Probability of Union of 3 or More Sets

Probability of Union of 3 or More Sets When two events are mutually exclusive, the probability of their union can be calculated with the addition rule. We know that for rolling a die, rolling a number greater than four or a number less than three are mutually exclusive events, with nothing in common. So to find the probability of this event, we simply add the probability that we roll a number greater than four to the probability that we roll a number less than three. In symbols, we have the following, where the capital P  denotes â€Å"probability of†: P(greater than four or less than three) P(greater than four) P(less than three) 2/6 2/6 4/6. If the events are not mutually exclusive, then we do not simply add the probabilities of the events together, but we need to subtract the probability of the intersection of the events. Given the events A and B: P(A U B) P(A) P(B) - P(A ∠© B). Here we account for the possibility of double-counting those elements that are in both A and B, and that is why we subtract the probability of the intersection. The question that arises from this is, â€Å"Why stop with two sets? What is the probability of the union of more than two sets?† Formula for Union of 3 Sets We will extend the above ideas to the situation where we have three sets, which we will denote A, B, and C. We will not assume anything more than this, so there is the possibility that the sets have a non-empty intersection. The goal will be to calculate the probability of the union of these three sets, or P (A U B U C). The above discussion for two sets still holds. We can add together the probabilities of the individual sets A, B, and C, but in doing this we have double-counted some elements. The elements in the intersection of A and B have been double counted as before, but now there are other elements that have potentially been counted twice. The elements in the intersection of A and C and in the intersection of B and C have now also been counted twice. So the probabilities of these intersections must also be subtracted. But have we subtracted too much? There is something new to consider that we did not have to be concerned about when there were only two sets. Just as any two sets can have an intersection, all three sets can also have an intersection. In trying to make sure that we did not double count anything, we have not counted at all those elements that show up in all three sets. So the probability of the intersection of all three sets must be added back in. Here is the formula that is derived from the above discussion: P (A U B U C) P(A) P(B) P(C) - P(A ∠© B) - P(A ∠© C) - P(B ∠© C) P(A ∠© B ∠© C) Example Involving 2 Dice To see the formula for the probability of the union of three sets, suppose we are playing a board game that involves rolling two dice. Due to the rules of the game, we need to get at least one of the die to be a two, three or four to win. What is the probability of this? We note that we are trying to calculate the probability of the union of three events: rolling at least one two, rolling at least one three, rolling at least one four. So we can use the above formula with the following probabilities: The probability of rolling a two is 11/36. The numerator here comes from the fact that there are six outcomes in which the first die is a two, six in which the second die is a two, and one outcome where both dice are twos. This gives us 6 6 - 1 11.The probability of rolling a three is 11/36, for the same reason as above.The probability of rolling a four is 11/36, for the same reason as above.The probability of rolling a two and a three is 2/36. Here we can simply list the possibilities, the two could come first or it could come second.The probability of rolling a two and a four is 2/36, for the same reason that probability of a two and a three is 2/36.The probability of rolling a two, three and a four is 0 because we are only rolling two dice and there is no way to get three numbers with two dice. We now use the formula and see that the probability of getting at least a two, a three or a four is 11/36 11/36 11/36 – 2/36 – 2/36 – 2/36 0 27/36. Formula for Probability of Union of 4 Sets The reason why the formula for the probability of the union of four sets has its form is similar to the reasoning for the formula for three sets. As the number of sets increases, the number of pairs, triples and so on increase as well. With four sets there are six pairwise intersections that must be subtracted, four triple intersections to add back in, and now a quadruple intersection that needs to be subtracted. Given four sets A, B, C and D, the formula for the union of these sets is as follows: P (A U B U C U D) P(A) P(B) P(C) P(D) - P(A ∠© B) - P(A ∠© C) - P(A ∠© D)- P(B ∠© C) - P(B ∠© D) - P(C ∠© D) P(A ∠© B ∠© C) P(A ∠© B ∠© D) P(A ∠© C ∠© D) P(B ∠© C ∠© D) - P(A ∠© B ∠© C ∠© D). Overall Pattern We could write formulas (that would look even scarier than the one above) for the probability of the union of more than four sets, but from studying the above formulas we should notice some patterns. These patterns hold to calculate unions of more than four sets. The probability of the union of any number of sets can be found as follows: Add the probabilities of the individual events.Subtract the probabilities of the intersections of every pair of events.Add the probabilities of the intersection of every set of three events.Subtract the probabilities of the intersection of every set of four events.Continue this process until the last probability is the probability of the intersection of the total number of sets that we started with.

