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.

President Dwight D. Eisenhower And The Vietnam War Essay

On the 7th of April 1954, President Dwight D. Eisenhower gave one historic press conference [1]. In that conference, he announced his so called â€Å"Domino Theory†. It was a substantial aspect of the US involvement in the Vietnam War, involvement in terms of its military support and methods used in Vietnam. Even though it was greatly enhanced by other factors, such as the Gulf of Tonkin Incident, it was the trigger, and therefore the source of all US involvement in the Vietnam War. The main rationale of the theory was that if one country fell down to communism, all the surrounding countries would do the same, hence why it was called the ‘Domino’ Theory. Eisenhower thought that Vietnam’s fall to communism would lead to consequential communist uprisings in neighboring countries, such as Laos, Cambodia and Thailand. He also that that it could possibly extend as far as India, Japan, the Philippines, Indonesia, and Australia and New Zealand -[1]. This gave The US enough motivation, and more importantly, justification as to why they greatly increased military presence in Vietnam. Eisenhower said, â€Å"The possible consequences of the loss [of Indochina] are just incalculable to the free world.† [2]. Even though the theory was imprecise, and only Laos and Cambodia joined communism after the US lost the war, it was significant as the foundation of the US involvement in the conflict. The United States started increasing their involvement in Vietnam, partly due to the French failure toShow MoreRelatedThe War Of The Vietnam War983 Words   |  4 Pages â€Æ' The Vietnam War, a war involving the invigorating power of the superpower America and its democracy and freedom versus the almighty Russia and its tight fist of communism. This infamous war would go on to be one of the most controversial wars to date, but people rarely tend to have the knowledge behind everything that took place in the Vietnam war. 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The U.S. was at ends with the Soviet Union, and this tension manifest ed itself into the population through the fear of nuclear missiles, and communism, and thanks to President Dwight D. Eisenhower, and his administration, many of these fears were resolved, or at least minimized. A very influential fear on the American PeopleRead MoreAmerican Involvement in the Vietnam War1078 Words   |  5 Pagesinternational affairs. In this particular case communism in Vietnam was the flame that leered American bugs in, not knowing that they would be brutally burned by communism in the end. From 1953 to 1961, all the initial decisions involving Vietnam were made by President Dwight D. Eisenhower, who once served as the Supreme Commander of the Allied forces in Europe as well as the first Supreme Commander of NATO. Thus, Eisenhower was very knowledgeable about war issues and was prepared to tackle pending conflicts

A Glimpse at Bernard Maybeck Architectural Works Free Essays

Just imagine looking at a full scale drawing? Well that was just one of Maybeck’s ways of studying his design. His works are blend of possibility and actuality and the reason why the blending of unrelated styles became a success in his works (Matthews, 2008). His love on experimenting designs is very much seen in his Buena Vista Way studio because it looks like a laboratory and at the same time a house. We will write a custom essay sample on A Glimpse at Bernard Maybeck Architectural Works or any similar topic only for you Order Now This visionary Architect and described as the Gothic man of the 20th Century believes that every architectural problem requires an original solution. He addresses the housing crisis during the World War 1 by making generic floor plans called small houses or the workers houses. Most of his house design works infused the garden with the house structure. He never uses one type of specifications for the finishes but leave it to his Clients to make the decision. However, most of his works are made up of wood and timber (see photo of a community hall on page 3) for which reason why some of his drawings did not came into reality because of the scarcity of lumber and failure on the part of the timber supplier. Design Studio Small House The gauge of success for a famous work is its revival when Bill Buchanan an architect whom he mentored enlivens his heritage in Oregon by adopting Maybeck standards for the design of 1000 dwellings on a steep hill on a 630 Acre land parcel near Harbor Hill. Bill Buchanan believes that his mentor ideas are still in proportion of meeting the requirements for affordable housing. Maybeck’s flexibility 80 years ago can now be answered by the existing technology when at the time of his mentor’s life is quite difficult to undertake for example, the inclusion of garden on rooftops which makes a plant difficult to tend and the bedroom oriented towards the sky when ninety years ago is inconvenient due to the lack of technology with regards to thermal protection for buildings. Keith Pepper Brooklyn city council member also believes in the potential of the revival of the famous Californian Architect by persuading that good designs are an economic potential (Week, 2000). Recently the Oregon’s Department of Transportation rerouted part of Highway 101 which will allow part of downtown to return to Maybeck’s original plan and provision of funding for the reconstruction of the design. How to cite A Glimpse at Bernard Maybeck Architectural Works, Papers