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What Are the Chances? Probability Made Clear

What Are the Chances? Probability Made Clear

Professor Michael Starbird Ph.D.
The University of Texas at Austin

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What Are the Chances? Probability Made Clear

What Are the Chances? Probability Made Clear

Professor Michael Starbird Ph.D.
The University of Texas at Austin
Course No.  1474
Course No.  1474
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Course Overview

About This Course

12 lectures  |  30 minutes per lecture

Life is full of probabilities. Every time you choose something to eat, you deal with probable effects on your health. Every time you drive your car, probability gives a small but measurable chance that you will have an accident. Every time you buy a stock, play poker, or make plans based on a weather forecast, you are consigning your fate to probability.

What Are the Chances? Probability Made Clear helps you understand the random factors that lurk behind almost everything—from the chance combinations of genes that produced you to the high odds that the waiting time at a bus stop will be longer than the average time between buses if they operate on a random schedule.

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Life is full of probabilities. Every time you choose something to eat, you deal with probable effects on your health. Every time you drive your car, probability gives a small but measurable chance that you will have an accident. Every time you buy a stock, play poker, or make plans based on a weather forecast, you are consigning your fate to probability.

What Are the Chances? Probability Made Clear helps you understand the random factors that lurk behind almost everything—from the chance combinations of genes that produced you to the high odds that the waiting time at a bus stop will be longer than the average time between buses if they operate on a random schedule.

In 12 stimulating half-hour lectures, you will explore the fundamental concepts and fascinating applications of probability.

High Probability You Will Enjoy This Course

Professor Michael Starbird knows the secret of making numbers come alive to non-mathematicians: he picks intriguing, useful, and entertaining examples. Here are some that you will explore in your investigation of probability as a reasoning tool:

  • When did the most recent common ancestor of all humans live? Applying probabilistic methods to the observed mutation rate of human genetic material, scientists have traced our lineage to a female ancestor who lived about 150,000 years ago.
  • How much should you pay for a stock option? Options trading used to be tantamount to gambling until about 1970, when two economists, Fischer Black and Myron Scholes, found a method to quantify those risks and to create a rational model for options pricing.
  • What do you do on third down with long yardage? In football, a pass is the obvious play on third down with many yards to go. Of course, the other team knows that. Probability and game theory help decide when to run with the ball to keep your opponent guessing.

What You Will Learn

The course literally begins with a roll of the dice, as Professor Starbird demonstrates that games of chance perfectly illustrate the basic principles of probability, including the importance of counting all possible outcomes of any random event. In Lecture 2, you probe the nature of randomness, which is famously symbolized by monkeys randomly hitting typewriter keys and creating Hamlet. In Lecture 3, you explore the concept of expected value, which is the average net loss or gain from performing an experiment or playing a game many times. Then in Lecture 4, you investigate the simple but mathematically fertile idea of the random walk, which may seem like a mindless way of going nowhere but which has important applications in many fields.

After this introduction to the key concepts of probability, you delve into the wealth of applications. Lectures 5 and 6 show that randomness and probability are central components of modern scientific descriptions of the world in physics and biology. Lecture 7 looks into the world of finance, particularly probabilistic models of stock and option behavior. Lecture 8 examines unusual applications, including game theory, which is the study of strategic decision-making in games, wars, business, and other areas. Then in Lecture 9 you consider two famous probability puzzles guaranteed to cause a stir: the birthday problem and the Let's Make a Deal® Monty Hall question.

Finally, Lectures 10–12 cover increasingly sophisticated and surprising results of probabilistic reasoning associated with Bayes theorem. The course concludes with probability paradoxes.

Take the Weather Forecasting Challenge

One of the most familiar experiences of probability that we have on a daily basis is the weather report, with predictions like, "There is a 30 percent chance of rain tomorrow." But what does that mean? What do you think? Choose one:

  • (a) Rain will occur 30 percent of the day.
  • (b) At a specific point in the forecast area, for example, your house, there is a 30 percent chance of rain occurring.
  • (c) There is a 30 percent chance that rain will occur somewhere in the forecast area during the day.
  • (d) 30 percent of the forecast area will receive rain, and 70 percent will not.
  • (e) None of the above.

