and intuitive setting in which independence arises — the toss of a coin and, more generally, a sequence of independent trials.

In sub-Tableau 9.3 we turn finally to a collection of quite splendid applications in which independence runs wild. This rich sub-tableau consists of 20 lectures labelled 9.3: a – t. These lectures begin with a detailed analysis of the casino game of craps, concluding with an understanding of the reason why the rules are so arcane and a salutary moral on the perils of gambling. The next sumptuous application is drawn from genetics and the viewer is taken on a voyage of discovery culminating in an understanding of the sad persistence of lethal genes notwithstanding the best efforts of natural selection, and a feeling for the time scales involved in evolution. The concluding lectures in this sub-tableau return to the hot hand phenomenon in sports psychology that we had used as a springboard to launch this indulgent suite of lectures. These lectures take a deep dive into an analysis of fluctuations and provide the viewer with a principled resolution of the hot hand paradox — while creating new questions to ponder.

### Tableau 9

### Tableau 10

### Tableau 10

From polls to bombs: the binomial & Poisson distributions

Tableau 10 consists of 39 lectures organised into two substantive sub-tableaux. These lectures introduce two of the three fundamental distributions in the theory of chance: the binomial and the Poisson distributions. The third of the fundamental distributions, the normal, makes its appearance in the next tableau.

Sub-Tableau 10.1 consists of 17 lectures labelled 10.1: a – q. In these video lectures the viewer will see the binomial distribution emerge in its natural setting: the ubiquitous opinion poll. Beginning with a discussion of dichotomous populations in the settings of political parties, invasive species, defective genes, and opinions, the binomial emerges as the focal point of a natural problem in estimation and the rest of the lectures in this sub-tableaux flesh out the key properties of the distribution.

Sub-Tableau 10.2 consists of 22 lectures labelled 10.2: a – v. As a prelude to discovery, these lectures begin with a fundamental question on the proper size of a jury — a historical question from the time of the French Revolution with echoes in jurisprudence in the modern day. The lectures in this sub-tableau begin by tracing the development of a model for jury selection dating to post-Revolution France — with an aside on two twentieth century Supreme Court rulings and a nod to the film adaptation of Harper Lee's “To Kill a Mockingbird” — and target a curious approximation to the binomial discovered by Siméon Denis Poisson in 1837 as a starting point for a fecund trail of discovery leading to the Poisson distribution. In these lectures the viewer will see weird and wonderful observations fitting the Poisson distribution — including the blitz of London, equine-caused fatalities in the Prussian army, radioactive emissions, misprints in a book, the distribution of wars in history, and the sad story of a birthday cake — and flesh out the properties of the distribution. Lectures 10.2: p – u at the end of the sub-tableau constitute a “dangerous bend” detour into the subtleties of arrival processes and queues in a discussion of traffic patterns and the distribution of stars in the cosmos.

### Tableau 10

### Tableau 11

### Tableau 11

The fabulous limit laws: the laws of large numbers & the central limit theorem

Tableau 11 consists of 37 lectures organised into three substantive sub-tableaux. These lectures introduce the viewer to the fundamental limit laws of probability: the laws of large numbers and the central limit theorem. These limit laws contribute to much of our common intuition for the subject and underlie the major applications of the theory. Our long journey through this suite of lectures has, in many ways, built towards this climax.

Sub-Tableau 11.1 consists of 14 lectures labelled 11.1: a – n. In these lectures the viewer is transported back to the curious case of the opinion poll. In an age where polls have sometimes failed spectacularly, the viewer will explore the art of the random sample and the subtlety of bias, and see why there are yet reasons for optimism. The mathematical dance leads via the subtle inequality discovered by Pafnuty Chebyshev to the sweeping law of large numbers. And along the way the viewer sees why a properly conducted poll works. The latter half of the sub-tableau takes lessons from polls to another topical and fraught setting in a time of pandemic: the new drug approval process. The viewer is led through the Food and Drug Administration’s regulatory process to an understanding of the key issues and why the approval mechanism for a vaccine or drug provides an ex post, mathematically principled rationale for why the lay public can have trust in a properly implemented process of approval.

Sub-Tableau 11.2 consists of 15 lectures labelled 11.2: a – o. This entire sub-tableau is a “dangerous bend” segment which needs exposure to the calculus; the individual who does not have much in the way of the integral calculus in his background can safely skip this sub-tableau and move directly to sub-Tableau 11.3 and Lecture 11.3: a. The lectures in sub-Tableau 11.2 build upon the lessons that may be drawn from the analysis of opinion polls leading to a general form of the laws of large numbers. While the level of abstraction looks unpromising for a casual viewer, if there are any such watching these lectures, the practical sway of the mathematical theory is illustrated via a sumptuous application which has fundamentally changed our views of computation in the late twentieth and twenty first centuries in this the era of big data. This is the story of Monte Carlo computation.

Sub-Tableau 11.3 consists of 8 lectures labelled 11.3: a – h. Fittingly, we have reserved the best for the last: in these, the concluding lectures of this tableau, the bell curve enters stage left and the viewer is walked through the development of the éminence grise of probability, the central limit theorem, which establishes the bell curve (or more formally, the normal curve or Gaussian density) as the one curve to rule them all. And along the way, the viewer will discover why a (properly conducted) opinion poll *really* works.

These lectures complete our development of the mathematical principles that undergird the basic theory of probability. Tableau 12 concludes our explorations but is entirely devoted to a sequence of sumptuous applications of the fundamental limit laws. These may be taken in entirely at a student’s discretion depending on time, inclination, and need.