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    Probability Theory


    Introduction:

    The word ‘probability’ is commonly used in our day-to-day life. For example “Probably the rain comes tomorrow” or “team A have chances to win the certain match”etc.All these words like chances, probably,possibility etc convey the same sense that there is no certainty for happening of such event.

    The theory of probability has its origin in games where there is uncertainty of happening like throwing a die, tossing a coin etc.Cardon(1501-1576), an Italian Mathematician, was first person to writinga book “Game of Chances”.

    Probability Defined:

    The probability of such a given event is an expression of occurrence of that event.The probability of an event ranges from (0-1).Probability is zero when there is no chances of occurring such event and probability is one when is there certainty of occurring such event.

    Importance of Concept of Probability:

    The theory of probability helps to treat and solve weighty problems.It helps to take managerial decisions on planning and controlling.It becomes a most important tool for all types of formal studies which involves uncertainty.It should be noted that theory of probability is introduced not only for solving scientific investigations but also for solving problems of our day-to-day life.


    Calculation of Probability:

    Experiment and Event:The word experiment refers to processes that result in different possible outcomes or observations.If in an experiment all outcomes are known in advance and none of them is predicted with certainty,then such an experiment is called random experiment.

    Events are generally denoted with capital letters A,B,C etc.An event which may or may not occur while performing a random experiment is known as random event.

    Independent and Dependent Events: Independent events are those events where outcome of one event is not affected by the other event.Dependent events are those in which occurrence and non-occurrence of event will affect the result of other events in other trial.

    Mutually Exclusive Events or Cases: Two events are said to be mutually exclusive or disjoint when both events cannot happen simultaneously in a single trial.For ex. Head and tail both cannot occur in an single trial of tossing a coin.

    Equally Likely Events: Events are said to be equally likely when one event does not occur more often than the others.

    Exhaustive Events: Exhaustive events are those events where totality includes all possible outcomes of a random experiment.For ex. while tossing a coin, the possible outcomes are head and tail, so total number of cases is 2.

    Complementary Events: Let there be two events X and Y .X is a complementary event of Y(and vice versa) if X and Y are mutually exclusive and exhaustive.For ex. when die is thrown, occurrence of an even number(2,4,6) and odd number (1,3,5) are complementary events.


    Theorems of Probability:

    1. Addition Theorem: Addition theorem states that if two events X and Y are mutually exclusive, probability of occurrence of either X or Y is:

    P(X OR Y) = P(X) + P(Y)

    2. Multipletheorems: Multiple theorem states that if two events X and Y are independent, the probability that they are both will occur is equal to:

    P(X and Y) = P(X) + P(Y).

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