This course begins with the basics of discrete probability. That is, what is the probability of a certain event occurring out of a finite set of possibilities? For example, what is the probability that the sum of two dice is even? Later in the course we discuss continuous probabilities. For example, what is the probability that we will have at least 1 inch of rain tomorrow? The amount of rain we get could be any nonnegative real number.
We study properties of both kinds of probabilities. We learn how to
derive some more complex probabilities out of simple ones. For
example, we study conditional probability and Bayes Rule. Conditional
probability asks the question, what is the probability that 
will
happen given that 
happens?
In order to understand the mathematical structures of probability
theory we talk about random variables. For example, if is the the
number of students who apply to Smith next year, then is a random
variable. The distribution
of a random variable may be described as a function that contains all
the relevant information about the probability of a random variable.
We study several known distributions, including the Normal or Gaussian, the
Binomial, the Poisson, and the Exponential.
For some purposes we do not need all the information contained in the distribution function, and a few descriptive properties of the distribution will serve. Two particularly useful properties are the expected value and the variance of the distribution. The first tells us where the distribution is located on the number line and the second tells us how spread out it is.
Sometimes we can use the distribution of one random variable to find the distribution of another, related variable. We study specific techniques for finding distributions of random variables that are mathematical transformations of other random variables whose distributions are known.