How to Calculate Backgammon Probabilities

How to Calculate Backgammon Probabilities Backgammon is a game that employs the use of two standard dice.   The dice used in this game are six-sided cubes, and the faces of a die have one, two, three, four, five or six pips. During a turn in backgammon a player may move his or her checkers or draughts according to the numbers shown on the dice. The numbers rolled can be split between two checkers, or they can be totaled and used for a single checker. For example, when a 4 and a 5 are rolled, a player has two options: he may move one checker four spaces and another one five spaces, or one checker can be moved a total of nine spaces. To formulate strategies in backgammon it is helpful to know some basic probabilities. Since a player can use one or two dice to move a particular checker, any calculation of probabilities will keep this in mind. For our backgammon probabilities, we will answer the question, â€Å"When we roll two dice, what is the probability of rolling the number n as either a sum of two dice, or on at least one of the two dice?† Calculation of the Probabilities For a single die that is not loaded, each side is equally likely to land face up. A single die forms a uniform sample space. There are a total of six outcomes, corresponding to each of the integers from 1 to 6. Thus each number has a probability of 1/6 of occurring. When we roll two dice, each die is independent of the other. If we keep track of the order of what number occurs on each of the dice, then there are a total of 6 x 6 36 equally likely outcomes. Thus 36 is the denominator for all of our probabilities and any particular outcome of two dice has a probability of 1/36. Rolling At Least One of a Number The probability of rolling two dice and getting at least one of a number from 1 to 6 is straightforward to calculate. If we wish to determine the probability of rolling at least one 2 with two dice, we need to know how many of the 36 possible outcomes include at least one 2. The ways of doing this are: (1, 2), (2, 2), (3, 2), (4, 2), (5, 2), (6, 2), (2, 1), (2, 3), (2, 4), (2, 5), (2, 6) Thus there are 11 ways to roll at least one 2 with two dice, and the probability of rolling at least one 2 with two dice is 11/36. There is nothing special about 2 in the preceding discussion. For any given number n from 1 to 6: There are five ways to roll exactly one of that number on the first die.There are five ways to roll exactly one of that number on the second die.There is one way to roll that number on both dice. Therefore there are 11 ways to roll at least one n from 1 to 6 using two dice. The probability of this occurring is 11/36. Rolling a Particular Sum Any number from two to 12 can be obtained as the sum of two dice. The probabilities for two dice are slightly more difficult to calculate. Since there are different ways to reach these sums, they do not form a uniform sample space. For instance, there are three ways to roll a sum of four: (1, 3), (2, 2), (3, 1), but only two ways to roll a sum of 11: (5, 6), (6, 5). The probability of rolling a sum of a particular number is as follows: The probability of rolling a sum of two is 1/36.The probability of rolling a sum of three is 2/36.The probability of rolling a sum of four is 3/36.The probability of rolling a sum of five is 4/36.The probability of rolling a sum of six is 5/36.The probability of rolling a sum of seven is 6/36.The probability of rolling a sum of eight is 5/36.The probability of rolling a sum of nine is 4/36.The probability of rolling a sum of ten is 3/36.The probability of rolling a sum of eleven is 2/36.The probability of rolling a sum of twelve is 1/36. Backgammon Probabilities At long last we have everything we need to calculate probabilities for backgammon. Rolling at least one of a number is mutually exclusive from rolling this number as a sum of two dice. Thus we can use the addition rule to add the probabilities together for obtaining any number from 2 to 6. For example, the probability of rolling at least one 6 out of two dice is 11/36. Rolling a 6 as a sum of two dice is 5/36. The probability of rolling at least one 6 or rolling a six as a sum of two dice is 11/36 5/36 16/36. Other probabilities can be calculated in a similar manner.

