Let us now conclude with a fun problem, which is also a little bit of a puzzle. We are told that the king comes from a family of two children. What is the probability that his sibling is female? Well, the problem is too loosely stated, so we need to start by making some assumptions. …
Independence is a very useful property. Whenever it is true, we can break up complicated situations into simpler ones. In particular, we can do separate calculations for each piece of a given model and then combine the results. We’re going to look at an application of this idea into the analysis of reliability of a …
We will now consider an example that illustrates the difference between the notion of independence of a collection of events and the notion of pairwise independence within that collection. The model is simple. We have a fair coin which we flip twice. So at each flip, there is probability 1/2 of obtaining heads. Furthermore, we …
Probability – L03.8 Independence Versus Pairwise Independence Read More »
Suppose I have a fair coin which I toss multiple times. I want to model a situation where the results of previous flips do not affect my beliefs about the likelihood of heads in the next flip. And I would like to describe this situation by saying that the coin tosses are independent. You may …
Probability – L03.7 Independence of a Collection of Events Read More »
We have already seen an example in which we have two events that are independent but become dependent in a conditional model. So that [independence] and conditional independence is not the same. We will now see another example in which a similar situation is obtained. The example is as follows. We have two possible coins, …
Probability – L03.6 Independence Versus Conditional Independence Read More »
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 …