Eyewitness error Assignment Example | Topics and Well Written Essays - 1000 words

Eyewitness error - Assignment Example Since, the attained information is either inadequate or poorly collected whereby those who facilitate the entire process before litigation commences present questions to the interviewee towards a certain direction when interrogating eyewitnesses (Wise, Dauphinais & Safer, 2007). This prompts eyewitnesses to err while delivering the essential information meant to help judges and those arbitrating the case impose unnecessary charges on the defendant (Tillman v. State, 2011). Therefore, information collection as well as representation during ligation process ought to consider proper legislative process, which this study seeks to present based on certain given cases (Kenneth Adams v. State, 1993). Mainly, this legislative action entails eliminating eyewitness errors coupled with considering essential issues that characterize criminal cases, for instance, crime control and due process (Wise, Dauphinais & Safer, 2007). To eliminate eyewitness errors, legal system ought to adopt the most ef fective method meant to extract the required information. This will ensure the convicted serve just sentences. In addition, it will ensure the entire process observes the involved parties’ rights, which the law dictates despite their role pertaining to the case. Therefore, this entails utilizing proficient officers to interrogate eyewitness and not excluding the service of eyewitness experts, for instance, psychologists (Tillman v. State, 2011). However, psychologists ought to be proficient in their field to the extent of being able to proffer the required information based on reliable scientific background and pertinent to the case supposed to undergo litigation process (Tillman v. State, 2011). In my opinion, the best solution or approach in eliminating numerous eyewitness errors that characterize litigation process is by employing Tripartite solution (Wise, Dauphinais & Safer, 2007). In this solution, the initial step entails considering eyewitness’ testimony during the litigation process, which is also supported in the Tillman v. State case (Tillman v. State, 2011). Since, this enables the jury and those arbitrating cases to understand essential facts that may be difficult especially in the field judges are not conversant with, for instance, psychology (Martire & Kemp, 2009). Therefore, the eyewitness expert ought to be capable of proffering adequate information coupled with being proficient in the field, which will help him or her deliver reliable and pertinent informatio, hence deliver information or essential facts regarding the eyewitness from psychological perspective which may pose a task especially to the law enforcement officers. This is evident in Tillman v. State’s case though Dr. Malpass failed to present clear and convincing facts intended to prove the testimony he claimed to have a strong scientific base and be pertinent to the case (Tillman v. State, 2011). Therefore, to ensure reliability of any information by eyewitness expert, the proponent ought to prove his or her (1) expertise employed is legitimate, (2) testimony lies within the required scope, and (3) uses the field’s principles (Tillman v. State, 2011). The second step entails adopting effective procedures when extracting the required information to act as eyewitness evidence (Wise, Dauphinais & Safer, 2007). This is via employing scientifically proven procedures that will not only encourage the eyewitness to deliver the

Compare and contrast reform under Khrushchev and Gorbachev Essay

Compare and contrast reform under Khrushchev and Gorbachev - Essay Example 2009). In 1934 Khrushchev became a member of the partys central committee. He had a close link to Soviet leader Joseph Stalin, and after Stalins death in 1953 Khrushchev surfaced as the new leader. He started restructuring Stalins most atrocious extremes, and as he criticized some crimes of Stalin which was considered as a dramatic development. Khrushchev as well tried to create fresh dealings with the United States; in 1959 he visited the U.S. and met with President Eisenhower (answers.com. 2009). Khrushchev delivered his well-known secret speech in 1956 wherein he exposed the reality of a letter written by Lenin prior to his death. The letter was significant of Stalin’s misdeeds, and Khrushchev made use it as weapon to condemn Stalins rules and practices, mainly the purges that he supported. This denigration led to de-Stalinization, a course that led to reforms, for instance leniency towards government censorship of the press, transference of economic decision making, and streamlining of the cooperative farm. Concerning foreign policy, Khrushchev promoted ‘peaceful coexistence,’ and reduction of strain involving the United States and the Soviet Union. At the same time Khrushchev was condemned from the start for the recommended reforms, and his political and armed forces failure in the Cuban Missile Crisis made certain his loss of control. In addition, the majority of his reforms were not working by the early 1960s (phs.prs.k12.nj.us/ewood., N.D) The Khrushchev’s duration of rule did not limit itself to the financial system. Where as he went on to demonstrate how changing financial requirements were mirrored in the social and political ‘superstructure’. The most remarkable aspect after the Stalin period was the reduction of scare and the majority of the labour camps were out of terror and more relaxed. Essential basics of the rule of law were reinstated, as the