The Difference Between Hispanic and Latino

The Difference Between Hispanic and Latino Hispanic and Latino are often used interchangeably though they actually mean two different things. Hispanic refers to people who speak Spanish or are descended from Spanish-speaking populations, while Latino refers to people who are from or descended from people from Latin  America. In todays United States, these terms are often thought of as racial categories and are often used to describe race, in the way that we also use white, black, and Asian. However, the populations they describe are actually composed of various racial groups, so using them as racial categories is inaccurate. They work more accurately as descriptors of ethnicity, but even that is a stretch given the diversity of peoples they represent. That said, they are important as identities for many people and communities, and they are used by the government to study the population, by law enforcement to study crime and punishment, and by researchers of many disciplines to study social, economic, and political trends, as well as social problems.  For these reasons, its important to understand what they mean literally, how they are used by the state in formal ways, and how those ways sometimes differ from how people use them socially. What Hispanic Means and Where  It Came From In a literal sense, Hispanic refers to people who speak Spanish or who are descended from Spanish speaking lineage. This English word evolved from the Latin word  Hispanicus, which is reported to have been used to refer to people living in Hispania - the Iberian Peninsula in todays Spain -   during the Roman Empire. Since Hispanic refers to what language people speak or that their ancestors spoke, it refers to an element of culture. This means that, as an identity category, it is closest to the definition of ethnicity,  which groups people based on a shared common culture. However, people of many different ethnicities can identify as Hispanic, so its actually more broad than ethnicity. Consider that people who originate from Mexico, the Dominican Republic, and Puerto Rico will have come from very different cultural backgrounds, excepting their language and possibly their religion. Because of this, many people considered Hispanic today equate their ethnicity with their or their ancestors country of origin, or with an ethnic group within this country. Reports indicate that it came into use by the United States government during Richard Nixons presidency, which spanned 1968‒1974. It first appeared on the U.S. Census in 1980, as a question prompting the Census taker to determine whether or not the person was of Spanish/Hispanic origin.  Hispanic is most commonly used in the eastern U.S., including Florida and Texas. People of all different races identify as Hispanic, including white people. In todays Census people self-report their answers and have the option to choose whether or not they are of Hispanic descent. Because the Census Bureau recognizes that Hispanic is a term that describes ethnicity and not race, people can self-report a variety of racial categories as well as Hispanic origin when they complete the form. However, self-reports of race in the Census indicate that some identify their race as Hispanic. This is a matter of identity, but also of the structure of the question about race included in the Census. Race options include white, black, Asian, American Indian or Pacific Islander, or some other race. Some people who identify as Hispanic may also identify with one of these racial categories, but many do not, and as a result, choose to write in Hispanic as their race. Elaborating on this, Pew Research Center wrote in 2015: [Our] survey of multiracial Americans finds that, for two-thirds of Hispanics, their Hispanic background is a part of their racial background – not something separate. This suggests that Hispanics have a unique view of race that doesn’t necessarily fit within the official U.S. definitions. So while Hispanic might refer to ethnicity in the dictionary and governmental definition of the term, in practice, it often refers to race. What Latino Means and Where It Came From Unlike Hispanic, which refers to language, Latino is a term that refers to geography. It is used to signify that a person is from or descended from people from Latin America. It is, in fact, a shortened form of the Spanish phrase latinoamericano -   Latin American, in English. Like Hispanic, Latino does not technically speaking refer to race. Anybody from Central or South America and the Caribbean can be described as Latino. Within that group, like within Hispanic, there are varieties of races. Latinos can be white, black, indigenous American, mestizo, mixed, and even of Asian descent. Latinos can also be Hispanic, but not necessarily. For example, people from Brazil are Latino, but they are not Hispanic, since Portuguese, and not Spanish, is their native language. Similarly, people may be Hispanic, but not Latino, like those from Spain who do not also live in or have lineage in Latin America. It was not until the year 2000 that Latino first appeared on the U.S. Census as an option for ethnicity, combined with the response Other Spanish/Hispanic/Latino. In the most recent Census, conducted in 2010, it was included as Another Hispanic/Latino/Spanish origin. However, as with Hispanic, common usage and self-reporting on the Census indicates that many people identify their race as Latino. This is especially true in the western United States, where the term is more commonly used, in part because it offers a distinction from the identities of Mexican American and Chicano  - terms that specifically refer to descendants of people from Mexico. Pew Research Center found in 2015 that 69% of young Latino adults ages 18 to 29 say their Latino background is part of their racial background, as does a similar share of those in other age groups, including those 65 and older. Because Latino has come to be identified as a race in practice and associated with brown skin and origin in Latin America, black Latinos often identify differently. While they are likely to be read simply as black within U.S. society, due to their skin color, many identify as Afro-Caribbean or Afro-Latino  - terms which serve to distinguish them both from brown-skinned Latinos and from descendants of the North American population of black slaves. So, like with Hispanic, the standard meaning of Latino  often differs in practice. Because practice differs from policy, the U.S. Census Bureau is poised to change how it asks about race and ethnicity in the coming 2020 Census. The possible new phrasing of these questions would allow for Hispanic and Latino to be recorded as the respondents self-identified race.

