Probability of a Large Straight in Yahtzee in One Roll

Yahtzee is a dice game that uses five standard six-sided dice. On each turn, players are given three rolls to obtain several different objectives. After each roll, a player may decide which of the dice (if any) are to be retained and which are to be rerolled. The objectives include a variety of different kinds of combinations, many of which are taken from poker. Every different kind of combination is worth a different amount of points. Two of the types of combinations that players must roll are called straights: a small straight and a large straight. Like poker straights, these combinations consist of sequential dice. Small straights employ four of the five dice and large straights use all five dice. Due to the randomness of the rolling of dice, probability can be used to analyze how likely it is to roll a large straight in a single roll. Assumptions We assume that the dice used are fair and independent of one another. Thus there is a uniform sample space consisting of all possible rolls of the five dice. Although Yahtzee allows three rolls, for simplicity we will only consider the case that we obtain a large straight in a single roll. Sample Space Since we are working with a uniform sample space, the calculation of our probability becomes a calculation of a couple of counting problems. The probability of a straight is the number of ways to roll a straight, divided by the number of outcomes in the sample space. It is very easy to count the number of outcomes in the sample space. We are rolling five dice and each of these dice can have one of six different outcomes. A basic application of the multiplication principle tells us that the sample space has 6 x 6 x 6 x 6 x 6 65 7776 outcomes. This number will be the denominator of all of the fractions that we use for our probabilities. Number of Straights Next, we need to know how many ways there are to roll a large straight. This is more difficult than calculating the size of the sample space. The reason why this is harder is because there is more subtlety in how we count. A large straight is harder to roll than a small straight, but it is easier to count the number of ways of rolling a large straight than the number of ways of rolling a small straight. This type of straight consists of five sequential numbers. Since there are only six different numbers on the dice, there are only two possible large straights: {1, 2, 3, 4, 5} and {2, 3, 4, 5, 6}. Now we determine the different number of ways to roll a particular set of dice that give us a straight. For a large straight with the dice {1, 2, 3, 4, 5} we can have the dice in any order. So the following are different ways of rolling the same straight: 1, 2, 3, 4, 55, 4, 3, 2, 11, 3, 5, 2, 4 It would be tedious to list all of the possible ways to get a 1, 2, 3, 4 and 5. Since we only need to know how many ways there are to do this, we can use some basic counting techniques. We note that all that we are doing is permuting the five dice. There are 5! 120 ways of doing this. Since there are two combinations of dice to make a large straight and 120 ways to roll each of these, there are 2 x 120 240 ways to roll a large straight. Probability Now the probability of rolling a large straight is a simple division calculation. Since there are 240 ways to roll a large straight in a single roll and there are 7776 rolls of five dice possible, the probability of rolling a large straight is 240/7776, which is close to 1/32 and 3.1%. Of course, it is more likely than not that the first roll is not a straight. If this is the case, then we are allowed two more rolls making a straight much more likely. The probability of this is much more complicated to determine because of all of the possible situations that would need to be considered.

Analysis Of The Book Five Faces Of Oppression By...

