# Probability Theory In Mathematics Sample Paper

### Probability Theory

In mathematics, probability theory is used as a vast field that studies the laws associated with random events. There are no unequivocal views as to when precisely the foundations of probability theory came into being. According to one version, it originated thousands of years ago when censuses were conducted; from this paradigm, probability has sociological origins (Porter, 2022). From another point of view, probability theory is thought to have been solidified in the seventeenth century in an attempt to study gambling. The leading names in this history are the Frenchmen Pierre de Fermat and Blaise Pascal. Both Pascal and de Fermat attempted to solve the problem of the Chevalier de Mairet, who asked about a fair division of bets for two players in the event that the game was interrupted. Both researchers formulated the essential vision of modern probability theory in an attempt to find a solution.

In probability theory, the prevailing principle is the actual probability of the occurrence of a random event. To calculate this probability, a formula is used in which the number of successful cases is divided by the total number of trials. For example, if a coin was tossed fifty times and only twenty of them produced tails, then the probability of tails was 0.4 or 40 percent. The advantage of this principle is the relative simplicity of calculating the probability of a random event occurring. Among other things, it can be used to calculate the probabilities of two events at once, sequential or parallel, as well as to calculate the inverse probability. However, the probability should not be taken literally, which is its main disadvantage. For example, a tails probability of 40% does not mean that in a hundred trials, precisely 40 of them will be tails. At the same time, if an event has a 50% chance of happening, such as the probability of winning the lottery, that does not mean that of the two tickets purchased, the second will necessarily be the winning one. In other words, the main drawback of probability theory is the need for a deep understanding of the principle in order to calculate and interpret it correctly.

### The Fundamental Nature of Probability

In probability theory research, there are two diametrically opposed approaches to understanding what probability really is. The first, the objective one, is based purely on actual observations and does not use assumptions (Kenton, 2022). For example, the calculations of the probability of tails or winning the lottery in the last paragraph are examples of objective probability because it is based on facts and numbers. In contrast, the subjective probability is based on personal experience and unscientific assumptions (Kenton, 2022). An example of this paradigm is, for example, an individual’s personal belief that he or she does not get into a traffic accident because he or she has not been in one before; that is, the individual believes this probability is zero. Or, for example, an individual has noticed that a public company’s stock has been rising for the past two days, so he expects its probability of rising on the third day to be high, although in a dynamic market, this need not necessarily be the case.

### Calculations

The probability that a randomly selected American has never had an HIV test is:

Accordingly, the probability that a randomly selected American in the sample of 212,556 has never been tested for HIV is 63.4%. It is worth clarifying that the conclusions only work for the sample because no statistical test was performed to extrapolate the results to the general population. At the same time, the proportion of Americans aged 18 to 44 who have never been tested:

That is, there is a possibility that 26.5% of 212,556 Americans were between the ages of 18 and 44 and had never had a diagnostic test.

### References

Kenton, W. (2022). Objective probability definition. Investopedia. Web.

Porter, T. M. (2022). Probability and statistics. Britannica. Web.

## Amazon: Information System Analysis

The top online shopping destination in the world is Amazon.com. The organization oversees businesses in practically every economic sector and sells products from various manufacturers. The potential for a large audience to shop from home ushered in a new era in marketing history. The company’s CEO, Jeff Bezos, has identified customer service as the most critical component of Amazon’s operation. Most customers purchase in physical stores, but Amazon’s market share is still increasing. An effective information system is a foundation for building an e-commerce site like Amazon, and it should be thoroughly investigated because it determines the company’s financial success.

The system’s users are from the middle class and of various ages. These could include secretaries, economists and lawyers, physicians and educators, engineers, and highly trained personnel. They are the ones who select and buy daily stuff like clothing, children’s items, furniture, home appliances, and technology online. These people range in age from youth to adults with a reliable income source. Conversely, the users can be young individuals with subcultures and values that elders find incomprehensible. However, the target market relies on product quality, expedited shipment, and website usability.

To satisfy its billions of users—representatives of various nations with distinctive tastes—Amazon has made its navigation as easy and convenient as possible. Users can search for items in the field at the top of the page after joining up. The customer inputs product-related data before clicking the “search” button to get a list of particular products. Another feature allows the system to select possible results as the user type keywords and presents them in a drop-down menu. The option for goods sorting is located in the right corner after entering terms in the search area. It suggests categorizing products in the catalog based on many factors, including low and high prices and vice versa, newness, and a rating. Selecting products involves deciding on them based on relevant criteria, such as department, brand, color, size, and price. As a result, it aids in increasing the offer even though there is not a perfect match with the item description.

In the top right corner, there is the traditional cart button. The online store must have this feature so customers can add one or more things to their shopping carts (Pappas et al., 2017). By clicking the “Add to Cart” button, selected items are added to the cart. The product name is shown on the page that follows the jump page. The item photo is posted to the product card because the product type is not often evident from the title. The user can modify the contents by removing extraneous items. Additionally, information on delivery and payment is provided in the purchasing cart.