Public Management Reform in the UK Essay Example | Topics and Well Written Essays - 2500 words

Public Management Reform in the UK - Essay Example This paper gives a comparison of power generation regulation in UK and India, with a number of insights on their drivers as well as ongoing strategy initiatives within the energy sector. The paper also point out a number of key issues along with policy limitations, both in meeting growth targets as well as in controlling emissions increase.The main objective here is to offer a reasonable as well as insightful classification of the factors that shape public management in each UK and India. This paper balances other research attempts of abatement prospective and prospects by offering an evaluation of basis - the framework of state of growth, future requirements and existing policy course. The people of the â€Å"world’s poor† reduced from 3 billion during 1991 to 2.4 billion during 2010, with almost this complete drop occurring in the UK. Majority of the world has not arrived at such sharp declines in poor standard of living. The population of the world’s poor apar t from UK has stayed on more or less 2.1 billion from 1991, regardless of a drop in proportional terms. According to the World Bank estimates, by 2017, this amount would decrease merely to 2 billion. Of the 2.4 billion poor during the year 2010, more than a third resides in India, who consists of almost 40 percent of the nation’s population. More than 50 percent of India’s inhabitants do not have access to electrical energy and they utilize kerosene for light. Merely one fifth of those utilizing electricity consume more than 100 kWh each month, in comparison with the average UK domestic utilization of more than 900 kWh each month (Barnett and Biornsgaard, 2000, p. 82). Therefore, broadening access as well as â€Å"meeting other development goals† (Kiameh, 2009, p. 95) will involve a considerable raise in production ability, even by developments in consumption efficiency. This earnings inconsistency has to be taken into consideration while expanding a load contri bution contract. â€Å"As the philosopher Henry Shue and others have argued, luxury emissions and not subsistence emissions should be the appropriate target of emissions reductions† (Keljik, 2008, p. 78). While one openly justifies this distinction, the comparative burden of India is minute, and is most likely activated just further than 2020. For instance, a current scheme for a burden contribution management for environment improvement supported by nations’ growing discharges from 2000 as well as their assets needs that the UK accept more than 30 percent of the universal improvement prerequisite during 2020, whereas India chips in more or less one percent (Kehlhofer et al, 2009, p. 66). A growth responsive analysis of burden sharing in countries does not mean development should be categorical or that improvement as well as expansion intrinsically diverges. In addition, both countries should focus on to the possibility of â€Å"getting locked† (Grigsby, 2007, p . 93) into flawed technologies and models of growth that will make it tougher to decrease emissions afterwards. Even a quicker look on India exposes discriminatory development in addition to a number of prospects for improvement in co-advantages. Despite the fact that India wants to develop, this sort of development will not chip in satisfactorily to growth unless it is comprehensive and prioritizes the requirements of the neglected. India has diverse evidence with respect to this issue. At the same time, as its scarcity level has dropped since 1991, â€Å"absolute poverty and income inequality† (Blume, 2007, p. 129) have remained more or less the

Steamboal Bill, Jr Essay Example | Topics and Well Written Essays - 750 words

Steamboal Bill, Jr - Essay Example An analysis of the film will address themes, sound tracts, and effectiveness to the audience. The plot of the movie begins with the arrival of a new steamer in the river junction. The owner of the new steamer is J.J. King (Wickstrà ¶m). He is shown being happy with other captains of the ship waiting for its arrival. On the other hand, Steam Boat Bill is shown with his mate Tom Lewis in the old steamer (Wickstrà ¶m). Bill is shown annoyed because of the new steamer, as he perceives it as a business rivalry. King is depicted as a wealthy business with some few banks and hotels under his name (Wickstrà ¶m). King notes of the old steamer as a floating ‘thing’ (Wickstrà ¶m). Bill also receives a telegram of his impending son visit. They go with his mate to get the son at the railway station. They both expect a well-built man. However, the contrary happens, as the son appears small and dressed in a funny way. This makes his father begin a journey of his transformation in a new life. At the same time, King Daughter (Marion Byron) arrives from school (Wickstrà ¶m). Both Bill and King get annoyed as they realize that both Willie and Byron were lovers. They engage in a journey of ensuring that the relationship does not exist. On the other hand, Stonewall Jackson is condemned from public safety committee from carrying out transactions (Wickstrà ¶m). This annoys Bill who confronts J.J. King. The confrontation lands Bill to jail. The son tries to rescue him without any success. In the end, Willie is shown saving father, Marion Byron, and King from ravaging weather (Wickstrà ¶m). He ends up being a hero in the film despite his initial weaknesses. The theme of the film appears as that of transformation. This has been well articulated and developed by the main characters in the play. For example, Willie is shown as a weak character in the railway station. The father and his mate Tom Lewis are ashamed of Willie as he

