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Introduction to free probability math

Let us learn some concepts in probability math for free.

We often hear phrases such as "Probably it will rain today" or "It will probably be a hot day tomorrow" or "Most probably I will stand first in the examination" etc. These phrases involve an element of uncertainty. Now the problem is, how can we measure this uncertainty? A measure of uncertainty is provided by a branch of Mathematics called " Theory of Probability". In this theory, we deal with those situations in which a particular result or outcome is not certain, but it can be any one of the several possible outcomes.

The theory had its beginning in the 16th century. It originated in the games of chance, for instance, throwing of dice or coins, drawing cards from a well-shuffled deck or balls from an urn etc. The first book on the subject was written by the Italian mathematician, J.Cardan (1501-1576). The title of the book was "Book on Games of Chance" (Liber de Ludo Aleae), published in 1663. Notable contributions were also made by French mathematicians, B.Pascal(1623 - 1662), Pierre de Fermat (1601 - 1665), ...
... Swiss mathematician J.Bernoulli (1654 - 1705)etc.

The theory of probability has wide and important applications in the fields of natural sciences and social sciences.

free probability math- as a Measure of Uncertainty

We turn our attention to one of the problems that was responsible for the development of the theory of probability, namely, that of throwing a die. A die is a well-balanced cube with its six faces marked with numbers (dots) from 1 to 6, one number on one face as shown in figure.

When we play a game witha die, we are generallly interested in the number coming up after the toss on its uppermost face. Let us throw a die once. What are the possible outcomes? Clearly, a die can fall with any of its faces uppermost. The number of each of the faces is therefore a possible outcome. Since the die is well-balanced, therefore it is as likely to show up a number, say '2', as any other number 1,3,4,5,or 6.

Since there are six equally likely outcomes : 1,2,3,4,5,or6 ina single throw of a die and there is only one way of getting a particular outcome '2' , therefore, the chance of the number 2 coming up is 1 in 6. In other words, we say that the probability of getting 2 is 1/6.

We write it as P( 2) = 1/6. Similarly, when an ordinary coin is tossed, it may show up head (H) or tail(T). We see that in this case there are only two equally likely outcomes of which only one is favourable to the occurrence of head. Hence, the probability of getting a head in a single toss of a coin is given by P(H) = 1/2 .

free probability math-Definition of probability

The above examples suggest the following definition of Probability (assuming that outcomes are equally likely).

Probabilityof an event E, written as P(E), is defined as

P(E) = Number of outcomes favourable toE / Total number of possible outcomes.

In the above example of throwing a die, the event E was getting a number 2 on the die. Similarly , in the example of tossing a coin, the event E was getting a head (H). Lets try to find the answers to the following two questions related to throwing of a die once.

(i) What is the probability of a die coming up with the number 8?

We know that there are only six possible outcomes in a single toss of a die. It may show any number from 1 to 6. Since no face of the die is marked with 8, it is obvious that we will never get the number 8, i.e., getting the number 8 is impossible. Such event is called an impossible event. P( getting 8 in a single throw of a die) = 0/6 = 0.

Hence, we say theat the probability of an impossible event is zero.

(ii) What is the probabilityof getting a number less than 7?

Since every face of a die is marked with a number less than 7, it is evident that we wil always get a number less than 7, i.e., getting a number less than 7 is a sure event. P(getting a number 7) = 6/6 =1. Thus, the probability of a sure event is 1.

Hence, the probability P(E) of any event E takes any value from 0 to 1,

i.e., 0 `=` P(E) `=` 1.

We have learnt some concepts in probability math for free.

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