A basketball coach has 20 players available. Out of them, he needs to choose five for the starting lineup, and seven who would be sitting on the bench. In how many ways can the coach choose these 5 plus 7 players? It is certainly a huge number, but what exactly is it? In this lecture, …
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 …