Probabilities for Rolling Two Dice

One popular way to study probability is to roll dice. A standard die has six sides printed with little dots numbering  1, 2, 3, 4, 5, and 6. If the die is fair (and we will assume that all of them are), then each of these outcomes is equally likely. Since there are six possible outcomes, the probability of obtaining any side of the die is 1/6. The probability of rolling a 1 is 1/6, the probability of rolling a 2 is 1/6, and so on. But what happens if we add another die? What are the probabilities for rolling two dice? Dice Roll  Probability To correctly determine the probability of a dice roll, we need to know two things: The size of the  sample space or the set of total possible outcomesHow often an event occurs In probability, an event is a certain subset of the sample space. For example, when only one die is rolled, as in the example above, the sample space is equal to all of the values on the die, or the set (1, 2, 3, 4, 5, 6). Since the die is fair, each number in the set occurs only once. In other words, the frequency of each number is 1. To determine the probability of rolling any one of the numbers on the die, we divide the event frequency (1) by the size of the sample space (6), resulting in a probability of 1/6. Rolling two fair dice more than doubles the difficulty of calculating probabilities. This is because rolling one die is independent of rolling a second one. One roll has no effect on the other. When dealing with independent events we use the multiplication rule. The use of a tree diagram demonstrates that there are  6 x 6 36 possible outcomes from rolling two dice. Suppose that the first die we roll comes up as a 1. The other die roll could be  a 1, 2, 3, 4, 5, or 6. Now suppose that the first die is a 2. The other die roll again could be a 1, 2, 3, 4, 5, or 6. We have already found 12 potential outcomes, and have yet to exhaust all of the possibilities of the first die. Probability Table of Rolling Two Dice The possible outcomes of rolling two dice are represented in the table below. Note that the number of total possible outcomes is equal to the sample space of the first die (6) multiplied by the sample space of the second die (6), which is  36. 1 2 3 4 5 6 1 (1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6) 2 (2, 1) (2, 2) (2, 3) (2, 4) (2, 5) (2, 6) 3 (3, 1) (3, 2) (3, 3) (3, 4) (3, 5) (3, 6) 4 (4, 1) (4, 2) (4, 3) (4, 4) (4, 5) (4, 6) 5 (5, 1) (5, 2) (5, 3) (5, 4) (5, 5) (5, 6) 6 (6, 1) (6, 2) (6, 3) (6, 4) (6, 5) (6, 6) Three or More Dice The same principle applies if we are working on  problems involving three dice. We multiply and see that there are 6 x 6 x 6 216 possible outcomes. As it gets cumbersome to write the repeated multiplication, we can use  exponents  to simplify work. For two dice, there are 6^2  possible outcomes. For three dice, there are 6^3  possible outcomes. In general, if we roll  n  dice, then there are a total of 6^n  possible outcomes. Sample Problems With this knowledge, we can solve all sorts of probability problems: 1. Two six-sided dice are rolled. What is the probability that the sum of the two dice is seven? The easiest way to solve this problem is to consult the table above. You will notice that in each row there is one dice roll where the sum of the two dice is equal to seven. Since there are six rows, there are six possible outcomes where the sum of the two dice is equal to seven. The number of total possible outcomes remains 36. Again, we find the probability by dividing the event frequency (6) by the size of the sample space (36), resulting in a probability of 1/6. 2. Two six-sided dice are rolled. What is the probability that the sum of the two dice is three? In the previous problem, you may have noticed that the cells where the sum of the two dice is equal to seven form a diagonal. The same is true here, except in this case there are only two cells where the sum of the dice is three. That is because there are only two ways to get this outcome. You must  roll a 1 and a 2 or you must roll a 2 and a 1. The combinations for rolling a sum of seven are much greater (1 and 6, 2 and 5, 3 and 4, and so on). To find the probability that the sum of the two dice is three, we can divide the event frequency (2) by the size of the sample space (36), resulting in a probability of 1/18. 3. Two six-sided dice are rolled. What is the probability that the numbers on the dice are different? Again, we can easily solve this problem by consulting the table above. You will notice that the cells where the numbers on the dice are the same form a diagonal. There are only six of them, and once we cross them out we have the remaining cells in which the numbers on the dice are different. We can take the number of combinations (30) and divide it by the size of the sample space (36), resulting in a probability of 5/6.