Priscilla Afriyie Phi – 173 5/25/16 Final Paper Charles W. Mills states â€Å"Whiteness is not really a color at all, but a set of power relations.† (The Racial Contract 127). meaning that being white is not a color but rather it places a position where you have the power to set rules. Iris Marian Young discusses oppression brought on by racism having five faces that constructs a society and shapes an individual in the book â€Å"Five Faces of Oppression† While some might say it is a social construct, it does not exist and if it does it is only meant to encourages people of a lower class to work harder, racism is a form of discrimination and needs to be put to an end. Racism causes inequality and oppresses certain groups of people. It leads a belief that one’s own race is superior, creates a society where people do not trust or respect each other, and leaves them struggling to be a part of prosperous society. First racism causes inequality and oppresses certain groups of people. Inequality can be found as early as kindergarten, within our education system. Here in the United States, schools are partially funded by taxes collected from each neighborhood and given to the school within its zone. Therefore, if a school is in a rich neighborhood filled with affluent parents, they would have more money to spend on their kids’ education than a school in a poorer neighborhood. This is an unfair advantage giving kids in richer neighborhoods a leg up in education. More resources such as modernShow MoreRelatedAlternative Learning Systems9735 Words   |  39 Pagesconcepts of doing gender, interaction order, and emotion work to the prison classroom. Based on their analysis, the authors examine the challenges and opportunities for critical education in prison. They aimed to use their teaching efforts to reach out to marginalized students and develop students’ sociological imaginations t o assist them through the challenges of confinement and reentry. 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Tma B322 Free Essays

Question 1 A) Innovation is about finding new or better method of making something, a product or a service. Innovations are born due to several reasons . Peter Drucker has outlined seven sources of innovative chances or opportunities . We will write a custom essay sample on Tma B322 or any similar topic only for you Order Now I am going to identify them below. 1. Unexpected happenings, as a sudden event that can be a indication of a distinctive chance for innovating or entrepreneurship. 2. Incongruity, as the divergence between what everyone think of a thing and the reality of it , this difference can create an innovation. 3. Innovation born when a new process is needed is a chance to fill the gap and create something new. . Changes happened to the industry structure can be a possibility for an innovative product to pop out 5. Demographics change such as population size, employment level and income ranges. 6. Perception variation along with mood and meaning of general assumptions can form an innovation. 7. New knowledge acquired based on fresh advances in scientific and technological fields can produce new goods to the market. [Tetteh,2008] Gary was a risk taker, his innovations were the result of unexpected happenings that combined with his great mind generated new and interesting products. It is said that his first step of being innovative and risky was an end result after reading a newspaper ad about the availability of large quantities of powder soap. He evaluate the news as a great one since the countries was just getting out of a war state and people will find this product pretty attiring. This unexpected happening has made Gary a deal that created a tiny wealth of a 30,000$ at that time when he contacted the right people and informed them about his product and made the sale from scratch. Word count: 288? B) According to Schumpeter, innovation is the process of introducing new products in addition to techniques to the market. These products need to be new or radically improved. Schumpeter identified five types of possible innovation: 1. Producing new kinds of goods or modifying the of attributes of the existing ones. 2. Introducing new way of production which might be based on innovative technical and scientific findings. 3. opening new market 4. Using new resources and raw materials in the inputs. 5. new organization of production and industry . What Gary has done is finding products and making them more attractive by adding some options or features to them, what he said about his products is that there are normal products with a twist. This twist can be new look or design or new attributes added to them. We can see that Gary’s type of innovation is product innovation, which consists of changing a feature or a look of an available product. Furthermore, Gary as an adventurer has introduced new products from toys and other amusing stuff to the market. For example: the plastic farm of ants , the spud gun and the fly cake which have made great successes and rewards. For instance, he released balloons that were designed as giant animals , this change in the look of the balloons attracted people and made them buy them in huge quantities . Mixing his original ideas with the product that is presented have resulted in new product that is a real money maker. Word count: 251? Question 2 A) Gary as shown in the case is a very successful entrepreneur. He combined great traits of successful and honored men in the business world. Gary was a normal person who hasn’t a degree in business but what he has is the idea and the initiative to take this idea to the last level and making it worthwhile. He has marketing skills, he knew what attract customers and also he knew how to convince people to buy his products that were not ordinary products, but with his skills and creativity he changed the actual product and made it more appealing in the addition of the fun part for ex: animal look balloons and plastic ant farm. Gary forever looked for high accomplishments and wanted to achieve his dream to be a millionaire from the beginning and he reached his goal. [Davison,2008] Being an ambitious person who is to a great extent motivated towards success, knew how stay