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Probability


Introduction

Probability is a measure of uncertainty of various phenomenon and to measure it. There are three approaches of probability i.e statistical approaches, classical approaches and Axiomatic approaches.

Probability is find on the basis of of observations and collected data in statistical approach, which  is not used in general.

Mathematical definition of Probability

Let the outcomes of an experiment consists of n exhaustive mutually exclusive and equally likely cases. Then the sample spaces S has n sample points. If an event A consist of m sample points, (0 ≤ m ≤ n). then the probability of event A, denoted by P(A) is defined to be m/n i.e. P(A) = m/n.

  • Let S = a1, a2 …………, an be the sample space
  • P(S ) = n/n = 1 corresponding to the certain event.
  • P(∅) = 0/n = 0 corresponding to the null event ∅ or impossible event.
  • If Ai = {ai}, i = 1, ……, n then Ai is the event corresponding to a single sample point ai. then P(Ai) = 1/n.
  • 0 ≤ P(A) ≤ 1

Some Basis Definition:

Deterministic Experiment

An experiment is an action or operation resulting in two or more outcomes is called an experiments. Deterministic experiments are those experiments which when repeated under identical conditions produce the same results or outcome are known as deterministic experiments. When experiments in science or engineering are repeated under identical conditions, we get almost the same result every time.

Random experiment

When  repeated under identical conditions do not produce the same outcome every time but the outcome in a trial is one of the several possible outcomes. Such experiments is known as a random or probabilistic experiment.

Sample Space

The set of all possible outcomes of an experiment is called the sample space. It is denoted by S. An element of S is called a sample point.

e.g. The sample space for the experiment of tossing a die is given by

S = {1, 2, 3, 4, 5, 6}

Event

Event in probability is the any subset of sample space in a event. Events can be classified into various types on the basis of elements. i.e.

Simple Event

An event is called  simple event if it is a singleton subset of the sample space S. e.g. In the experiment of tossing a coin twicely getting both head i.e. E = {H, H} 

Compound Events

In Simple terms, It is the joint occurrence of two or more simple events. i.e. If an event has more than one sample point, It is called a compound event. e.g In the experiment of tossing a coin thrice the events.
E = {HTT, THT, TTH} (Exactly one head appeared)
F = {HTT, THT, TTH, HHT, THH, HTH, HHH} (Atleast one head appeared)
G = {HTT, THT, TTH, TTT}  (Atmost one head appeared, etc)

Equally Likely Events

A number of simple events are said to be equally likely if there is no reason for one event to occur in preference to any other event. e.g. When an unbiased die is thrown, all the six faces 1,2,3,4,5,6 are equally likely to come up.

Exhaustive Events

All the possible outcomes taken together in which an experiment can result are said to be exhaustive or disjoint. e.g When a die is thrown, events 1,2,3,4,5,6 form an exhaustive set of events.

Mutually Exclusive or Disjoint Events

If two events cannot occur simultaneously, then they are mutually exclusive. If A and B are mutually exclusive, then A ∩ B = ∅.

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