A Comparison of Methods in Business Research - 3572 Words

A Comparison of Methods in Business Research By Song Gao This essay compares the research method of case study with five other methods in business research. The five methods are focus group, in-depth interview, observation, questionnaires and field experiment. This essay provides the definition, advantages, disadvantages/limitations for each of the six methods. Moreover, five published articles related to case study method and two published articles related to each of the other five methods were found to further discuss their application in research. Case Study Definition It is an understatement that there is confusion among students, teachers and researchers about the definition of case study research. In this essay, I use the†¦show more content†¦The authors collected data from a major business communication provider, a networking infrastructure solutions provider, and a GPS technology. The topic of this article is very professional and difficult to understand for the audience who have no relevant background. But one advantage to their case study is it helps audience to better understand a complex organisational phenomenon. The authors also illustrated the limitations of their study. Article 4 – DANAHER, P. J. (2013). Comparing the Relative Effectiveness of Advertising Channels: A Case Study of a Multimedia Blitz Campaign. Journal of Marketing Research (JMR)., 50(4), 517–534. 18p. 1 Black and White Photograph. In this research, the authors developed an inexpensive method to help firms assess the relative effectiveness of multiple advertising media. Specifically, they used a firms loyalty program database(Case Study) to capture media exposure, through an online media survey. At the end, not only did the authors demonstrate how to use their advertising response model to determine the optimal budget allocation across each advertising media channel, but also they pointed out the limitations of their studies. Article 5 – Fahimnia, B., Farahani, R. Z., amp; Sarkis, J. (2013). Integrated aggregate supply chain planning using memetic algorithm – A performance analysis case study. International Journal of Production Research, 51(18), 5354–5373.Show MoreRelatedChartered Management Institute Report Review1062 Words   |  4 Pagesï » ¿CMI Report Review Introduction A research can be defined as the systematic inquiry conducted with an objective of testing a hypothesis, evaluating knowledge or searching for new information. To ensure that the outcome of a research is valid, researchers must ensure that the processes followed minimize errors, inconsistencies, and anomalies that are capable of impairing the findings. 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Design and Analysis of Experiments System †MyAssignmenthelp.com

Question: Discuss about the Design and Analysis of Experiments System. Answer: Introduction It is critical to understand students perception about environmental issues and how they take part in saving both energy and water. This can help in understanding future generation and whether a sustainable environment can be attained through this young generation. In accordance with (UNICEF, 2015), youth and children are an important part of society that can help in creating a sustainable future. UNICEF targets this group on their sensitization program about creating a sustainable future. Severn Cullis-Suzuki, in the 1992 speech silenced the world and pointed that the only way to create a sustainable future is through educating the youths (Orfalea, 2017). Therefore, this report will investigate how sensitive the students from the NSW are about environmental issues. The research will evaluate some of the students characteristics and how different aspects of life can impact their academic achievement. The key components that this report will highlight include the hours of sleep during a school night, the time taken to travel to school (minutes), time spent on different issues such as doing homework, doing things with friends/family among others. The variables measuring time is on an interval scale, and those asking for an opinion about environmental issues are nominal scale. Lastly, how important the students value some issues both personal and environmental are on an ordinal scale. The data analysis was performed to illustrate the distribution of various variables under investigation. First, it was determined the distribution of gender in the sample obtained from the NSW state. Second, the distribution of how student spent their time, especially playing video games was assessed. The summary of the descriptive statistics and measure of dispersion is as summarized below. Summary measures for selected variables Q9 Doing homework Q9 Doing things with family Q9. Playing computer/video games Q9. Watching TV Count 40.000 40.000 40.000 40.000 Mean 5.900 11.825 10.150 8.700 Median 4.500 7.000 5.500 6.500 Standard deviation 5.486 10.476 11.023 7.803 Minimum 0.000 0.000 0.000 0.000 Maximum 