In Lecture 5, Dr. Starbird puts this particular forecast under the microscope to demonstrate that probabilistic statements have very precise meanings that can easily be misinterpreted—or misstated. He explains why the answer is (e) and not one of the other choices. He also explains why the official definition from the National Weather Service is subtly but decidedly wrong.

He even wagers that within five years the phrasing of the official definition will change because somebody at the National Weather Service will hear this lecture!

Games People Play

The formal study of probability was born at the dice table. Gambling continues to provide instructive examples of the principles of chance and probability, including:

  • Gambler's ruin: A random walk is a sequence of steps in which the direction of each step is taken at random. In gambling, the phenomenon assures that a bettor who repeatedly plays the same game with even odds will eventually—and invariably—go broke.
  • St. Petersburg paradox: A famous problem in probability involves a hypothetical game supposedly played at a casino in St. Petersburg. Though simple and apparently moderately profitable for the gambler, the expected value of the game is infinite! Yet no reasonable person would pay very much to play it. Why not?
  • Gambler's addiction: Randomness plays a valuable role in reinforcing animal behavior. Changing the reinforcement in an unpredictable, random way leads to behaviors that are retained for a long time, even in the absence of rewards. Applied to humans, this observation may help explain the compulsiveness of some gamblers.

Probability to the Rescue

One approach to probability, developed by mathematician and Presbyterian minister Thomas Bayes in the 18th century, interprets probability in terms of degrees of belief. As new information becomes available, the calculation of probability changes to take account of the new data. The Bayesian view reflects the reality that we adjust our confidence in our knowledge as we gain evidence.

The world of fluctuating probabilities, under continual adjustment as new evidence comes to light, captures the way the world works in realms like medicine, where a physician makes a preliminary diagnosis based on symptoms and probabilities, then orders tests, and then refines the diagnosis based on the test results and a new set of probabilities.

If you think about it, it's also the way you work when you're on a jury. At the outset, you have a vague impression of the likelihood of guilt or innocence of the defendant. As evidence mounts, you adjust the relative probabilities you assign to each of these verdicts. You may not do a formal calculation, but your informal procedure is nonetheless Bayesian.

Randomness is all around us. "Many or most parts of our lives involve situations where we don't know what's going to happen,"; says Professor Starbird. Probability comes to the rescue to describe what we should expect from randomness. It is a powerful tool for dispelling illusions and uncertainty to help us understand the true odds when we roll the dice in the game of life.

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12 Lectures
  • 1
    Our Random World—Probability Defined
    The concept of randomness and its quantification through probability is central to understanding the world of science, games, business, and other endeavors. This lecture introduces the basic laws of probability. x
  • 2
    The Nature of Randomness
    Randomness refers to situations in which given results are unpredictable, but a large enough collection of results is predictable. The goal of probability is to describe what it is to be expected from randomness. x
  • 3
    Expected Value—You Can Bet on It
    Expected value is a useful measure for making decisions about probabilistic outcomes. It provides a numerical way to judge whether to bet on a particular game or make a particular investment. x
  • 4
    Random Thoughts on Random Walks
    A random walk is a description of random fluctuations. It aids the analysis of situations ranging from counting votes to charting pollen on a fishpond, and it explains the sad fate of persistent bettors. x
  • 5
    Probability Phenomena of Physics
    Quantum mechanics describes the location of subatomic particles as a probability distribution. Weather predictions also give probabilistic descriptions; but what is the meaning of a statement like "There is a 30 percent chance of rain tomorrow"? x
  • 6
    Probability Is in Our Genes
    Because randomness is centrally involved in passing down genetic material, probability can be used to model the distribution of genetic traits and to describe how traits of whole populations alter through a random process called genetic drift. x
  • 7
    Options and Our Financial Future
    By characterizing the expected behavior of a stock in the future and describing a probability distribution of its likely future price, mathematicians can quantify sophisticated risks in options contracts. However, the practice can be a very dangerous game. x
  • 8
    Probability Where We Don't Expect It
    What does probability have to do with determining if a number is prime, or deciding football strategy, or training animals? More than you might think—probability often plays a central role where we least expect it. x
  • 9
    Probability Surprises
    No course on probability could be complete without a discussion of two of the most famous examples of counterintuitive probabilistic scenarios: the birthday problem and the Let's Make a Deal® Monty Hall question. x
  • 10
    Conundrums of Conditional Probability
    Conditional probability refers to a situation where the probability of one event is affected by some other event or piece of information. Principles of dealing correctly with conditional probability are tricky and highly nonintuitive. x
  • 11
    Believe It or Not—Bayesian Probability
    This lecture looks at probability from a different point of view: namely, probability associated with measuring a level of belief as opposed to measuring the frequency with which the results of a random process occur. This is the Bayesian view of probability. x
  • 12
    Probability Everywhere
    A pair of paradoxes shows the power of the Bayesian approach in analyzing counterintuitive cases in probability. The course concludes with a review of the topics covered and the importance of probability in our world. x

