Probability

D.A.V. Public School, Gurugram, Haryana

Chapter - ℘robability

-Manya Jain

X-E

Hey everyone, Today, we're going to understand probability, in just no time.

Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true.

In other words, The probability of an event is a number between 0 and 1, where, roughly speaking, 0 indicates the impossibility of the event and 1 indicates certainty.

The higher the probability of an event, the more likely it is that the event will occur. A simple example is the tossing of a fair (unbiased) coin.

Since the coin is fair, the two outcomes ("heads" and "tails") are both equally probable; the probability of "heads" equals the probability of "tails"; and since no other outcomes are possible, the probability of either "heads" or "tails" is 1/2 (which could also be written as 0.5 or 50%).

These concepts have been given an axiomatic mathematical formalization in probability theory, which is used widely in areas of study such as statistics, mathematics, science, finance, gambling, artificial intelligence, machine learning, computer science, game theory, and philosophy to, for example, draw inferences about the expected frequency of events.

Probability theory is also used to describe the underlying mechanics and regularities of complex systems.

Terminology of the probability theory

Experiment: An operation that can produce some well-defined outcomes, is called an Experiment.

Example: When we toss a coin, we know that either head or tail shows up. So, the operation of tossing a coin may be said to have two well-defined outcomes, namely, (a) heads showing up; and (b) tails showing up.

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Random Experiment: When we roll a die we are well aware of the fact that any of the numerals 1,2,3,4,5, or 6 may appear on the upper face but we cannot say that which exact number will show up.

Such an experiment in which all possible outcomes are known and the exact outcome cannot be predicted in advance is called a Random Experiment.

Sample Space: All the possible outcomes of an experiment as a whole, form the Sample Space.

Example: When we roll a die we can get any outcome from 1 to 6. All the possible numbers can appear on the upper face from the Sample Space(denoted by S). Hence, the Sample Space of a dice roll is S={1,2,3,4,5,6}

Outcome: Any possible result out of the Sample Space S for a Random Experiment is called an Outcome.

Example: When we roll a die, we might obtain 3 or toss a coin to obtain heads.

Event: Any subset of the Sample Space S is called an Event (denoted by E). When an outcome that belongs to the subset E occurs, it is said that an Event has occurred. Whereas, when an outcome that does not belong to subset E occurs, the Event has not occurred.

Example: Study the trial of throwing a die. Over here the Sample Space S={1,2,3,4,5,6}. Let E indicate the event of a number resembling less than 4.' Thus the Event E={1,2,3}. If digit1 appears, we say that Event E has befallen. Likewise, if the results are 2 or 3, we can say Event E has occurred since these outcomes belong to subset E.

Trial: By a trial, we mean performing a random experiment.

Example: (i) Tossing a fair coin, (ii) rolling an unbiased die^[4]

Interpretations

• When dealing with random and well-defined experiments in a purely theoretical setting (like tossing a fair coin), probabilities can be numerically described by the number of desired outcomes, divided by the total number of all outcomes. For example, tossing a fair coin twice will yield "head-head", "head-tail", "tail-head", and "tail-tail" outcomes. The probability of getting an outcome of "head-head" is 1 out of 4 outcomes, or, in numerical terms, 1/4, 0.25, or 25%. However, when it comes to practical application, there are two major competing categories of probability interpretations, whose adherents hold different views about the fundamental nature of probability:

Objectives

• Objectivists assign numbers to describe some objective or physical state of affairs. The most popular version of objective probability is frequentist probability, which claims that the probability of a random event denotes the relative frequency of occurrence of an experiment's outcome when the experiment is repeated indefinitely. This interpretation considers probability to be the relative frequency "in the long run" of outcomes.[5] A modification of this is propensity probability, which interprets probability as the tendency of some experiment to yield a certain outcome, even if it is performed only once.

Subjective

• Subjectivists assign numbers per subjective probability, that is, as a degree of belief.[6] The degree of belief has been interpreted as "the price at which you would buy or sell a bet that pays 1 unit of utility if E, 0 if not E."[7] The most popular version of subjective probability is Bayesian probability, which includes expert knowledge as well as experimental data to produce probabilities. The expert knowledge is represented by some (subjective) prior probability distribution. These data are incorporated in a likelihood function. The product of the prior and the likelihood, when normalized, results in

a posterior probability distribution that incorporates all the information known to date.[8] By Aumann's agreement theorem, Bayesian agents whose prior beliefs are similar will end up with similar posterior beliefs. However, sufficiently different priors can lead to different conclusions, regardless of how much information the agents share.[9]