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, …
Probability – L01.10 Interpretations & Uses of Probabilities Read More »
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 …
Probability – L01.6 More Properties of Probabilities Read More »
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 …
Probability – L01.5 Simple Properties of Probabilities Read More »
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 …