focused and identify the opportunity when he sees it as he did when reading the soap powder availability in the market and took risk to make his first great sale, that showed how clever he was by perceiving this opportunity and evaluating it to decide that it will be a great offer for people after the war. Having the mental ability to think in an intelligent and creative way for doing all kinds of business while keeping his eyes on the objective and not losing the energy however remaining persistent to take risks and new challenges by introducing high number of innovative products to the market without fear of failure are the reason behind the enormous success of this inspirational entrepreneur. Word count: 276 ? B) An entrepreneur is a person who has the potential to do something different. Being an entrepreneur is about having the right skills and the right idea along with the right experience and in the end being able to generate a totally unique innovation Gary as an entrepreneur is a risk taking individual who enjoys taking the chances and entering uncertain situations to do business. Many characteristics that he owned helped him be the entrepreneur he is now. The many experiences he has from his childhood, in hi father grocery’s store and the experience he gained in the army and combining them all together taught him that being risky can be quite fun and very profitable. Furthermore, having an idea and staying focused on the goal of making that idea an innovative creation next to realizing good opportunities and chances for a fresh venture were strengths for Gary and reason behind his success. Having the potential is about generating ventures and succeeding from scratch, the entrepreneur is innovative and inventive and knows how to develop his ideas and turn it into a success and money. Gary is self confident and multi-skilled, he didn’t focus on one field for business, and his interest was in marketing. He still remained confident when he faced of difficulties and failures and discouraging circumstances. Gary gathered in his personality the potential to be a successful entrepreneur. The risky trait, the experience, the intelligence, the initiative, the creativity, the know-how, the complete commitment and dedication to achieve his objectives. Word count: 252 ? Question 3 Innovation is spreading vey fast in the economic world. Innovation permits nations and countries to grow rapidly better than cultures that aren’t adopting innovation. Acquiring ideas of technological and innovation is actuality important for achieving organization’s competitive advantage. Everett roger has discussed the importance of customers adopting and using innovation, since customers are the most vital part of the business process. That is what the purchasing pathway network that includes several stages of introducing the products until the consumer purchase them . The purchasing network perspective got great interest from different business people, among managers, consultants, and decision makers are aiming to promote the innovative capability for organizations. Firstly innovators asses what is needed and wanted under the competitive pressure and the need to produce different products. After that it comes releasing the product, making the ads for it to attract new customers and position the image of the product in the mind of customers to make them move and buy the product. Mankin, E 2007] People are divided into several levels of innovation adoption , some of them are innovators who like to purchase the product first, and then comes the later adopters who wait for another people to try the product before them because they fear it will not satisfy them for example purchasing an iphone The purchasing pathway network is greatly advised for organization or firms when designing a project since the innovator need to know that people’s percep tion of the innovative product differ where there is a crucial need to identify different users adoption behavior. Word count: 255 Question 4 The six markets model illustrates the view of where the marketing of goods can be applied. The six domains appearing above are the ones that organizations direct their marketing strategies by. Those markets are: the referral market where the organization has good communication links with some experts and professionals in business world who can recommend others regarding purchasing products and other decisions. The supplier markets, recruitments markets, influence markets and internal markets or employees. For example The Word of mouth communication can be recognized as a valuable method in the marketing mix. Each of these interactions must be managed with a strategic plan. Gary’s marketing activities were shown firstly when he marketed the powder soap from the supplier market and offered them for people he already met (referral market) and he succeeding in making the deal . Then he used the mail-order business and published an ad in the newspaper, and customers saw the ad and were attracted by it (customer market). He introduced also new products in national trade fairs also considered as customers market. He has set up a billboard next to his house that declared the image and the marketing intelligence he had. Big companies and executives were attired by the billboard and became known among businessmen. Moreover, Gary has published two books that showed his knowing in promoting his experiences and stories to his students( internal markets). SO we can see that Gary has used the whole framework of the six markets in his business life. That helped him becoming successful and a winner in the entrepreneurship and marketing world. Word count: 263 ? References Davison, ED 2008, ‘Social Entrepreneurship’, Social Entrepreneurship — Research Starters Business, p. 1, Research Starters – Business, EBSCOhost, viewed 7 May 2012. Mankin, E 2007, ‘Measuring Innovation Performance’, Research Technology Management, 50, 6, pp. 5-7, Research Starters – Business, EBSCOhost, viewed 7 May 2012. Tetteh, VA 2008, ‘Entrepreneurship Business Planning’, Entrepreneurship Business Planning — Research Starters Business, p. 1, Research Starters – Business, EBSCOhost, viewed 7 May 2012. How to cite Tma B322, Essay examples