Probability of Randomly Choosing a Prime Number

Probability of Randomly Choosing a Prime Number Number theory is a branch of mathematics  that concerns itself with the set of integers. We restrict ourselves somewhat by doing this as we do not directly study other numbers, such as irrationals. However, other types of real numbers are used. In addition to this, the subject of probability has many connections and intersections with number theory. One of these connections has to do with the distribution of prime numbers. More specifically we may ask, what is the probability that a randomly chosen integer from 1 to x is a prime number? Assumptions and Definitions As with any mathematics problem, it is important to understand not only what assumptions are being made, but also the definitions of all of the key terms in the problem. For this problem we are considering the positive integers, meaning the whole numbers 1, 2, 3, . . . up to some number x. We are randomly choosing one of these numbers, meaning that all x of them are equally likely to be chosen. We are trying to determine the probability that a prime number is chosen. Thus we need to understand the definition of a prime number. A prime number is a positive integer that has exactly two factors. This means that the only divisors of prime numbers are one and the number itself. So 2,3 and 5 are primes, but 4, 8 and 12 are not prime. We note that because there must be two factors in a prime number, the number 1 is not prime. Solution  for Low Numbers The solution to this problem is straightforward for low numbers x. All that we need to do is simply count the numbers of primes that are less than or equal to x. We divide the number of primes less than or equal to x by the number x. For example, to find the probability that a prime is selected from 1 to 10 requires us to divide the number of primes from 1 to 10 by 10. The numbers 2, 3, 5, 7 are prime, so the probability that a prime is selected is 4/10 40%. The probability that a prime is selected from 1 to 50 can be found in a similar way. The primes that are less than 50 are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43 and 47. There are 15 primes less than or equal to 50. Thus the probability that a prime is selected at random is 15/50 30%. This process can be carried out by simply counting primes as long as we have a list of primes. For example, there are 25 primes less than or equal to 100. (Thus the probability that a randomly chosen number from 1 to 100 is prime is 25/100 25%.) However, if we do not have a list of primes, it could be computationally daunting to determine the set of prime numbers that are less than or equal to a given number x. The Prime Number Theorem If you do not have a count of the number of primes that are less than or equal to x, then there is an alternate way to solve this problem. The solution involves a mathematical result known as the prime number theorem. This is a statement about the overall distribution of the primes and can be used to approximate the probability that we are trying to determine. The prime number theorem states that there are approximately x / ln(x) prime numbers that are less than or equal to x. Here ln(x) denotes the natural logarithm of x, or in other words the logarithm with a base of the number e. As the value of x increases the approximation improves, in the sense that we see a decrease in the relative error between the number of primes less than x and the expression x / ln(x). Application of the Prime Number Theorem We can use the result of the prime number theorem to solve the problem we are trying to address. We know by the prime number theorem that there are approximately x / ln(x) prime numbers that are less than or equal to x. Furthermore, there are a total of x positive integers less than or equal to x. Therefore the probability that a randomly selected number in this range is prime is (x / ln(x) ) /x 1 / ln(x). Example We can now use this result to approximate the probability of randomly selecting a prime number out of the first billion integers. We calculate the natural logarithm of a billion and see that ln(1,000,000,000) is approximately 20.7 and 1/ln(1,000,000,000) is approximately 0.0483. Thus we have about a 4.83% probability of randomly choosing a prime number out of the first billion integers.