Lecture Titles

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Michael Starbird
Ph.D. Michael Starbird
The University of Texas at Austin

Dr. Michael Starbird is Professor of Mathematics and University Distinguished Teaching Professor at The University of Texas at Austin, where he has been teaching since 1974. He received his B.A. from Pomona College in 1970 and his Ph.D. in Mathematics from the University of Wisconsin-Madison in 1974. Professor Starbird's textbook, The Heart of Mathematics: An Invitation to Effective Thinking, coauthored with Edward B. Burger, won a 2001 Robert W. Hamilton Book Award. Professors Starbird and Burger also collaborated on Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas, published in 2005. Professor Starbird has won many teaching awards, including the Mathematical Association of America's 2007 Deborah and Franklin Tepper Haimo National Award for Distinguished College or University Teaching of Mathematics, which is the association's most prestigious teaching award. It is awarded nationally to 3 people from its membership of 27,000. Professor Starbird is interested in bringing authentic understanding of significant ideas in mathematics to people who are not necessarily mathematically oriented. He has developed and taught an acclaimed class that presents higher-level mathematics to liberal arts students.

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Reviews

Rated 3.6 out of 5 by 54 reviewers.
Rated 1 out of 5 by Old school style Worse buy, kick myself for not reading feedback b4 purchase. Does not mention combination or permutation when covering that topic. He is knowledgable, not an effective teacher. I learned more from Jason Gibson of MathTutorDVD.com. He is the best! I had subscribed for more that 3 years, still active. Highly recommend for this topic. I thought I would explore other lectures, am disappointed. April 20, 2014
Rated 2 out of 5 by Not Very Advanced I tutor high school and college students. I was looking for something that could help me better explain some of the formulas. Unfortunately, the material on these lectures is very basic and was not sophisicated enough for anything beyond the third week on a basic high school level stats class. If you just want to learn about stats, you might enjoy this. If you want to learn stats at a high school course level, you will need to find something more advanced. April 15, 2014
Rated 5 out of 5 by Clear presentation This is a well-organized course, presented with clear, understandable examples. It provides a good basis for the study of statistics, and highlights the idea that the reality of probability may be at odds with intuition or "common sense", providing an important critical thinking tool. March 6, 2014
Rated 4 out of 5 by Useful and revealing! I have truly been enjoying this video and may watch parts of it again to cement my understanding. You don't need to be a mathematician to understand it, and there are many real-life examples. I only have two minor gripes: 1, The speaker really needs to look at the camera. It's almost as if he is deliberately avoiding eye contact. I find this very distracting. 2. While there are visual aids and charts, they border on the bland and boring. Certainly, with today's technology, the visuals could have much more impact. August 28, 2013
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