Subject: Mathematics

Introduction:

The theory of probability originated in the middle of the seventeenth century. Its origin came in the context of correspondence of Pascal and Fermat while dealing with a problem of a game of chance posed by a gambler Chevalier de mere. Such game’s present situations where that under given conditions more than one result are possible and the occurrence of a particular result is unpredictable and remains uncertain. As for example consider the tossing of a coin. When it falls , the result may be either a head or a tail. We know what the possible results are, but we are not quite sure which one of these result will actually turn up. A similar situation arises when we roll a dice whose six faces are marked with 1, 2, 3, 4, 5, 6.

A numerical measure of such uncertainty is known as the probability. Sometimes we use “most likely”, probably ,almost certain or chance as the synonym of the word probability by saying almost it is certain that it may rain today”, “probably you may pass +2 this year”.

Some basic terms used in probability :

Before giving the definition of probability , we should learn about some terms used in the definition of probability:

**Experiment :**The processes, which when performed result from different possible outcomes or cases, are known as an experiment. While performing an experiment repeatedly under the same condition if the result obtained is not unique but may be any one the possible outcomes, then the experiment is known as a random experiment, The set of all possible outcomes is called sample space of the random experiment.**Trial and Event:**Performing of a random experiment is called a trial and outcome or a Combination of outcomes of an experiment is termed as an Event. Drawing a card from a deck of 52 cards is a trial or an experiment and getting any one of the cards is an event. An event is said to be a sure event if its occurrence is certain and an event which can never occur is known as an impossible event. We use the term “success” whenever an event of an experiment under consideration, takes place and failure whenever it does not. There are two types of events : (i) Simple event (ii) Compound event. An event is said to be simple if it relates to the occurrence or non-occurrence of a single event. On the other hand, when two or more events occur in connection with each other, their simultaneous occurrence is called a compound event. Getting a head or a tail when a coin is tossed is a simple event. The event that the “sum of two numbers shown on the upper faces of two dies when the two dies are thrown simultaneously is a compound event.**Exhaustive cases :**The number of cases which include all possible outcomes of a random experiment is said to be the exhaustive cases for the experiment. In throwing a dice, the turning up of 1, 2, 3, 4, 5 and 6 marked in a dice made cases. Thus the total number of exhaustive cases in throwing a dice is 6.**Equally likely cases:**While performing Experiments if any one of the possible outcomes may occur but no one case can be expected to occur more than the other then the cases are said to be equally likely. If a dice is rolled, any one of the six numbers may turn up. So, there are 6 equally likely cases in throwing a dice.**Mutually exclusive cases (events):**Two or more events are said to be mutually exclusive if their simultaneous occurrence is not possible. If a coin is tossed either head or tail will occur, so head and tail are two mutually exclusive events.**Favourable cases :**The cases or the outcomes of a random experiment which entail the happening of an event are known as the cases favourable to that event. In throwing a dice , the cases favourable to “getting an odd number” are 3.**Independent cases (Events) :**Two events are said to be independent if the occurrence of one event does not affect the occurrence of the other. If a coin and dice are thrown, the turning head up in a coin will not affect the getting 1 on the dice.

Knowledge of these terms is really important when solving problems related to probability.

Taken reference from

( Basic mathematics Grade XII and A foundation of Mathematics Volume II and Wikipedia.com )

- The terms Experiment, Trial and events, Exhaustive cases, favourable cases etc should be well known about.

- It includes every relationship which established among the people.
- There can be more than one community in a society. Community smaller than society.
- It is a network of social relationships which cannot see or touched.
- common interests and common objectives are not necessary for society.

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