Conditional probabilities are like ordinary probabilities, except that they apply to a new situation where some additional information is available. For this reason, any concept relevant to probability models has a counterpart that applies to conditional probability models. In this spirit, we can define a notion of conditional independence, which is nothing but the notion …
Let us now discuss an interesting fact about independence that should enhance our understanding. Suppose that events A and B are independent. Intuitively, if I tell you that A occurred, this does not change your beliefs as to the likelihood that B will occur. But in that case, this should not change your beliefs as …
Probability – L03.4 Independence of Event Complements Read More »
In the previous example, we had a model where the result of the first coin toss did not affect the probabilities of what might happen in the second toss. This is a phenomenon that we call independence and which we now proceed to define. Let us start with a first attempt at the definition. We …
As an introduction to the main topic of this lecture sequence, let us go through a simple example and on the way, review what we have learned so far. The example that we’re going to consider involves three tosses of a biased coin. It’s a coin that results in heads with probability p. We’re going …
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 …
Probability – L02.5 A Radar Example and Three Basic Tools Read More »
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 …
Probability – L02.4 Conditional Probabilities Obey the Same Axioms Read More »