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Probability theory is study of chance, it is the mathematical framework that makes it possible to study chance events in a logical manner. Probability of an event is represented through a number which suggests how likely that event is to occur, and this number is always between 0 and 1, where 0 suggests impossibility and 1 suggests absolute certainty. The chance of an event occurring is associated with the situation in which that event is to take place. Some essential concepts of basic probability theory are:

**RANDOM VARIABLES HOMEWORK HELP:** It is a variable whose value determine the outcome of a random event. Possible values of a random variable indicate possible results of an experiment which has not been performed yet. The results suggested might be that of an objective random process or a subject one where complete knowledge about the phenomenon is not available.

**PROBABILITY DISTRIBUTION HOMEWORK HELP:** This is the mathematical function which indicates what is the likelihood of occurrence of each of the different outcomes. It takes sample space as an input and offers probability as an output, where sample space is the set of all possible outcomes of the random event that is taking place.

**STOCHASTIC PROCESS HOMEWORK HELP:** A mathematical object, it is said to belong the family of random variables. It represents the values of a system which randomly changes over time, like growth of bacterial population or movement of gas.

Probability is the mathematical study of chance and statistics applies that study to real world events; for instance, when a company predicts it future growth curve or risks making use of statistics, it is depending on probability. Some concepts which are integral to probability and statistics are:

**CONDITIONAL PROBABILITY:** It is the measure of the chance occurrence of an event when another event has already occurred (either by presumption or evidence). Therefore, probability of second event’s outcomes would be influenced by the first event.

**INDEPENDENCE AND DEPENDENCE:** Two events are said to be independent of one another when the probability of occurrence of one event in no way affects the probability of occurrence of another. For instance, the probability that it might rain in no way affects who might win a board game being played indoors. Similarly, dependent events would be ones that affect each other’s outcomes.

**VARIANCE:** Variance of a random variable measures how concentrated is the distribution of the variable around its mean

**PROBABILISTIC CONDITIONING:** Probabilistic conditioning suggests how presumed beliefs are to be revised based on the new findings of probabilistic experiments.

Probability can be seen in common everyday phenomenon around us, like when a coin is tossed. A coin which has two sides has the chance of falling on either one of its faces when tossed, and therefore, probability of the coin landing either on head or tail is 50-50%

A dice has six sides, and when such a dice is rolled, the probability that it will land on one specific number (let’s assume 2) is 1/6. This is because it has six different numbers, and chance of each number is to be divided by the total numbers. Similarly, if two dices are rolled simultaneously, the chances that they will both land on the same number (say 2) becomes 1/36.

A deck contains 52 cards with 4 of them being aces. Therefore, when a card is drawn, chance for an ace’s occurrence in 4/52 and chance for the occurrence of any card which is not an ace is 48/52 (52-4). Chance of the occurrence of a specific digit card of a specific colour and group when drawn from 52 cards would be 1/52 (Black Spade number 4 for instance).

Bayesian probability interprets basic probability in which expectations regarding the frequency of a phenomenon or probability is regarded as a reasonable one, representing a state of knowledge. Therefore, according to Bayesian theory, when a hypothesis is being tested, it is assigned a probability before the experiment is conducted. Under frequentist inference, however, the hypothesis would be tested without assigning any probability to it in advance. Bayesian probability falls under the category of evidential probability, where the hypothesis is tested with a prior probability, and depending on the results it is updated to the posterior probability. A given set of formulae and procedure is followed in Bayesian interpretation. Bayesian probability can be objective as well as subjective; in objective interpretation it is an extension of logic where the probability quantifies the logical expectations that every individual with the common knowledge is likely to share, which should be in harmony with the rules of Bayesian statistics. In case of subjective interpretation, the probability is related to personal beliefs and not everyone with the same knowledge might share the view. Construction and interpretation of prior probability changes with objective or subjective interpretations.

Quantum probability theory was developed in 1980s with an aim to justify the mathematical foundation of quantum theory and its statistical interpretation. There are different existing approaches for solving problems in quantum mechanics, and each of these approaches have corresponding probability theories. These two fields of study, quantum mechanics and probability, complement one another. Probability has recently contributed to the study of quantum mechanics by providing a dynamic solution to the quantum measurement problem, and it has achieved the solution by offering constructive models of quantum observation that can settle well-known paradoxes of quantum mechanics.

Simulation theory is a hypothesis that suggests that all reality including the universe, planet Earth and all life forms on it is a simulation, or a computer simulation to be precise. This hypothesis presumes the development of simulated reality, which would be an artificial reality, that would seem like a real one to its inhabitants, much as one sees in video games. Astrophysicists like Neil DeGrasse Tyson has suggested that the probability of this simulation theory being true is more than 50-50%. Recently, Elon Musk commented on simulation theory, weighing in favour of the probability of our reality being a simulation. Probabilistic analysis of the probability that we might indeed be living in a simulation was first presented by Nick Bostrom in his 2001 paper, ‘Are you living in a computer simulation?’ Like all probabilistic claims, his hypothesis was based on certain assumptions, such as, if one could learn to model the mind down to the finest detail, it would be possible to create artificial minds which are capable of thoughts in the same manner that we are. In addition to this, it was assumed that if the entire world could be simulated in fine detail, and if this detail could be fed to the artificial mind through sensory inputs, the artificial mind would not be able to demarcate the world as a simulation unless this knowledge were fed to them by the creator of the simulation. Thereafter, Boston uses formal probability theory to support his claim. By reviewing the equations and propositions made by Boston, it is deduced that there are two probabilities, first one being that simulations are created in very small numbers by the creators or it is almost certain that we ourselves are part of the simulation.

Probability theory regarding simulation is a largescale extension informed by probability theory related to Artificial Intelligence (AI). Probability theory finds great application in machine learning, and especially in artificial intelligence, where outcomes need to be predicted or decisions need to be made. Softmax functions in computer science limit function outcomes to values between 0 or 1. Also referred to as squashing functions, these functions are effective in the algorithm process of assigning probability values to outcomes. Neural networks become capable of making better decisions owing to these assigned values, and they often mark the final step in the function of neural networks.

While AI is only a subset of machine learning, probability informs many other aspects of it. In supervised machine learning, the objective is to learn from labelled data, such as gaining knowledge like for input X an output Y is to be desired. This aspect can produce many results, one of them being medical diagnosis, that is to say, when information (X) regarding physical symptoms are fed to the said machine, it comes up with a certain diagnosis of a disease (Y), and the use of probability here also offers scope for uncertainty since there is always a chance that the diagnosis might be wrong. In similar ways, probability and machine learning can be combined in different fields to generate geospatial maps, to detect if a file is corrupted, or to predict future prices of the stock of a company when information about the company’s characteristics and details about past and present market conditions are provided. Probability theory combined with machine learning, therefore, has utility in a large number of fields.

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