25.000 40.000 40.000 29.000 First quartile 2.000 3.000 1.750 2.000 Third quartile 8.000 17.500 18.000 10.750 Mean absolute deviation 4.185 8.799 9.138 6.105 On average, students spent 5.90 hours doing their homework with a standard deviation of 5.48 hours. The middle 50% of the students spent 2 hours and 8.00 hours, doing their homework (Afifi Azen, 2014). Students spent, on average, 11.83 hours with their family members, with some spending no time and some up to 40 hours. On average, students spends 10.15 hours playing video games with a standard deviation of 11.02 hours. This shows that there might be a possibility that students spent more time playing video games than they do taking their homework and with their family (Gelman, et al., 2014). However, to ascertain if this is true, a hypothesis was tested to determine whether an on average student spent more time playing video games than they spend with their family and doing homework. Lastly but not least, the average time the students spent watching is 8.70 hours with a standard deviation of 7.80 hours. The p-value .05, which purport that the hypothesis that the average time spends doing homework and doing things with the family is equal (Montgomery, 2017). This implies that students spend less time reading than they do doing things with their family. The summary shows that there is enough evidence to support the claim that student spent more time playing computer/video games than they spent doing their homework (Gelman, et al., 2014). In particular, with 95% confidence, a student selected randomly from the population spends more time playing computer/video games than they spent doing their homework. Lastly, it was evaluated whether students spend more time watching TV than doing their homework. The p-value 0.034 is less than 0.05, supporting that the rejection of the null hypothesis (Peck, Olsen, Devore, 2015). Therefore, it should be noted that with the 95% confidence, we can claim that students spend more time watching than doing their homework. Environmental Issues The assessment on whether there was a relationship between gender and switching off the power at the main switch. First, the distribution of the data was illustrated in Figure 2. The chart shows that most of the female students do not power off the main switch to save the electricity. On the other hand, most male student power off the main switch. Therefore, there might be an association between the gender and the powering off the main switch. Chi-square test was performed to determine whether the claim that there is a connection between gender and powering off the main switch to save energy. The p-value 0.1098 is greater than .05. Thus, fail to reject the null hypothesis (Schumacker Sara, 2015). This means that there is no association between gender and powering off the main switch. This means that both male and female have the same behavior of powering off the main switch to save energy. There is enough evidence to reject the null hypothesis at the level .10 (Schumacker Sara, 2015). This suggests that with 90% confidence, we can say that the gender of a student is associated with the installation of the water saving shower head. It was important to determine whether there exists an association between the number of paid working hours and the amount received as pocket money. Therefore, a linear regression model was run, and the direction and strength of the association noted. The model is not significant since the p-value is greater than 0.05 (Chatterjee Hadi., 2015). The coefficient of the number of hours is positive. The R2-value shows that the model can take into account to 0.02% sources of error. This is a very low source of variation since 99.98% could not be accounted. Conclusion The findings point important aspect of students time utilization. It was established that students spent more time, watching TV, playing computer/video games and with their families. Therefore, parents need to schedule their students well, so that they can adequately utilize their time well to improve their academic performance. For instance, playing video game long may have negative effects on academic achievement. The results also pointed that there is a correlation between students gender and installing water saving shower head to save water. Also, no connection was established between gender and powering off the main switch to save energy. This means that both male and female students were equally sensitive about water conservation, but not about power energy conservation. Lastly but not least, there was no relationship between the number of hours student spent on a paid work and the amount they receive as pocket money. It should be noted that there might be one gender that is more sensitive on water conservation than the other. Therefore, the government should conduct a sensitization program to increase water conservation strategies like installing water saving shower head to all people. The parents should discourage a lot of video game playing and Tv watching. Students should be encouraged to use most of their time improving their studies. References Afifi, A. A., Azen, S. P. (2014). Statistical analysis: a computer oriented approach. Academic press. Chatterjee, S., Hadi., A. S. (2015). Regression analysis by example. John Wiley Sons. Gelman, A., Carlin, J. B., Stern, H. S., Dunson, D. B., Vehtari, A., Rubin, D. B. (2014). Bayesian data analysis. 