Verb Agreement

7 Classes of Noun/Verb Agreement 7 Classes of Noun/Verb Agreement 7 Classes of Noun/Verb Agreement By Mark Nichol Below youll find seven classes of noun/verb agreement you need to understand. 1. Indefinite Pronouns Most indefinite pronouns correspond to singular verbs: â€Å"Someone has left her plate on the table.† â€Å"Everybody is entitled to his or her opinion.† â€Å"Each boy is responsible for his actions.† To confirm, test for the proper verb form by writing a simple sentence in which is follows the pertinent pronoun: â€Å"Someone is missing† (not â€Å"Someone are missing†). The proper verb form for some indefinite pronouns depends on the reference: â€Å"All of the soup is gone. (Soup is a single entity.) â€Å"Some of the comments are favorable. (The comments are counted as separate entities.) The indefinite pronoun none can be singular or plural depending on the context: â€Å"None of the jewels are missing.† (None of the components of the whole entity in question are missing.) â€Å"None of the jewelry is missing.† (Not one part of the whole entity is missing.) 2. Conjunctive Phrases The simple conjunction and cannot necessarily be replaced by such phrases as â€Å"along with,† â€Å"as well as,† and â€Å"together with†: â€Å"The doe along with its fawns is resting in the meadow.† (This sentence is correct, however, if â€Å"along with its fawns† is inserted into the sentence â€Å"The doe is resting in the meadow,† which requires bracketing commas. The same is true of the other phrases.) 3. â€Å"Either/Or† and â€Å"Neither/Nor† Neither and either refer to two compared or associated objects as individual entities and are therefore usually employed with singular verbs: â€Å"Neither she nor I are ready for that.† â€Å"Either option will work for me.† Informally, however, an exception is made in such constructions as â€Å"Are either of you ready?† In â€Å"either/or† and â€Å"neither/nor† constructions with a mixture of singular and plural nouns, the verb form is determined by whether the closest noun is singular or plural: â€Å"Either the captain or one of the lieutenants are leading the patrol.† â€Å"Neither the students nor the teacher remembers hearing anything.† However, because the plural noun and the singular verb still clash in the second sentence despite their lack of proximity, it is advisable to construct the sentence so that the singular pronoun precedes the plural one: â€Å"Neither the teacher nor the students remember hearing anything.† 4. Positive and Negative Subjects in Combination A subject consisting of positive and negative sentiments that differ in singular and plural form should be followed by a verb that corresponds with the positive element: â€Å"The delivery of the speech, not its contents, is the issue.† As with â€Å"either/or† and â€Å"neither/nor† constructions, perhaps it is best to rearrange the sentence so that the singular noun is in proximity with the verb: â€Å"It is not the contents of the speech, but its delivery, that is at issue.† 5. Expletives In sentences beginning with such expletives as here and there, the actual subject, which follows the verb, determines the verb form: â€Å"There is a word for that.† â€Å"Here are several choices.† 6. Plural Nouns for Single Objects Plural nouns that name single objects, such as scissors and pants, are matched with plural verbs unless the phrase â€Å"pair of† precedes the noun; in that case, pair is the subject: â€Å"Scissors are dangerous.† â€Å"A pair of scissors is required for this activity.† Some other nouns ending in s are also singular in meaning: â€Å"The mumps is a disease you don’t hear much about anymore.† Meanwhile, others stand for a single thing but call for a plural verb: â€Å"Thanks are in order.† 7. Fractional Phrases Phrases referring to a mathematical portion may, depending on the context, be singular or plural: â€Å"A small percentage of the employees are opposed.† â€Å"A large percentage of the cargo was damaged.† â€Å"Three-fourths of the land is forested.† â€Å"One-third of the trees are oaks.† Numbers expressed as part of a mathematical operation are linked with a plural verb, but the outcome of a computation is expressed as a single entity: â€Å"Ten and six are added together to equal sixteen.† â€Å"Ten minus six is four.† Want to improve your English in five minutes a day? Get a subscription and start receiving our writing tips and exercises daily! Keep learning! Browse the Grammar category, check our popular posts, or choose a related post below:The Letter "Z" Will Be Removed from the English AlphabetWhen Is a Question Not a Question?

Psychology Essay Example | Topics and Well Written Essays - 1250 words - 11

Psychology - Essay Example Considering the argument brought forth by then article on then relationship between the hippocampus and the acquisition of independent memories, which do not rely on the hippocampus systems. It further questions the interference of the hippocampus with the retrieval of perfectly normal hippocampus dependent memory, which has been established in non-hippocampus systems. The authors go further to make working assumptions relating to disruption of memory that is not stored in hippocampus systems during the resumption of functionality by the hippocampus. This is backed with evidence from experiments on the temporary inhibition of the hippocampus, and the resulting inconclusive results on the outcome. The main aim of the article because of the experiments conducted is revealed that the loss of memory is affiliated failure in the establishment of a lasting fear memory. In order to prove its findings in relation to loss of memory and the hippocampus; the results of experiments conducted on animals are included and the inferences revealed. As a result, it is found that inactivation or loss of functionality of the hippocampus does not result in loss of memory, retrograde amnesia, the same way that the hippocampus can interfere with retrieval of memory. In order to arrive at their conclusions, the authors of the article conducted an experiment, and based it on what is already known to cause amnesia or failure in the retrieval of memory. As such, the sponsoring institution, University of Lethbridge, approved their experiment. The approval means that the experiment was done according to set guidelines in order to acquire accurate results for their hypothesis. With this in mind, the sample of animals used in the experiment involved the use of rats that were housed in a standard laboratory and facing all similar treatment and conditions. However, in the