Food Cart Starting Mobile Catering Business Marketing Essay

Food Cart Starting Mobile Catering Business Marketing Essay Mobile catering business has become a ubiquitous feature of urbanization in many countries around the world. Countries such as the UK are a classic example of the popularity of this business, and a number of mobile catering vans can be spotted at the busy destinations of London and other cities. The good part about this business is that almost anyone with a basic knowledge and experience of food catering services may enter this business. It does not require much financial investment to start this business. At the same time, this business offers enough flexibility for the entrepreneur to scale up the scope of business along with the rise in market demand. Before going ahead with the steps of setting up the mobile catering business, it is important to understand what kind of vehicles can be used for this business. Food Cart: Starting Mobile Catering Business A food cart is a sort of trailer that has no motorized operation. It has to be manually hauled to the point of sale. Usually, such food carts operate in public areas such as parks, picnic spots, or busy sidewalks. Such carts normally maintain an automated heating or refrigerating equipment to keep the food absolutely ready to serve instantly. Mostly such carts sell snacks, breakfast items, or desserts such as Ice creams. Catering Van: Starting Mobile Catering Business A catering van is generally a motorized vehicle that can be moved from one place to another easily. It sells a substantially higher volume of food than a food cart, and caters to a bigger clientele. Ice cream vans are a common example of such catering vans in the UK. Their immense popularity among the customers has made mobile catering business into a highly lucrative business opportunity for many new entrepreneurs. Food Truck: Starting Mobile Catering Business A food truck is a larger vehicle converted into a mobile kitchen with a commercial built-in grill, a deep fryer, freezer, and other automated cooking gadgets. Due to the better availability of space and more equipment, this vehicle is able to offer a larger variety of food items to the customers, and manage larger volumes with efficiency. Fish and Chip vans in the UK are a good example of such food trucks. Concession Trailer: Starting Mobile Catering Business A concession trailer is not much different from a mobile kitchen, except that it is a stationary vehicle. Therefore, such trailers are ideally suited for large public events such as fairs and community celebrations where a large number of people are expected to converge over a period of few days consecutively. Such trailers add to the facilities and atmosphere of the event, providing the visitors with plenty of opportunities to enjoy different types of food at low cost. Anyone can launch a mobile catering business successfully, if one follows these basic steps to set up the business: Step 1: Setting up the kitchen facility The first step in this business is to decide whether the kitchen facility will be set up within the mobile van unit, or at another location. If the business owner decides to use his own home kitchen to prepare the food, it is important to obtain the approval of the local regulatory authorities in advance. Only after the conditions of the local law have been satisfied, the commercial kitchen can be operated out of home. Setting up the commercial kitchen may have to include separate provision for storage and refrigerating facility, dish-washing equipment, sufficient surface area for cooking, and larger size of cooking equipment. Step 2: Deciding the menu and the pricing A crucial part of the mobile catering business planning is to choose the menu, and decide the right pricing. This is the most sensitive business decision that may influence the destiny of the business over a period of time. A basic market survey and research can help to understand the local taste and preferences of food, and to identify the potential food items that are in high demand which is waiting to be satisfied. Similarly, the pricing must also be decided carefully to ensure that it competes successfully with other market players, and makes a reasonable profit margin for the business owner at the same time. Step 3: Organizing the mobile catering vehicle There are various options for vehicles available before the entrepreneur in a city like London or other cities within the UK. The choice of vehicle depends upon the ready availability of a vehicle, or the personal preference of the entrepreneur, and also keeping in mind the available financial budget for the business. The size of vehicle will also depend upon the scale of the business that the entrepreneur has planned. The heating, refrigerating, storage and preservation equipment within the vehicle will depend upon the kind of menu that has been decided for the mobile catering business. A fully functional kitchen may also be organized within the vehicle, depending upon the business plan of the owner. Step 4: Advertising and Marketing the Mobile Catering Business In the times of stiff competition, even a seemingly simple business such as mobile catering also cannot afford to ignore the importance of advertising and marketing. The menu of the business can be designed in such a way that it also serves as a brochure and advertisement for the business. Special brochures to advertise the business may also be printed and distributed to the local commercial establishments in the area to attract the potential office-goers to patronize the business. The mobile van itself can serve as the biggest advertisement for the business. The branding and slogan of the business must be displayed prominently and creatively on the vehicle. Usually the mobile vans in the UK and other countries can be seen parked in major locations where they can have high visibility for the local traffic in the area. Therefore, maximum advantage must be drawn by advertising on the mobile vehicle. Another key idea for many mobile catering business operators who are looking for alternative places to establish their kitchen can be to rent space at a local restaurant or a local church or hall. Such places do not normally use their commercial kitchens in the early morning hours, and many of them are happy to rent out their kitchens by the hour. This opportunity can be available by the mobile caterer who can make long-term tie-up with a local restaurant or other commercial kitchen, and use their facilities in the early morning hours to make his food ready for the day. With these innovative steps, it is possible to set up the mobile catering business successfully with a low investment. Over a period of time, as the business grows, the entrepreneur can expand the scope of the business by adding more facilities and more vehicles for the business.

