In this segment, we will discuss the multinomial model and the multinomial probabilities, which are a nice generalization of the binomial probabilities. The setting is as follows. We are dealing with balls and the balls come into different colors. There are r possible different colors. We pick a ball at random, and when we do …
We will now apply our multinomial formula for counting the number of partitions to solve the following probability problem. We have a standard 52-card deck, which we deal to four persons. Each person gets 13 cards as, for example, in bridge. What is the probability that each person gets exactly one ace? Well, before we …
We now come to our last major class of counting problems. We will count the number of ways that a given set can be partitioned into pieces of given sizes. We start with a set that consists of n different elements. And we have r persons. We want to give n1 items to the first …
Let us now put to use our understanding of the coin-tossing model and the associated binomial probabilities. We will solve the following problem. We have a coin, which is tossed 10 times. And we’re told that exactly three out of the 10 tosses resulted in heads. Given this information, we would like to calculate the …
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, …
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. …