L01.5 Simple Properties of Probabilities
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...