Teenage Abortion Research Paper Example | Topics and Well Written Essays - 1500 words

Teenage Abortion - Research Paper Example In olden days, ancient people used natural abortion practices when cases involving health complications and emotional immaturity to save a mother’s life (Soranau’s Gynecology) (Bull.  Acad.  Natl.  Med.  171  (8):1027–32). In 21st century, with the increasing urbanization and industrialization, the number of girls, who visit the clinic for abortion, is extremely high. According to the National Abortion Organization, nearly one million teenagers become pregnant each year and 78 percent of these teenage pregnancies are unintended. A practice, which is used to save a mother’s life on olden days, is now became big concern as teenagers started thronging into the clinics because of unwanted pregnancies. There are several reasons behind girls choosing abortion. It is the responsibility of all educated to empower teenage girls to stay away from the activities that helps unwanted pregnancies. As teenagers who are carrying an unwanted pregnancy are visitin g clinics without any parental notification for abortion, empowering teenagers regarding the implications of unwanted pregnancy will prevent teenage abortions. Teen Abortion has always been most controversial subject in nature. I feel, this is attributable to the well known fact that majority of the people have got different set of beliefs that are inherited from the religion and society. Some people say teenage abortion is wrong but some enunciate it is right under some circumstances. As per the statistics of National Abortion Federation, 80 out of 100 Americans have sex before 20 years of age. The average age of a girl, who loses her virginity is around 17 years. This is the main reason behind the teen abortions in America. There are a very few cases where girls under 15 years of age become pregnant due to forceful sex and they will not have any option but to choose abortion. Teen abortion was legal in all states before 1991. The law in a

Leadership Development and Self Awareness Research Paper

Leadership Development and Self Awareness - Research Paper Example This paper highlights  the Army’s existing Army curriculum pushes the most multifaceted subjects to the final stages of a soldier’s learning career. As a result, only some soldiers get an opportunity of expanding their intellectual limits through creative and critical thinking before their field grade experience. Doing business, in such a manner, denies junior level soldiers the opportunity to develop the necessary skills required to excel in the leadership area. The Army has to promote superior learning opportunities to a young soldier’s career in order to develop the essential leadership skills needed to manage the 21st century warfare scenery.From the discussion it is clear that  the education and evaluation methodologies should be modified to replicate the complexities of the current operating environment to encourage a change in the career progression of leaders equipped for the 21st century warfare. In order to achieve this, the Army should adjust its l eader development systems to identify and promote strategic thinking before it is too late.  The Army education system plays a vital role in creating strategic leaders for the logical challenges they might encounter against an equally resolute enemy. However, it does not stress much on the essential factors of creating able leaders who will lead the Army for many generations to come.  Strategic leadership education should offer soldiers options regarding how they might proceed decisively to overcome adversaries that know no limitations or regulations of war.