College Athletes Must be Held to the Same Standards as Full-Time Studen

Despite a tendency for institutions to be lenient with college athletes, as they enjoy the benefits of full-time-student status, they must be held to the same standards as non-athlete students. Every institution has an obligation to set attendance rules; students who attend these institutions have an obligation to abide by them, however, in actuality, professors chose whether or not they’re enforced. This may cause significant problems between other students targeting a player, rather than who is responsible, the Professor. Although institutions want their students equally treated, diversity is well accepted. Existence as a college athlete contributes to diversity as well as a substantial amount of favoritism. Issues arise pertaining to how one may view an institution as well as the institution itself, due to an athlete’s gift. Though athletics contribute to a decrease in anxiety and stress, in some cases, facts show otherwise. Furthermore, every institution has a set o f rules pertaining to attendance policies, that every student, including athletes need to follow. Most institutions have similar attendance polices. Their policies include: attending regular classes on time, fully comprehending the syllabus for every class, prompt notice to instructor regarding absences due to competition and/or travel, once a certain amount of days have been missed the student is dropped, and failure to properly communicate with instructor may result in a failing grade. Policies have acknowledged the student-athlete is fully responsible for communicating with their professors regarding their attendance. They have also acknowledged students are not entitled to a certain number of absences. Most importantly, policies have stated the importance of in... ...Q Educause Quarterly.NDP. Web. 28 November 2011. Educause Quarterly Magazine, Volume 30, Number 2. 2007. www.educause.edu/EDUCAUSE+Quarterly/EDUCAUSEQuarterlyMagazineVolum/clickerintheclassroomanactive/157458 Pros, Cons on pay for play: yahoo. USA Today. 31 August 2004. Web. 25 November 2014. www.usatoday.com/sports/2004-08-31-pros-cons-pay_x.htm Taylor, Donald L. "A Comparison Of College Athletic Participants And Nonparticipants Of Self-Esteem." Journal Of College Student Development 36.5 (1995): 444-51. ERIC. Web. 12 Dec. 2014. Student Athlete Class Attendance Policy: JEFFCO. Viking Intercollegiate Athletics. 14 October 2011. Web. 25 November 2014. Jefferson College Policy. www.jeffco.edu.edu/athletics/inex.php?option=com_content&task=view&id=287&Itemid=256 Worsnop, Richard L. "College Sports." CQ Researcher 26 Aug. 1994: 745-68. Web. 12 Dec. 2014.

The Most Important Day of My Life

The Day that Changed my Life There I was having an ordinary day, a barbecue with my friends where all went as normal and then I get a call from my daughter, I was so happy as she hadn’t called me in over 6 weeks but that happiness was only temporary. She wanted to have lunch with me at that afternoon so I abandoned my friends and the barbecue just for her.While there our conversation stays on only one topic, me, until she wants the favour that I regret giving- my permission to go to Kenya for some charity work. In hope of retrieving her trust, I agree to such thing as I thought she would be safe. I warned her about the danger of the world but I knew she didn’t pay attention. I went home worried and panicking, I couldn’t sleep or concentrate on anything else, I just waited for her call but when it came it wasn’t a good one.Kidnappers called with a voice so dark and horrifying that my heart almost stopped. Their words were â€Å"She is gone forever†, I cried and cried†¦ When I finally got full of anger, I got on the first plane to Kenya looked for someone who could help me, get a gun and get my daughter back. Then I found Rebecca, she told me that these kidnappers have been working for 3 months kidnapping innocent teens from the airport and taking them to the Old Stadium to auction them.

Impact of Dividend Policy on Share Price Free Essay Example, 1500 words

Nowadays, the business firm relates the dividend policy with the product life cycle (PLC). For an example, a firm with large cash flow, high growth rate and lower trends in project appraisal tries to pay more dividends to the shareholder, as the earnings of the firm are comparatively high. The firms with high growth rate follow different interesting patterns in order to change the pattern of such decisions and enhance the complexity (Lease, 56). Dividends totally depend on earnings of the business firm. If the earnings of business firms are increased, then the dividend should increase too. On the other side, the dividend cut denotes the decreasing trend of earnings. It is very essential to understand the situation, how the dividend payout policy affects the share price. Moreover, it will help to recognize the influence of dividend payout policy on share prices. The literature review starts with generalizing the overall scenario which called perfect capital market assumptions are as follows. Miller and Modigliani (MM) first introduced the contradiction of dividend announcement whether it effects on the share price or not. The augmentation of payout from the dividend needs to be combined with the increase of business growth according to the dividend discount model. We will write a custom essay sample on Impact of Dividend Policy on Share Price or any topic specifically for you Only $17.96 $11.86/page Moreover, tax on the dividend is more than the tax on capital gain, which affects the fiscal policy. This case also influences the partialities of the tax-paying investors. The average tax paying investors can hold a well-diversified portfolio (Balanced portfolio) of high yield and low yield shares that offer the same facilities on the rate of return after tax.