Open course ware

An important reason why conditional probabilities are very useful is that they allow us to divide and conquer. They allow us to split complicated probability modes into simpler submodels that we can then analyze one at a time. Let me remind you of the Total Probability Theorem that has his particular flavor. We divide our sample space into three disjoint events-- A1, A2, and A3. And these events form a partition of the sample space, that is, they exhaust all possibilities. They correspond to three alternative scenarios, one of which is going to occur. And then we may be interested in a certain event B. That event B may occur under either scenario. And the Total Probability Theorem tells us that we can calculate the probability of event B by considering the probability that it occurs under any given scenario and weigh those probabilities according to the probabilities of the different scenarios. Now, let us bring random variables into the picture. Let us fix a particular v...

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 concrete example. Consider a probability model with 12 equally likely possible outcomes, and so each one of them has probability equal to 1/12. We will focus on two particular events, event A and B, two subsets of the sample space. Event A has five elements, so its probability is 5/12, and event B has six elements, so it has probability 6/12. Suppose now that someone tells you that event B has occurred, but tells you nothing more about the outcome. How should the model change? First, those outcomes that are outside event B are no longer possible. So we can either eliminate them, as was done in this picture, ...

Now, conditional probability will let us explain a lot of the confused arguments that people brought up about Monty Hall. And we'll see that it is a little bit confusing and where there is some correct sounding arguments that give you the wrong answer. So let's go back and look at our Monty Hall tree that allowed us to derive the sample space and probability space for the whole process of the prize being placed and the contest picking a door and Carol opening a door. Now, this tree was way more complicated than we needed if all we were trying to do was figure out the probability of winning if you switch. But having the tree will allow us to discuss a whole bunch of other events in their probabilities that will get us a grip on some of the arguments that gave the wrong answer. So let's look at the event, first of all that the goat is at 2. Now, this is the branch where the prize is at 2. And so in all the other branches the goat is at 2, l which means that we h...

Conditional probability is an absolutely basic idea that we use all the time. It's the probability that some event occurs, given certain information about it. For example, an insurance company wants to know, what's the probability that you'll live for the next 10 years given your medical history? Or a typical investor wants to know, what's the probability that this stock is going to rise given its stock price gyrations for the past month? There are people who actually think you could do that, the chartists. That, not knowing anything about the nature of the company, or the business that the stock is part of, that just by watching the price gyration you can make a better guess on what the stock will do tomorrow than you could otherwise. Another good example is for a system engineer. What's the probability that the system is going to overload, given the recent history of the rate at which requests have been coming in? And finally, as a joke that I like t...