The coefficients n-choose-k that we calculated in the previous segment are known as the binomial coefficients. They are intimately related to certain probabilities associated with coin tossing models, the so-called binomial probabilities. This is going to be our subject. We consider a coin which we toss n times in a row, independently. For each one …
Let us now study a very important counting problem, the problem of counting combinations. What is a combination? We start with a set of n elements. And we’re also given a non-negative integer, k. And we want to construct or to choose a subset of the original set that has exactly k elements. In different …
We will now use counting to solve a simple probabilistic problem. We have in our hands an ordinary six-sided die which we are going to roll six times. So this is our probabilistic experiment. And we’re interested in the probability of a certain event, the event that the six rolls result in different numbers. So …
In this segment we introduce a simple but powerful tool, the basic counting principle, which we will be using over and over to deal with counting problems. Let me describe the idea through a simple example. You wake up in the morning and you find that you have in your closet 4 shirts, 3 ties, …
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, …
Probability – L04.1 Counting Lecture Overview Read More »
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 »