There are a number of problems associated with the rise of English as Essay

There are a number of problems associated with the rise of English as a world language - Essay Example Different factors work together in attaining such a privileged position. Research personals have pointed out various reasons that help to attain the position. In this regard David Crystal has the opinion that the military power is crucial. He rightly comments: â€Å"a language becomes an international language for one chief reason: the political power of its people- especially their military power† (Burns & Coffin 2001, p. 30). The international economy plays a vital role in determining the future of the English language. Due to globalization the patterns of trade is undergoing drastic changes and new working practices emerge powerfully. For example screen based labour is getting wider acceptance nowadays and it affects the use of the English language in composite ways. At present one can observe a perceptible increase in the number of persons learning and using the English language but a close examination will prove that the long term growth of English is comparatively lesser than its former status. The 20th century witnessed many scientific, economic, and cultural developments and in fact it boosted the relevance of English as well. But it cannot be expected that it will last long as there is a trend of wider mixed language use in the recent years. Mainly four sectors can be identified where English may loose its dominance. Say for the audio- visual market, computer based communication, the associated pro cesses in economic globalization and foreign language training in developing countries where the regional trade may use other language of economic importance. For hundreds of years the English language has been dominated over the mono lingual speakers but now it is expected to have a bilingual future. Language professionals have the opinion that an increasing proportion of the world’s population will be fluent speakers of more than one language in the coming years. It really will affect the dominance of English over the world population. Social value shifts also affect the language of English in a negative way. The actions of governments, institutions and individuals have helped the promotion of English in the recent years. But it was guided by the logic of economic rationalism. Now there is a great change in the public opinion that quality of life is as important as the income. These type values shifts will question the ethical issues associated with the dominance of a world language like English and cause a reassessment of its impact on other cultures (Graddol 1997). The advantages of a world language like English are several. For example in business it can be very important as a medium as the business personals working together may vary of their nationalities. There a common language has got such a great relevance and it decides the effects of their dealings. But at the same time it has got certain demerits or disadvantages as well. First of all the existence of a global language will make an elite monolingual linguistic class and they will have a dismissive and complacent attitude towards other languages. Moreover the accessibility of a common language will make the people lazy of learning other languages. They will be reluctant to utilize such an opportunity though they get a chance. Some times the global language can cause the disappearance for a minority language and it may belittle the necessity of such languages. Above all the practice of a sin gle language can be a threat to social harmony and mutual understanding. This has got historical evidence. The American civil war, the Spanish civil war, Vietnam War former Yugoslavia and contemporary Northern Ireland are

