Learn Probablility

By Author: Pierce Brosnan
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Introduction to learn probablility:

Chance plays an important role, in human life. For example it is common for a farmer, to expect the chance of raining, for a captain to estimate the chance of his team to win the game, and for the army chief, to estimate the chance of destroying the strategy of the enemy. Even though there is a lot of experience and logical thinking behind these expectations, one needs an appropriate measure, to estimate the chance. For this purpose the theory of Probability was originated. In other words, the branch of Mathematics, which deals with the estimation of chance is called the theory of Probability.

The theory of Probability has its origin in gambling, which was more popular in the 17th century in France. The Italian Mathematician Galileo tried to derive a Quantitative measure to estimate the results, when dice are rolled in gambling. But the foundation of probability was laid by Pascal and Fermat through their letters and discussions. Huggins , a Dutch Mathematician published the first book on Probability. Jacob Bernouli applied this theory to the other branches of Mathematics. In ...
... 1933 Kolmogorov, a Russian Mathematician made it more applicable, by the axiomatic approach.

learn probablility-Random Experiments and Events

Let us learn some technical terms to introduce probability.

Random experiment or trial : An experiment which can be conducted repeatedly under similar conditions, and the result of a particular trial is not known, even though all the possible outcomes are known, is called a Random experiment of a trial.

Random event : All possible outcomes of a random experiment are called random events. For example, when an unbiased coin is tossed, the possible outcomes are Head and Tail. This experiment can be repeated any number of times and the result of every trial is unpredictable. Hence this experiment is called a random experiment. The random events are getting a Head (H) or a Tail (T). Similarly rolling a die is a random experiment and getting an even number or an odd number, or getting '6' as outcome is a random event.

Elementary event : A random event which corresponds to a single possible outcome of a random experiment is called an elementary event or a simple event.

Event : A combination of two or more elementary events is called an event. For example, when a symmetric die is thrown, getting 1,2,3,4,5,6 are elementary events getting an even number is an event, which is a combination of three elementary events, namely getting 2,4 or 6.

List of exhaustive events : The list of all possible, elementary events of a random experiment is known as the list of exhaustive events. For example, In the experiment of tossing a coin, the exhaustive events are {H,T}. In the experiment of tossing two coins, the exhaustive events are {(H,H) (H,T) (T,H) (T,T)}. H=Head, T= Tail.

Successes and Failures : The number of cases, which are favourable to a particular event of an experiment, are called successes and the remaining cases are called failures, with respect o that event. For example, when a die is rolled, getting 2,4 or 6 are successes for the event of getting an even number and getting 1,3 or 5 are failures.

Equally likely events : Events of a random experiment are said to be equally likely if they have equal chances to occur. For example in the experiment of tossing a coin, getting a Head and getting a Tail are equally likely events. In the experiment of a student appearing for an examination to pass the examination, to get a first class, to get the first-rank, are not equally likely events, because the chances are different for the above events.

Mutually Exclusive Events : Events of a random experiment are said to be mutually exclusive, if the happening of one event, prevents the happening of other events. For example, when two teams A and B are playing a game, the events, A, winning the game, and B, winning the game are mutually exclusive. When A,B,C,D are appearing for an examination, the event, A passing in the examination does not prevent the events of B, C or D, passing int he examination. Hence these events are not mutually exclusive,

learn probablility-Classical definition and Axiomatic approach

Let us learn Classical definition and Axiomatic approach in probability:

Classic Definition :

If there are n mutually exclusive and equally likely events in a random experiment, out of which, m events are favourable for a particular event E, then we define the probability of E, as

P(E) = m/n = Number of favourable cases with respect to E / Number of elementary events of the experiment.

This probability is otherwise known as probability of success of E.

Certain event , Impossible event : If E is any event in a random experiment and its probability is P(E), then we know that 0`