## Probability

Probability is defined as the assessment of the possible outcomes of an experiment whose outcome is “random”. In this definition, the term “outcome” is not exclusive to outcome of an experiment, but also to an “explanatory variables” if that is not fixed. For example, in a drug study, experimenter decides the drug dose, but if the subjects are chosen randomly then the experimenter does not have a control over the age, and hence it is not fixed and can be classified as “outcome” under probability theory.

With experiments, there are possible outcomes, and the collection of the possible outcomes is called the sample space (S). These sample spaces should be unique, and normally very exhaustive, and possibly as simple as possible.

Any subset of sample space is called events. For example, in an dice experiment , there are six possible outcomes (1u,2u,3u,4u,5u,6u- with each term representing a possibility), a subset is {1u}, {1u,6u}, and so on. In probability theory, we compute the chance than an one of the above stated event will occur taking in consideration the probabilities of the elementary outcomes in the sample space.

However, we see that the outcome of most experiment, even in the case of the above mentioned dice experiment, the results are not numbers but situation like 1u, 2u and so on. Thus, for mostly convenience reasons these outcomes are mapped or represented by integer or real numbers, like 1 to 6 for dice experiment, instead of 1u to 6u. Technically, these numbers are called a random variable. These outcomes are commonly represented as X, Y,Z .

**Source:**

http://www.stat.cmu.edu/~hseltman/309/Book