Plato and Crito Essay Example for Free

Plato and Crito Essay In Plato’s Crito, Crito attempts to persuade Socrates to flee from his death sentence. However, Crito fails because Socrates presents a counter argument which invalidates much of Crito’s original pleas. Despite this, a fallacy of justice may have been created. Even so, the Republic’s conception of justice seems to have little impact on Socrates’ existing ideas on justice. The first argument presented is the fact that the majority will look down upon Crito and others for not preventing Socrates death; they will find it to be a â€Å"shameful thing both for you and for us† because it seems â€Å"that [Crito] let the opportunity slip because of some vice, such as cowardice† (46a). Another reason which he presents to Socrates is that Crito and the others are â€Å"justified in running the risk† of â€Å"further penalty† for helping him to flee from execution† (44e). While Socrates says that he fears for them, Crito goes on to elaborate that even the sum of money to help him escape is overall â€Å"not large† (45a). He expounds further that people are willing to support him wherever he might go (45c). Next, Crito goes on to mention Socrates two sons; Crito feels that by being executed when there is a possibility to escape, he is â€Å"betraying those sons† (45c), that â€Å"one ought to see their upbringing and education through to the end† (45d). Overall, Crito feels that Socrates would be â€Å"throwing away [his] life†, which would ultimately set him in his enemies own wishes instead of his own (45c). Socrates starts his counter arguments by stating that he cannot just start rejecting the arguments that he had stated before just because of his current, unfortunate situation (46b). Because of this, he begins an examination of the arguments presented by Crito. Socrates states that the majority’s opinion is not what matters because it is not well informed, but rather it is the opinion of â€Å"the person who understands just and unjust things† (48a) that is the most valid opinion to follow. To conclude this section of his argument, Socrates proposes that â€Å"the most important thing isn’t living, but living well†, which he says is the same as living justly (48b). So, by this logic, the concerns that Crito had should be determined just or unjust for Socrates to escape when he had not been acquitted. Socrates reasons that â€Å"doing injustice in any circumstances is bad† and therefore â€Å"one should never do injustice† (49b), â€Å"no matter what one has suffered at [injustice’s] hands† (49c). He further explains through his use of the Laws that violating the contract of the city (his â€Å"deeds† within it [52d]) is only going to undermine the laws in such a way that he would indeed be held to his original charges of corruption (53c). He concludes his argument by stating that in the afterlife, Socrates will have â€Å"all this to offer as [his] defense to the authorities there† (54b). Overall, these arguments that Socrates puts forth are generally sound. However, one overarching theme that could be debated is the conception of being just by submitting to the laws. In many ways, it could be said that Martin Luther King Jr. by resisting the unjust established laws he was himself being unjust and would ultimately end up leading to a worse society. It seems obvious, however, that his actions made a positive difference to the world. Perhaps if Socrates resisted, he could have changed the unjust laws to make them just in such a way the city more just. Crito may have been right when he said that Socrates was giving in – his full potential would ultimately never be realized. If one was to attempt to define the justness of Socrates actions here by using Plato’s Republic, then it may seem that there may be a series of different arguments for justice. However, it is my own conclusion that none of the arguments for the definition of justice in the Republic would really influence Socrates in the slightest. After all, Socrates has already said he would not reject any previous arguments, all of which made in Crito are to be considered previous arguments, so it could be inferred that Plato, as the writer of the Crito, used his ideas of justice as the foundation for the dialog and eventually the Republic. Even the case of supplementary information, justice is explained in part in the Republic as â€Å"doing one’s own work† (433b). Socrates indeed did his own work and was just, but this does not address justice in an unjust city where one’s own work might be considered to be unjust. Further, the three virtues of the soul, moderation, courageousness, and wisdom (435b), were also fulfilled to some extent. Even more, by the classes of the people in the city (435c-441c), it could be said that Socrates followed his class assignment in the just city, but this is not a correlation to Athens, a city with a different class structure. Largely, the expanded view of justice is just more refinement of the original view of justice presented all throughout Plato’s dialogs. Expanded or not, the base remains the same and so it follows that Socrates’ arguments against Crito are relatively the same, even in light of the Republic. All said, Socrates arguments in response to Crito seem to be mostly plausible. Despite the world losing a skillful philosopher, at least the laws and justice of the city were upheld and Socrates therefore lived a just life, regardless of his sentence.

Communication in Health Care Essay -- Healthcare Communication

Introduction This is an analysis of a taped interview between a nurse and a patient who is taking pre-employment medicals. The information given during the interview, including her name, Pink Cloud is fictitious because of the need of confidentiality. During the interview, objective and subjective data will be collected. The areas of communication focused on in the analysis are verbal, questioning and listening skills. Analysis will be made and later suggestions and recommendations will be made on how to make improvements in the future. To achieve this, direct quotations from the will be used to make references to the three theories being analysed and will be supported by the literature. Analysis of the interview An interview is a planned interpersonal conversation with a purpose between two people who interact largely through a question and answer format to achieve specific goals. Verbal communication is a way in which people communicate face to face. The key components of verbal communication are words and phrases, intonation and clarity and brevity. Different words and phrases have different meanings. The denotation meaning is shared by individuals who share a common language while connotative meaning is the interpretation of a word’s meaning influenced by feeling and thoughts of people. Therefore, Nurses should carefully select words that cannot be interpreted, especially when explaining a client’s medical condition (Taylor & Crisp, 2006). Additionally, other concepts of verbal communication skills such as questioning and probing, paraphrasing and understanding are used when talking to patients. Questioning and probing is where various types of questions and statements... ...Apler, J. P. (2006). Professional nurse communications skills sets in health care team interactions. Journal of Professional Nursing , 22 (2), 180-189. Boyler, D. &. (2004). Enhancing collaborative communication of Nurded and Psysican Leadership. Journal of Nursing Administartion , 34 (2), 60-70. Dixon, J. &. (2006). Skilled communication:making it real.Advances in critical care. Journal of Continuing Education In Nursing , 17 (4), 376-382. Robertson, K. (2004). Active listening:more than just paying attention. Australia: Australian Family Pysician. Roter, D. (2004). The enduring and evolving nature of the patient-physician relationship. Journal of clinical oncology , 22 (13), 10-15. Stein-Parbury, J. (2009). Patient and person. New York:Churchill Livingstone. Taylor & Crisp. (2006). Fundamental of Nursing. Sydney , New south wales, Australia: Vaughn Curtis.