033. Рrobability distributions. Principles of Probability

Statisticians often take samples for the purpose of gaining knowledge regarding the world around them. On the basis of these samples, they can frequently estimate the probability that specific evens will occur. Probability is the numerical likelihood of the occurrence of an uncertain event.

Experiments, Outcomes and Sets

The probability of an event is measured by values between 0 and 1. The probability of a certainty is 1. The probability of impossibility is 0, i. e.

P (certain event) = 1,

P (impossible event) = 0.

The process that produces the event is called an Experiment. An Experiment is well-defined action leading to a single, well-defined result. That result is called the Outcome. A Set is any collection of objects. The objects in a set are its Elements or Members. The set of all possible outcomes for an experiment is the Sample space (SS). For Example, the sample space for the experiment of flipping a coin is

SS = (heads, tails).

The occurrence of either a head or a tail is a certainty. So, the probability that a head or a tail occurs equals 1. That is,

P (head or tail) = 1.

Properties of probability.

1. The probability that some uncertain event will occur is between 0 and 1. If Ei, is any given event, then it can be said that

0 ≤ P (Ei) ≤ 1.

For Example, (1) the probability that the sun will rise tomorrow is very high – quite close to 1; (2) the probability a student will pass this course without studying it, at the other extreme, close to zero.

2. If Ei is an event representing some element in a sample space, then

ΣP(Ei) = 1.

There are only three generally accepted ways to approach probability: (1) the relative frequency (or posterior) approach, (2) the subjective approach, and (3) the classical (or a priori) approach.

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