Mathematics 5651 (Basic Theory of Probability and Statistics)

Description for prospective students

This course introduces the main mathematical concepts of probability theory, and some of the most important applications of probabilistic reasoning. It is a basic course in probability in the sense that it requires no prior course in probability, but it is a fairly challenging course since it covers probability concepts thoroughly. Students should be aware that probability theory requires a different kind of mathematical thinking from calculus, although a good knowledge of single and multi-variable calculus is also required, including the ability to change variables and set up regions of integration for double and occasionally triple integrals. Some prior review of calculus topics is recommended if skills have become a bit rusty.

Key ideas in probability theory include the sample space, which represents all possible outcomes of an experiment, random variables, which are quantities depending on the outcome of an experiment, the probability of an event, the expected value of a random variable, distributions and joint distributions of one or more random variables, as well the laws relating these concepts, all of which make it possible for us to understand the behavior of random systems. Another important topic is how to update probabilities when new information is obtained.

Students in Math 5651 also learn how obtain mathematical statements from verbal descriptions of problems involving randomness in the physical world, for example: choosing at random from a box of satisfactory and defective manufactured parts, frequency of eye colors of children and misprints on pages in a book, and the expected time until a particular random event occurs. Being able to apply probability to real situations deepens our understanding of the theory, so it is an integral part of this course.

Here are some sample problems to suggest the flavor of the subject.

• A player draws a hand of five cards at random from a standard deck of 52. Given that there are exactly two aces in the hand, find the probability that there are no hearts in the hand. (A common feature of Math 5651 is that small changes in the wording of an applied problem can have significant effects on answers. For instance, this sample problem becomes very different if the given information is that the two aces are the club and diamond aces.)
• When Sam tries to make a foul throw in basketball practice, she succeeds on two-thirds of her attempts. Today she decides to keep throwing until she succeeds two times in succession. Find the expected number of times that she will throw the basketball.
• A company manufactures electronic devices. The devices are manufactured from inexpensive materials, and each device has a 10 per cent chance of being defective. A buyer orders ten thousand of the devices, and the company agrees to pay a penalty if more than eleven hundred of the devices in the order turn out to be defective. Use the Central Limit Theorem to estimate the probability that the company will have to pay a penalty.

Connections with other courses:

Math 5651 (or alternatively Stat 5101), in combination with 2xxx-level linear algebra, constitutes a sufficient prerequisite for Math 5652 (Introduction to Stochastic Processes) and for Math 5654 (Prediction and Filtering). Math 5652, Math 5654, and Stat 5102 are very different from each other, and all three can be taken for credit.

There is one other undergraduate probability course, Math 4653. This course gives an introduction to probability which makes less use of calculus than Math 5651, and covers some different topics. Math 4653 is not a sufficient prerequisite for any of Math 5652, Math 5654, and Stat 5102. Moreover, Math 4653 cannot be taken for additional credit if a student has already received credit for Math 5651. However, credit for Mathematics 5651 can be earned after having earned credit for Mathematics 4653.

Programs using Math 5651:

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