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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 probabilities are non-negative. We will show that probabilities are also less than or equal to 1. We have another axiom that says that the probability of the entire sample space is 1. We will show a counterpart that the probability of the empty set is equal to 0. This makes perfect sense. The empty set has no elements, so it is impossible. There is 0 probability that the outcome of the experiment would lie in the empty set. We also have another intuitive property. The probability that an event happens plus the probability that the vendor does not happen exhaust all possibilities. And these two probabilities together should add to 1. For instance, if the proba...

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...