2. Boca Raton, FL: CRC press. Montgomery, D. C. (2017). Design and analysis of experiments. John Wiley Sons. Orfalea, M. (2017, June 26). Girl Who Silenced the World (25 yrs later). Retrieved from Youtube: https://www.youtube.com/watch?v=3ipOdsd1SmA Peck, R., Olsen, C., Devore, J. L. (2015). Introduction to statistics and data analysis. Cengage Learning. Schumacker, R., Sara, T. (2015). Chi-square test. Understanding Statistics Using R, 169-175. UNICEF. (2015, Februalt 13). Environment and climate change. Retrieved October 2, 2017, from UNICEF: https://www.unicef.org/environment/index_60524.html

Louis Armstrongs Influential Career Essays - Louis Armstrong

Louis Armstrong's Influential Career Louis Armstrong's Influential Career Louis Armstrong was the most successful and talented jazz musician in history. His influence and expansive career continues to make waves in the jazz world. That is what made him become what he is to many today ? a legend. Born on August 4, 1901, in the poorest section of New Orleans, Armstrong grew up with his grandparents due to his parents' separation. On January 1, 1913 he made a mistake which turned out to be the best thing that ever happened to him. At a New Year's celebration in downtown New Orleans, Louis Armstrong, also known as ?Satchmo? and ?Satch?, fired a pistol into the air and was placed in the Colored Waifs' Home. It was there that he was introduced to Peter Davis ? the brass band leader who taught him how to play the cornet (Brown 17). Soon after he began playing, Armstrong was made leader of the band ? something he was extremely proud of. In June of 1914, Armstrong was free to leave the Waifs' Home. He was hired by various cabarets throughout the city, as well as for picnics, dances, and funerals. It was at one of these places that he was spotted by the famous Joe ?King' Oliver. King Oliver found Armstrong stand-in slots at orchestras and other venues. In 1918, he was offered the vacant seat left by Oliver in the band the Brown Skinned Babies. Kid Ory, leader of the band, once said that after Louis joined them he, improved so fast it was amazing. He had a wonderful ear and a wonderful memory. All you had to do was hum or whistle a new tune to him and he'd know it right away? (Boujut 21). At the end of 1918 Armstrong married Daisy Parker, a prostitute he had met at a dance hall that he played on Saturday nights. The marriage ended only four years later due to her beating him regularly (Bergreen 87). Louis Armstrong was hired in May of 1919 to play on a riverboat that traveled the Mississippi River from New Orleans to St. Louis. Armstrong soon became very popular in St. Louis and was in high demand (Collier 124). Two and a half years later, he was thrown off the riverboat and fired due to a fight. After returning to New Orleans, he received a telegram from King Oliver in Chicago. It was an invitation to join The Creole Jazz Band ? an offer Armstrong couldn't refuse. The Jazz Band cut it's first record in the spring of 1923 and toured throughout Illinois, Ohio, and Indiana (Hadlock 64). A year later Armstrong married Lil Hardin, the pianist in the band. He soon grew tired of playing in Chicago and left Oliver's band to head for New York City. When Armstrong reached the City and began playing everyone went wild. No where had they ever heard anything like him. Before this time, he had only played instruments but he was soon encouraged to begin singing. It was then discovered that Armstrong had a natural extension of his trumpet abilities, which was singing. His gravelly voice was something new that had never been heard (Sadie 600). Armstrong formed a studio band in 1925 called the Hot Five. The band included Johnny Dodds, Kid Ory, John St. Cyr, Lil Armstrong, and himself. These recordings were some of the most famous made by Armstrong, including ?Mandy Make Up Your Mind? and ?Money Blues? (Bergreen 96). A year later the Hot Five made their only public appearance at the Chicago Coliseum. The Chicago Defender spoke of ?Louis Armstrong, the miracle with steel lips,? (Boujut 27). For the next few years Armstrong recorded with the Hot Five and played with other musicians in New York and Chicago. He then traveled to California, in July of 1930, where he starred in his first film, Flame. Only after he had been in California for a few weeks he was arrested at a nightclub for the possession of marijuana. The incident obviously caused a lot of controversy across the world. The sentence, six months in prison, was suspended after only three days. Mainly due to Armstrong's addiction to marijuana, Lil