In this lecture, we introduce and develop the concept of independence between events. The general idea is the following. If I tell you that a certain event A has occurred, this will generally change the probability of some other event B. Probabilities will have to be replaced by conditional probabilities. But if the conditional probability […]

We now come to the third and final kind of calculation out of the calculations that we carried out in our earlier example. The setting is exactly the same as in our discussion of the total probability theorem. We have a sample space which is partitioned into a number of disjoint subsets or events which

Let us now revisit the second calculation that we carried out in the context of our earlier example. In that example, we calculated the total probability of an event that can occur under different scenarios. And it involves the powerful idea of divide and conquer where we break up complex situations into simpler pieces. Here

As promised, we will now start developing generalizations of the different calculations that we carried out in the context of the radar example. The first kind of calculation that we carried out goes under the name of the multiplication rule. And it goes as follows. Our starting point is the definition of conditional probabilities. The

Let us now examine what conditional probabilities are good for. We have already discussed that they are used to revise a model when we get new information, but there is another way in which they arise. We can use conditional probabilities to build a multi-stage model of a probabilistic experiment. We will illustrate this through

I now want to emphasize an important point. Conditional probabilities are just the same as ordinary probabilities applied to a different situation. They do not taste or smell or behave any differently than ordinary probabilities. What do I mean by that? I mean that they satisfy the usual probability axioms. For example, ordinary probabilities must

This is a simple example where we want to just apply the formula for conditional probabilities and see what we get. The example involves a four-sided die, if you can imagine such an object, which we roll twice, and we record the first roll, and the second roll. So there are 16 possible outcomes. We

Conditional probabilities are probabilities associated with a revised model that takes into account some additional information about the outcome of a probabilistic experiment. The question is how to carry out this revision of our model. We will give a mathematical definition of conditional probabilities, but first let us motivate this definition by examining a simple

Suppose I look at the registry of residents of my town and pick a person at random. What is the probability that this person is under 18 years of age? The answer is about 25%. Suppose now that I tell you that this person is married. Will you give the same answer? Of course not.

We end this lecture sequence by stepping back to discuss what probability theory really is and what exactly is the meaning of the word probability. In the most narrow view, probability theory is just a branch of mathematics. We start with some axioms. We consider models that satisfy these axioms, and we establish some consequences,