The review feature is one of the most critical aspects of the item description page. Because Amazon receives hundreds of user reviews, the website analyzes all feedback scores and generates an evaluation list using decryption (Lee et al., 2018). Customers may see how many of each product were purchased and what rating it received. Customers can rate the reviews on Amazon by indicating whether they were helpful, and the most popular ones are shown at the top (Lee et al., 2018). People frequently check reviews before looking at product attributes, as evidenced by practice (Smith & Linden, 2017). Thus, the user does not need to look over all product details.

Talking about advantages, the fact that this online purchasing platform saves the customer’s time should be primarily mentioned. As long as users have access to the Network, they can make purchases from the online store whenever it is convenient for them and from any location. Customers do not have to travel to department stores because the search engine offers them an extensive list of websites with products. Choosing a few items and placing an order will take a few minutes.

Additionally, online shopping saves money because clients may buy things for a lot less, thanks to various discounts and discount schemes. Even without it, Amazon products are always less priced because an online retailer does not need to spend extra money on showrooms and pricey spaces in shopping malls. Typically, warehouse properties are outside of the city. Therefore, the rent is less expensive (Pappas et al., 2017). There are a lot of discounts and seasonal sales on the platform.

By entering keywords, the Amazon interface makes it simple for customers to find the products they are looking for so that the search engine can provide relevant results. The user can provide the requirements for the item, and the store will quickly display the appropriate goods. Since there is no longer a need to wait for sales assistants, the buyer is always given all the pertinent information about the acquired item, saving time (Pappas et al., 2017). On the Internet, the customer can rapidly analyze products, look at user evaluations, and read ratings from other customers. As a result, customers can choose the best purchase option.

The platform’s primary goal is to make users as comfortable as possible to encourage them to make a purchase. A helpful search engine can easily find a product by its related terms. Many product catalogs are provided to make it easier for customers to traverse the website. The business is rightfully regarded as prominent in the e-commerce market. Amazon now invests millions of dollars in improving the user interface constantly. The Internet is an environment that needs continuous adjustments. Amazon pays attention to the website’s mobile version. After reaching a certain point, the business started producing its goods and acquiring new services, which broadened its range of potential markets.

### References

Lee, S. G., Trimi, S., & Yang, C. G. (2018). Perceived usefulness factors of online reviews: a study of Amazon.com. Journal of computer information systems, 58(4), 344-352.

Pappas, I. O., Kourouthanassis, P. E., Giannakos, M. N., & Lekakos, G. (2017). The interplay of online shopping motivations and experiential factors on personalized e-commerce: A complexity theory approach. Telematics and Informatics, 34(5), 730-742.

Smith, B., & Linden, G. (2017). Two decades of recommender systems at Amazon.com. Ieee internet computing, 21(3), 12-18.

## Is Shakespearean Hamlet’s Madness Feigned?

### Introduction

Hamlet is a tragedy play written by Williams Shakespeare; it is set in Denmark during unsettle times. It was revealed to the young prince that his father has been murdered by his uncle King Claudius, so Prince Hamlet seeks revenge by acting insane. Hamlet’s comments about madness, his elaborate actions and his madness compared to Ophelia’s reveals Hamlet’s madness to be feigned.

Through the main character’s comments, a reader can note that Hamlet’s madness is a mask he puts on himself. For example, the last scene of the first act illustrates the most deliberate and firm decision of the protagonist to pretend to be insane after meeting with the Ghost: “How strange or odd soe’er I bear myself, as I perchance hereafter shall think meet to put an antic disposition on” (Shakespeare, 1599/2003, 1.5.170-172). However, Hamlet does not always put on the mask of a madman; he conveys his “transformations” in such a way: “I am but mad north-north-west: when the wind is southerly I know a hawk from a handsaw” (Shakespeare, 1599/2003, 2.2.378-379). Moreover, it becomes especially evident about the hero’s acting when the prince abruptly cuts off his speech, seeing the approach of the king and the entire court camarilla: “They are coming to the play; I must be idle” (Shakespeare, 1599/2003, 3.2.90). Therefore, due to a deep analysis of Hamlet’s sayings, one can remark that the prince did not go mad and did not lose his mind because his frenzy is imaginary.

### Elaborate Actions

Hamlet is a master of complex intrigues, deceptions, and manipulations when interacting with people. Thus, Hamlet longs for revenge, but his mind does not allow him to act blindly; he pretends to be insane to be able to provoke frankness, speaking in riddles as if out of place, leading to topics that would never have surfaced: “May be the devil, and the devil hath power” (Shakespeare, 1599/2003, 2.2.628). Hamlet is circumspect, self-critical, and distrustful; he needs to check everything and then strike a blow, as in the case when he arranges an “investigative experiment” in the role of a crazy: “O good Horatio, I’ll take the ghosts word for a thousand pound” (Shakespeare, 1599/2003, 3.2.260-261). Nevertheless, even while in the presence of the king and the entire court, Hamlet uses a specific tactical move, continuing to play an insane, occasionally regaining consciousness: “Was ’t Hamlet wronged Laertes? Never Hamlet […] Who does it, then? His madness.” (Shakespeare, 1599/2003, 5.2.203-207). Consequently, one should assume that Hamlet is not mad because his every step is weighed and considered, and in the play, there is only a game of the mind, not madness.