Queen Elizabeth I Essay Example for Free

Queen Elizabeth I Essay The play A Midsummer Nights Dream was written by William Shakespeare in 1594. From the theme and context of the play, I can conclude that it was probably written for a wedding. It would be similar to the epithalamion written at weddings at this time, but a more elaborate version. Many of the lines rhyme and it all has the feel of poetry: some beautiful and some seeming supernatural and spell-like. It canvasses all aspects of marriage through plot, characters and symbolism, touching on both light and dark areas. The tragic side is shown more through references that actually spelling it out; Pyramus and Thisby is one example of a union which ended in catastrophe. However, despite this, the main idea of marriage is perceived as harmonious; all is shown in a comic frame, with a humorous touch and a happy ending. This would be an appropriate feel for a wedding play because it gives the concept of marriage an idyllic feel, telling the newlyweds what they want to hear: that marriage is a good idea. If you search the play in depth, you will find that not only is it about the union of people in a marriage, but also about the unity of opposing forces in nature and bringing harmony from chaos. It is also, in an even broader sense, about the Elizabethan ideals about God and the universe. To clarify exactly what these ideals were and how A Midsummer Nights Dream defines them, I must start at the beginning with the story of Adam and Eve. Elizabethans believed in everything fitting into its natural place. After studying the sky, they thought the universe was made up of spheres, one containing the stars, one containing planets and so on. They thought the Earth was at the centre of the universe; the spheres they believed in were a way of dealing with the fact that different bodies in the sky moved in different ways. It was thought that these spheres made the world harmonious, but humans couldnt see the true beauty of it because of Adam and Eve. The legend goes that Eve was tempted by the serpent and ate the forbidden fruit in the Garden of Eden. This, apparently, caused the whole human race to fall from Gods grace. Now, the relevance of this to A Midsummer Nights Dream and the theme of marriage is that when this happened, it meant man kind could not the music of the spheres; they were only told by God that everything had its place. Even the evil in the world has a reason for being there. Humans cannot see quite how everything fits in because of their sin. They do not fully understand why some things happened because they cant see the big picture. This was illustrated by the Great Chain of Being. God came at the top of this chain, with angels under Him. Then came nature (represented by Oberon and Titania), people (shown in all the human characters), and below them animals, plants and rocks. Humans came between animals and Angels, a relationship personified by Bottom. He is a typical human, not very bright or beautiful, but just a normal working class person. He then has a link with both the animal and angel world; he is turned into an ass and he has a relationship with Titania. This establishes the link between animals, angles and humans, and also tells you about the human nature. They are always trying to find something greater that themselves (for example fairies or angels) but to the greater beings, they must look as animals do to us far beneath them and having little or no brains. Also exhibiting the same way of thinking was the fact that men came above women in the category of humans. If you were to sum up the play with one idea, this is the single idea addressed most eloquently. Many events in A Midsummer Nights Dream suggest the superiority of men, through the media of imagery, plot and characters, which will be tackled later. The Elizabethan idea of the cosmos is also addressed by the union of opposites connected with marriage. Their idea of the universe is that everything joins together perfectly and everything has its place. Things which may not seem to have a purpose will have been created by God for a reason; we just dont know what that is. The prime example is the joining of a man and a woman in marriage. It may seem to make more sense to join two women or two men, as these will have more in common, but this play is trying to say a little of both the male and female persona is needed to make the match united. They are like opposite forces, but marriage brings them together. It is as is they have only half of the qualities needed to bring harmony and they must be brought together in marriage to make a whole. The same is also being expressed through the union of other opposite forces. There are many contrasting pairs which link hand in hand in this play: night and day; sleeping and waking; supernatural and mortal; discord and harmony; passion and reason; and ultimately female and male. These pairs are all the exact opposite of each other; and yet Shakespeare is presenting the notion that we need each half of the pair to make the world harmonious.

Exposing Boundaries in Wilsons Fences :: Wilson Fences Essays

Exposing Boundaries in Fences Fences is a play that deals with boundaries that hold people back and the trials and tribulations of those who try or wish to cross them. The characters are African-Americans in a time before the civil rights movement, living in an industrial city. The main character, Troy Manxson, is a talented baseball player who never had the chance to let his talent shine, with restrictions on race and his time in jail as the main obstacles that held him back. He is now hard working and loves his family. However, he tends to exaggerate and has his faults, most prevalent a wandering eye when it comes to women. His wife, Rose, is younger than him and loyal, but she may not have known about all of his faults when she married him. At the beginning of the play, Troy has a son from a previous marriage, Lyons, and a son with Rose, Cory. Also appearing are Bono, Troy’s drinking buddy, and Gabriel, his brother. All of the characters are "fenced in," by various barriers. Troy is working in a job where African Americans can get the lowest and most difficult tasks. On the home front, he has responsibilities to his family. Rose has chosen life with Troy as an alternative to "a succession of abusive men and their babies, a life of partying, or the Church." Troy’s son, Lyons, is supposedly a musician but is going nowhere. Cory has potential but has his dream of playing college football extinguished by both protective and jealous Troy. The characters must deal with hardships of daily life, racial discrimination, straining relationships with each other, and the feeling that this is all their lives are: somewhat of a confined space with no escape; fenced in. Troy’s brother Gabriel, although minor, is important to the play for many reasons. The most important is that while Gabriel perhaps has highest and most impassable fence around him, he is the only one who ignores it; he is not bounded, at least not in his own mind. Gabriel is seven years younger than Troy. They were both in a large family with a frustrated and abusive father. Gabe is the only family Troy speaks with now. Injured in World War II, Gabriel had part of his head blown away. He now has a metal plate and is confused and somewhat delusional.