What is probability ?

Aug 09, 2022, 16:45 IST

History on probability theory

In 1654, a French gambler Chevalier De-Mere, approached the well known 17th century French philosopher and mathematician Blaise Pascal (1623–1662) regarding some dice problems. Pascal discussed this problem with another French mathematician, Pierre De-Fermat
(1601–1665). This work of Pascal and Fermat laid the foundation of the probability theory.

The first book on the subject was published in 1663. The title of the book was “Book on Games of Chance”. Significant contributions in the field were also made by mathematicians
J. Bernoulli (1654 – 1705), P. Laplace ((1749 – 1827), A.A. Markov (1856 – 1922) and A.N. Kolmogorov (in the 20^th century).

DIFFERENT APPROACHES

There are following three types of approaches to theory of probability :

(i) Experimental approach, empirical approach or observed required approach.

(ii) Classical approach.

(iii) Axiomatic approach.

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In this chapter, we will study only the Empirical approach.

IMPORTANT TERMS AND DEFINITIONS

Experiment : An activity which gives some well-defined outcomes is called experiment “Tossing a coin” gives either head (H) or tail (T), is an experiment.

Random experiment : It is an experiment, in which we know about all the possible outcomes but not sure about a specific outcome.

For e.g. in throwing a dice, possible outcomes are 1, 2, 3, 4, 5, 6 but we are not sure that the no on dice is ‘4’.

Event : The possible outcomes of a trial are called events. e.g., when a die is rolled, showing the number 1 or 2 or 3 or 4 or 5 or 6 is an event.

Sure event : When all the outcomes of a random experiment favour an event, the event is called a sure event and its empirical probability is 1.

Impossible event : When no outcome of a random experiment favours an event, the event is called an impossible event and its empirical probability is 0.

Sample space : The collection of all possible outcomes in an experiment is called sample space.

Favourable events : The cases, which ensure the occurrence of an event are called favourable cases to that event.

Bayesian probability is an interpretation of probability suggested by Bayesian theory, which holds that the concept of probability are often defined because the degree to which an individual believes a proposition. Bayesian theory also suggests that Bayes' theorem are often used as a rule to infer or update the degree of belief in light of latest information.On the Bayesian interpretation, the theorems of probability relate to the rationality of partial belief within the way that the theorems of logic are traditionally seen to relate to the rationality of full belief. Bayesian probability is meant to live the degree of belief a private has in an uncertain proposition, and is therein respect subjective. Some people that call themselves Bayesians don't accept this subjectivity. Advocates of logical (or objective epistemic) probability, like Harold Jeffreys, Rudolf Carnap, Richard Threlkeld Cox and Edwin Jaynes, hope to codify techniques whereby any two persons having an equivalent information relevant to the reality of an uncertain proposition would calculate an equivalent probability. Such probabilities aren't relative to the person but to the epistemic situation, and thus lie somewhere between subjective and objective. However, the methods proposed are controversial. Some regard Bayesian inference as an application of the methodology because updating probabilities through Bayesian inference requires one to start out with initial beliefs about different hypotheses, to gather new information (for example, by conducting an experiment), then to regulate the first beliefs within the light of the new information. Adjusting original beliefs could mean (coming closer to) accepting or rejecting the first hypotheses

Some events seem to have certainity about their outcome; while few are certain not to happen. There are others, which, with regard to their outcome, vary between the two extreme situations referred to the above. In a rough way, a measure of the happening or non-happening of an event may be given by the term Probability. The word probability and the word chance are synonymous and may be taken, in this context, to be indistinguishable

Table of Content probability

·      Concept of Probability

·      Definition of Probability

·      Formula of Probabilty

·      Types of Probability

·      Basic Theories of probability

·      Types of Experiment

·      What is Event

·      Types of Events

·      Conditional Probability

·      Baye’s Theorem or Inverse Probability

·      Baye’s Theorem Proof

·      Baye’s Theorem Formula

·      Binomial Probability Distribution

·      Mean and Variance of the Binomial Distribution

·      Use of Multinomial Expansion

Concept of probability in set Theoretic Language

When an experiment is repeated under similar conditions and it doesn't give an equivalent result whenever but may end in anybody of the several possible outcomes, the experiment is named an attempt and therefore the outcomes are called cases. the amount of times the experiment is repeated is named the amount of trials

Probability Definition in Math

Probability may be a measure of the likelihood of an occasion to occur. Many events can't be predicted with total certainty. we will predict only the prospect of an occasion to occur i.e. how likely they're to happen, using it. Probability can home in from 0 to 1, where 0 means the event to be an impossible one and 1 indicates a particular event. Probability for sophistication 10 is a crucial topic for the scholars which explains all the essential concepts of this subject. The probability of all the events during a sample space adds up to 1.For example, once we toss a coin, either we get Head OR Tail, only two possible outcomes are possible (H, T). But if we toss two coins within the air, there might be three possibilities of events to occur, like both the coins show heads or both show tails or one shows heads and one tail, i.e.(H, H), (H, T),(T, T)

Formula for Probability

The probability formula is defined because the likelihood of an event to happen is capable the ratio of the quantity of favourable outcomes and thus the entire number of outcomes. This is often often the essential formula. But there are some more formulas for various situations or events.

Probability of event P(E)= Number of favourable outcomes/Total Number of outcomes

Solved Examples

1.     There are 8 pillows in a bed, 4 are red, 2 are yellow and 1 is blue. What is the probability of picking a red pillow

Ans: The probability is equal to the number of red pillows in the bed divided by the total number of pillows, i.e. 4/8 = 1/2.

2.     There is a container full of coloured bottles, red, blue, green and orange. Some of the bottles are picked out and displaced. Ram did this 100 times and got the following results

No. of blue bottles picked out: 30

No. of red bottles: 20

No. of green bottles: 45

No. of orange bottles: 5

a.     What is the probability that Ram will pick a green bottle

Ans: For every 100 bottles picked out, 45 are green.

Therefore, P(green) = 45/100 = 0.45

b.     What is the probability that Ram will pick a blue bottle?

Ans: For every 100  bottles picked out, 30 are blue .

P(blue) = 30/100 = 0.3

Types of Probability

There are three types of probabilities:

·       Theoretical Probability

·       Experimental Probability

·       Axiomatic Probability

Theoretical Probability

It is supported the possible chances of something to happen. The theoretical probability is especially supported the reasoning behind probability. for instance, if a coin is tossed, the theoretical probability of getting a head are going to be ½

Experimental Probability

It is supported the idea of the observations of an experiment. The experimental probability are often calculated supported the amount of possible outcomes by the entire number of trials. for instance, if a coin is tossed 10 times and heads is recorded 6 times then, the experimental probability for heads is 6/10 or, 3/5.

Axiomatic Probability

In axiomatic probability, a group of rules or axioms are set which applies to all or any types. These axioms are set by Kolmogorov and are referred to as Kolmogorov’s three axioms. With the axiomatic approach to probability, the probabilities of occurrence or non-occurrence of the events are often quantified. The axiomatic probability lesson covers this idea intimately with Kolmogorov’s three rules (axioms) along side various examples.Conditional Probability is that the likelihood of an occasion or outcome occurring supported the occurrence of a previous event or outcome.