This blog site offers a quick overview of statistics issues with illustrations to help the audience understand and appreciate the value, importance, and applications of statistics in everyday life. In today's society, it is common knowledge that in order to understand about something, you must first gather data. The skill of learning from data is known as statistics. It is involved with the gathering of information, its subsequent description, and analysis, which frequently leads to conclusions.
We can use some postulates or axioms to define this probability. Axioms or postulates are forms of basic assumptions that we make to characterize anything that are logically coherent and non-overlapping. Typically, such postulates are derived by considering feasible features that we would like to see in the defined thing. There is no way to prove or disprove these fundamental assumptions. The probability or chance of occurrence of the event A will be defined by the following three postulates.
(i) 0 ≤ P(A) ≤ 1 or the probability of an event is a number between 0 and 1, both
inclusive;
(ii) P(S) = 1 or the probability of the sure event is 1;
(iii) P(A1 ∪ A2 ∪ ⋯) = P(A1) + P(A2) + ⋯ whenever A1,A2,... are mutually exclusive [The events may be finite or countably infinite in number]
The characteristic 0≤P(A)≤1 corresponds to the requirement that a relative frequency be between 0 and 1. The fact that an outcome from the sample space happens on every trial of an experiment results in the property P(S) = 1.
On this topic, your comments/suggestions are highly appreciated.
In a random experiment, randomness is linked with the possible outcomes, not with the experiment's conduct. Any activity or procedure whose outcome is uncertain is considered an experiment. Although the term "experiment" usually conjures up images of planned or meticulously controlled laboratory testing, we use it here in a far broader sense. A random experiment is one in which the probable outcomes of interest, or the items you're seeking for, are not predictable or predefined in any way. Tossing a coin once or several times, selecting a card or cards from a deck, weighing a loaf of bread, determining the commuting time from home to work on a particular morning, obtaining blood types from a group of people, or measuring the compressive strengths of different steel beams are all examples of experiments that might be of interest.
The Sample space is the collection of all possible outcomes indicated by the letter S. Let A be a part of the collection of outcomes in S; that is, A is a subset of S denoted by A⊂S. Given an outcome space S, let A be a part of the collection of outcomes in S. Then A is referred to as an event. When a random experiment is done and the result is in A, we say event A has occurred.
Examining a single weld to discover if it is faulty is one of these experiments. S= {N, D} denotes the sample space for this experiment, where N denotes not defective, D denotes defective, and braces are used to enclose the members of a set. Another experiment might be tossing a thumbtack and recording whether it landed point up or point down, with sample space S = {U, D}, and monitoring the gender of the next kid born at the local hospital, with S = {M, F}.
Some C++ programs produced at a corporation compile on the first attempt, but others do not (a compiler is a program that converts source code, in this case C++ programs, into machine language so that programs can be executed). Assume that an experiment consists of selecting and compiling C++ programs one by one at this address until you find one that compiles on the first try. S (for success) denotes a program that compiles on the first run, while F (for failure) denotes one that does not (for failure). Although it's unlikely, one possible conclusion of this experiment is that the first five (or ten, or twenty, or...) are Fs, and the following one is a S.
Any subset A of the sample space S of a random experiment is referred to as an event or a random event. We're talking about a random event described in a sample space or a subset of a sample space when we talk about an event in the future. An event is a collection (subset) of outcomes contained within the sample space S. It is simple if an event has exactly one outcome; it is compound if it has multiple outcomes. When an experiment is carried out, a specific event A is said to have occurred if the experimental result is contained in A. In general, only one simple event will happen at a time, while multiple compound events will happen at the same time.
When a sample space has n individual components, for as when a coin is thrown twice and there are 4 elements or 4 points in the sample space S, the elementary events are the singleton elements in S.
There are an endless number of simple events in the sample space for the program compilation experiment because there are an infinite number of outcomes. Compound events include
A = {S, FS, FFS} = the event that at most three programs are examined.
E = {FS, FFFS, FFFFFS,…} = the event that an even number of programs are examined.
On this topic, your comments/suggestions are highly appreciated.
Probability as a branch of mathematics has a long history, dating back over 300 years, when it was first applied to situations concerning games of chance. Many books are dedicated solely to probability, but our goal is to focus on the aspects of the subject that have the greatest immediate influence on statistical inference difficulties.
The present mathematical theory of probability can be traced back to attempts by Gerolamo Cardano in the sixteenth century and Pierre de Fermat and Blaise Pascal in the seventeenth century to examine games of chance (for example the "problem of points"). Their motivation stemmed from an issue regarding games of chance provided by the chevalier de Méré, a notably philosophical gambler. When a game of chance is stopped, De Méré inquires about the right allocation of stakes. Let's say two players, X and Y, are playing a three-point game with 32 pistoles each, and they're interrupted when X has two points and Y has one.
Pascal thought Fermat's solution was too complicated, so he recommended solving the problem in terms of the quantity now known as "expectation," rather than probability.
Games of chance like this one served as model problems for the theory of chances in its early stages, and they are still used in textbooks today. Pascal's posthumous work on the "arithmetic triangle," which is now associated with his name (see binomial theorem), demonstrated how to calculate numbers of combinations and combine them to solve basic gambling difficulties.
Girolamo Cardano, an Italian mathematician, physician, and gambler, estimated chances for games of chance by counting up equally likely occurrences more than a century ago. However, his small work was not published until 1663, by which time the elements of the theory of chances were well known among European mathematicians.
Probability is the study of calculating the chances of something happening. At its most basic level, it is concerned with the roll of a dice or the fall of cards in a game. Probability, on the other hand, is critical to both science and everyday life. It's used for a variety of things, like weather forecasting and figuring out how much your insurance premiums would cost. Probability is the scientific study of randomness and uncertainty. The study of probability gives methods for calculating the chances, or likelihoods, of various outcomes in any situation where one of a number of possible outcomes could occur.
In both written and spoken contexts, the language of probability is frequently utilized in an informal manner. For example, “It is likely that the Dow Jones average will increase by the end of the year,” or “It is likely that the Dow Jones average will climb by the end of the year.” “The incumbent has a 50–50 likelihood of seeking reelection,” says the expert. “It's likely that at least one component of that course will be given next year,” says the professor. “The odds favor a rapid resolution of the strike,” and “at least 20,000 concert tickets are expected to be sold.”
