 Probability in Statistics

Probability in Statistics

Probability is the study of random or unpredictable experiments, where it is helpful in investigating important features of random experiments. The origin of probability theory was based on the analysis of certain games of chance that was well known in 17th century. Many branches of science and engineering have found an extensive scope of application which makes probability as one of important studies.

Sample Space and Events

In probability theory, the set of all possible outcomes of an experiment, denoted as Ω is known as Sample Space. An element in Ω is called a Sample Point or a Sample Element. An Event A is a set of particular type of outcomes of an experiment in probability theory. It is a subset of a sample space.

A simple formula explains Probability Theory

Probability (P) = Number of Events/ Total Number of outcomes

A Venn diagram of an event B is the sample space and A is an event.

By the ratio of their areas, the probability of A is approximately 0.4.

Types of Sample Spaces:

There are 3 types of Sample Spaces in probability theory are listed and explained with illustrations such as,(i) finite, (ii) countable infinite and (iii) uncountable

Illustration1. Finite: A coin is tossed in air for three times and observed the sequence of heads (H) and tails (T) that appear. Here the sample space is Ω = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} consist of eight elements and this becomes an example of finite sample space.

Illustration 2. Countable infinite: A coin is tossed and counted until the outcome appears as heads. The sample space is Ω = {1, 2, 3, 4…Infinity}. Here Infinity refers to the case when a head never appears and so the coin is tossed an infinite number of times. This is an example of Countable Infinite.

Illustration 3. Uncountable: Inthis example, let’s take a pencil, head first and drop it in a rectangular box, note the point on the bottom of the box that the pencil first touches. Now here Ω consists of all points on the bottom of the box.

Properties of Probability Functions:

Here are the few common properties of a probability functions which are proved by using the definition of probability function.

Theorem 1If ø is the empty event, then P (ø) = 0

Proof:If A is an event, then A and ø are disjoint. Again A = A ø.

P (A) = P (A ø)

= P (A) + P (ø)

Hence,

P (ø) = P (A) – P (A)

= 0

Theorem 2 If A is any event in sample space Ω then P (Ă) = 1- P (A).

Proof:If A is any event, then A and Ă are disjoint and Ω = A Ă

1 = P (Ω)

= P (A U Ă)

= P (A) + P (Ă)

Hence,

P (Ă) = 1- P (A)

The two events A and B are called mutually exclusive if they are disjoint, that is, A B =ø

In other words A and B cannot occur simultaneously. A sample space, where each sample point has the same probability is called an equiprobable sample space.

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