We have seen so far an example of a probability law on a discrete and finite sample space as well as an example with an infinite and continuous sample space. Let us now look at an example involving a discrete but infinite sample space. We carry out an experiment whose outcome is an arbitrary positive […]

We will now go through a probability calculation for the case where we have a continuous sample space. We revisit our earlier example in which we were throwing a dart into a square target, the square target being the unit square. And we were guaranteed that our dart would fall somewhere inside this set. So

Let us now move from the abstract to the concrete. Recall the example that we discussed earlier where we have two rolls of a tetrahedral die. So there are 16 possible outcomes illustrated in this diagram. To continue, now we need to specify a probability law, some kind of probability assignment. To keep things simple,

We will now continue and derive some additional properties of probability laws which are, again, consequences of the axioms that we have introduced. The first property is the following. If we have two sets and one set is smaller than the other– so we have a picture as follows. We have our sample space. And

The probability axioms are the basic rules of probability theory. And they are surprisingly few. But they imply many interesting properties that we will now explore. First we will see that what you might think of as missing axioms are actually implied by the axioms already in place. For example, we have an axiom that

We have so far discussed the first step involved in the construction of a probabilistic model. Namely, the construction of a sample space, which is a description of the possible outcomes of a probabilistic experiment. We now come to the second and much more interesting part. We need to specify which outcomes are more likely

Let us now look at some examples of sample spaces. Sample spaces are sets. And a set can be discrete, finite, infinite, continuous, and so on. Let us start with a simpler case in which we have a sample space that is discrete and finite. The particular experiment we will be looking at is the

Putting together a probabilistic model- that is, a model of a random phenomenon or a random experiment- involves two steps. First step, we describe the possible outcomes of the phenomenon or experiment of interest. Second step, we describe our beliefs about the likelihood of the different possible outcomes by specifying a probability law. Here, we

Probability – L01.1 Overview Probability – L01.2 Sample Space Probability – L01.3 Sample Space Examples Probability – L01.4 Probability Axioms Probability – L01.5 Simple Properties of Probabilities Probability – L01.6 More Properties of Probabilities Probability – L01.7 A Discrete Example Probability – L01.8 A Continuous Example Probability – L01.9 Countable Additivity Probability – L01.10 Interpretations

Welcome to the first lecture of this class. You may be used to having a first lecture devoted to general comments and motivating examples. This one will be different. We will dive into the heart of the subject right away. In fact, today we will accomplish a lot. By the end of this lecture, you