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Uniform probability

Uniform probability
One of the major applications of combinatorics is uniform probability. That is a probability function on a set that is finite and where the possible outcomes have the same opportunity to occur. More precisely, let omega be a finite set. A uniform probability on this set omega is a function that assigns to every subset of omega, the value, P of A, which is equal to the quotient of the number of elements of A divided by the number of elements of omega. There is a particular terminology in probabilities. Omega is called the sample space. The subsets of omega are called events, and every element of omega is called an elementary event.
Of course, if you’re assigned a uniform probability on omega, this means that every elementary is equal probability and equal to 1 divided the number of elements of omega. As an example, consider a dice that is thrown. The possible outcomes are 1, 2, 3, 4, 5, 6. Well, if who throws the dice is not a magician, and if the dice is regular, all the faces have the same opportunity to occur. Then it is reasonable to assign to every face, the probability one over six. That is to assign to the space of the possible outcomes from one to six, the uniform probability. As an example, consider an extraction of two numbers between three numbers.
Two of them are red, and the other one is black. We are asking, which sample space is more appropriate for the uniform probability? Well, first, the two collections of RB, red, black. R, number two, the two sequences of R1, R2, and B, were R1, R2 stand for the two red numbers, which has now a label, R1, R2. Well, of course, the second choice is the appropriate one, because the collection of RB is twice the possibility with respect to RR. It may occur if the first is red and the second is black, or if the first is black and the second is red. Whereas the collection RR may occur just if both are red.
Let’s see some properties of the uniform probability. Of course, the probability of the empty set is zero. Let us call omega the sample space and P the probability of this space, where P of omega is equal to 1. This is because P of omega is the cardinality of omega divided by the same cardinality. Now, let us take two events, A and B of the sample space. Well, the probability of the union of A with B is the sum of the probabilities P of A plus P of B minus the probability of the intersection. This follows directly from the similar property of the number of elements of a set.
Let us consider the complement of A, which is denoted by A to the C. That is omega minus A. Well, the probability of omega minus A is the complement of A is is 1 minus the probability of A. This is useful, because sometimes, it is easier to find the probability of the complement with respect to the probability of the initial event. As an example, assume we want to find the probability that among 10 students, at least two of them have the same birth date. As a sample space, we can consider the 10 sequences of the birth dates that we may assume being from one to 365. We do not include the February 29. I’m sorry.
Our event, A, corresponds to the 10 sequences that have at least a repetition. It turns out that it is not so easy to count directly the number of elements of this set, whereas it is quite easy to count the number of elements of the complement. Just because the complement is the set of 10 sequences of I 365 with no repetition. So the property of the probability tells us that the probability that we are looking for is 1 minus the probability of the complement. That is of the set of 10 sequences without repetition.

What is the probability to win at the lottery? Or to write your name if you choose randomly one by one as many letters as your name needs?

Carlo will explain what is the uniform probability on a finite set, where every single outcome has the same “possibility” to occur: in this case probabilities are an application of combinatorics. We will be able to answer to questions, however we will not cover the wide filed of probabilities (and questions like: what is the probability to have a 6 with a rigged dice?)

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Combinatorics: Strategies and Methods for Counting

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