Economics Term Paper About Probability Theory (Term Paper Sample)

PROBABILITY THEORY
Name
Institution
Probability Theory
Introduction
The term probability theory refers to mathematical discipline whose main concern is the execution of the random chances. In this branch, a prediction of an event happening is made. In ideal situation, the actual expected outcome could not be accurately pre-determined. However, a range of tall the potential outcomes may be drawn and the outcome will definitely be one of the many expected outcomes. Indeed, the science of probability is based on the chance where an event may have very many chances of occurring. However, the actual one is only determined through the chance. The events may be several and the chances of them happening or a certain order of events may be determined from the precise knowledge of probability. This forms the most basic usage of the probability. There are only two potential expectations from an even, which are one or zero. A zero expectation means that the event does not happen totally while 1 refers to certain expectation of the event happening. If analysis is conducted using probability of the events, then the data is referred to as statistics (Shafer, 2016). If the percentage criterion is used to map the probability space, then a range of 0% to 100% is used. In this regard, use of probability is a statistical device or model, which differs from the other statistical tools only by the concept of chance. This phenomenon has practical application, which may not only involve figures but a representation of ideal chances in the life.
A company may have the need to determine the chances of their products selling after a certain production process are improved. A prediction on the chances that a hypothesis is true or false therefore gives a basic outlook on the probability scope. Almost all scientific disciplines use the probability in the description of the concepts. For instance, biological researches use it in the determination of the potential outcome from the random researches they may be conducting on their scientific quest for knowledge. In probability, the term experiment refers to the basis process leading to the outcome. This may include the rolling of a die, tossing a coin or a practical event like giving genetic variance (Barlow & Proschan, 2015). The other commonly used term in this chapter, include sample space. This is the entire options or alternatives of the expected outcomes of an event. It must be exhaustive enough to be complete probability space. A good example is the sample space for a doll, which will be 1, 2, 3, 4, 5, 6. The term event could be defined as the single subset of a all potential outcomes from an experiment. This study will therefore give concrete approach to the probability theory and explore the broad history, which it traces. Moreover, basic concepts entailed in the [probability with relevant examples have been put forward. The life application will also be subjected to scrutiny to show the significance of the concept to the society.
Strand 1: History
The probability theory began as a tool that was used to predict the chances of winning a game back in the 17th century. This has been today developed to accommodate the statistical data analysis and prediction of the chance. This includes the use of experiments to investigate the outcomes. The sample spaces are also used to express the number of all potential outcomes of the event. Some of these experiments included the coin tossing experiments as well as die rolling. The first records of probability were written in 1550 and once again, in 1654, both times it was over the probability of rolling dices in gambling games. French mathematicians, Blaise Pascal and Pierre de Fermat are credited with creating probability theory that resulted from gambling disputes. In 1654, there was a dispute over a gambling problem with dices; this dispute had several correspondences between them (Durrett, 2014). It was also associated with stochastic processes and relative frequency dealing with this element of randomness and chance. That a specific outcome cannot be 100% guaranteed. The fundamental law of large number: that when done a large amount of times, an outcome can be predicted.
The early forms of probability theory entailed description and explanation of discrete experiments and events and often used a variety of methods, which were used in combination. These were independent of each other and had independent outcomes. As time went by, the probability approach extensively extended to integrate the analysis and other complex variables, which had continuous nature. These are the pioneer versions, which have led to the modern probability theory. This was founded by Andrew Nikolaevich. He came up with this notion after a careful integration of the sample space, which had been brought out by Richard Von Mises. The other critical component was the measure theory, which he used in conjunction with axiom space as at 1933. Following this advancement, the axiomatic basis became a ground and stereotype for the modern probability, which was later made to have various alternatives. These other alternatives include the abject adoption of finite, which trace its founder as Bruno Finetti and it had been modified to exclude the countable addition capacity.
Strand 2: Mathematics
In probability, we draw a clear boundary between the sample space from all the anticipated outcomes. These are usually characterized by exhaustive approach and the mutually exclusive events. An exhaustive criterion confers that at least an event will meet one expectation in its outcome. On the other hand, mutually exclusive events often have only a single anticipated outcome and there are no chances of having more results. For example, one is either alive or dead and cannot be both. A coin had either the head of tail and cannot exhibit both characteristics at the same time. Before an event happens, the possible outcomes may be several but once the actual event occurs, they are referred to as realized outcome. If a die is tossed, there are chances that any of the six faces may land (Jeffreys, 2013). However, none of the faces has better chance than failing. Therefore, the sample point is six since all faces have equal chances. In this case, the event will be defined as mutually exclusive since it cannot land on more than one face. It may be as well regarded as exhaustive owing to the fact that only one face will be witnessed to fall facing downward. Characteristics of probability therefore include range, surety, as well as countable additively.
Continuous Probability Distribution
Continuous probability distributions entail a randomized figure with a continuous nature of the variable. This therefore demonstrates uniqueness from the discrete probability due to the assumption of a zero value, lacks the capability to be drawn to a table and it can only be expressed in form of an equation. The values are often defined by the capacity to be countable and occur in a certain range of values. Through the equation, a probability density function is arrived at and it is often known to satisfy a randomized variable. In this case, Y may be used as X function to arrive at an equation;
Y=k(x)
This implies that the value that is assigned to y can only ne either equal or greater than 0 for any value given to x. Using the above formula to draw a curve, the diagram that results has the area below it summing to 1. A graph drawn using the probability equation of y=1 is a straight line parallel to the y axis. Considering a range of 0 to 1, then y=0 at any other point..

y = 1

A graph which indicate a probability density equation as y=1-0.5x and whose x values range 0 and 1, would yield to a graph of this nature.
It is therefore area under the curve always give the probability of the event.
A calculation may be induced to investigate that a man’s weight is between 160 and 170.
The shaded part in the graph indicates the range of the 160 to 170 and these accounts to 1.0.
Discrete Probability Distributions
This category is uniquely defined by the composition of the mass function. These often entail the binomial figures, Bernoulli distribution as well as the Poisson distribution.
The summation of all values of discrete probability is always 1.If two coins are tossed, the discrete probability would be given by: SS = HH,TH,HT,TTHH,TH,HT,TT.
If the discrete values for x is 0, 1, 2, then  xx = 0, P(Y)P(Y) = 1414 xx = 1, P(Y)P(Y) = 2424 xx = 2, P(Y)P(Y) = 1414summation of the three events gives 1414 + 2424 + 1414 = 1.General formulas:
∑eP(X=e)∑eP(X=e)
Central limit theorem
This concept helps in the determination of whether a circular variable conform to the normal variable distribution.
The formula used is;
1n−−√∑i=1nXimod2π1n∑i=1nXimod2π
Strand 3: Real-World Applications
The banking sector has been identified to have wide application of the probability concept in the determination of the credit worthiness of a person. It is a common phenomenon in insurance: the risk rate of insuring a customer. The probability that a claim will or will not be made Moreover, this have been adopted by the insurance companies to assess the risk factor to insure. For those risks, which are very high, the insurance company may decide to request very high premium. Moreover, a rare case would have low amounts on the premium. Some risk factors may not be insured by the bank and they may include those events, which occur often. A car accident would be more risky than the flood case on a highland. This is because, it may be close to impossible for a loss to be incurred after a flood, which has been witnessed, and sweeps even the mountain slopes.
Probabilities are applied to ...