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.

George Douglass s Psalms 34 17-20 It - 876 Words

In Psalms 34:17-20 it says â€Å"When the righteous cry for help, the Lord hears and delivers them out of all their troubles. The Lord is near to the brokenhearted and saves the crushed in spirit. Many are the afflictions of the righteous, but the Lord delivers him out of them all. He keeps all his bones; not one of them is broken.† I believe this is a great symbolic allegory of the darkness that slavery birthed into our world. Each of these were bonded together in hopeless but in each case and region had different difficulties that had to face. Each using their life story to be a beacon of hope to others that freedom was within reach When we look at the biographies of Douglass, Ball, and Turner we see a common theme that connected each of these individuals: hopelessness. Frederick Douglass explained his hopelessness when he said â€Å"It opened my eyes to the horrible pit, but to no ladder upon which to get out In moments of agony, I envied my fellow-slaves for their stupidity. I have often wished myself a beast. I preferred the condition of the meanest reptile to my own. Any thing, no matter what, to get rid of thinking! It was this everlasting thinking of my con-dition that tormented me. There was no getting rid of it (1)† Charles Ball defined is hopelessness when he stated â€Å"I slept but little this night, feeling a restlessness when no longer in chains; and pondering over the future lot of my life, which appeared fraught only with evil and misfortune.† Nat Turner got into such aShow MoreRelatedBiblical Models of Servant Leadership13223 Words   |  53 Pagesleadership†¦and shepherding is considered a training ground for those destined to lead.†[3] On the other hand, Moses’ humble beginning that resulted in a drastic change in his early status from a slave‘s house to the palace (Exodus 2:1-10; Acts 7:20-22) may have had a drastic effect upon his personality. He grew up to be the proud child of the Pharaoh (Exodus 2:10), yet in his heart burnt the Hebrew love (Exodus 2:11-12)! But considering the way and manner he went about, one can not help but notice

Character Analysis Lady Macbeth free essay sample

Set in 1603, Shakespeare’s play, Macbeth, follows the life of what started out to be a normal married couple. When the couple, Macbeth and Lady Macbeth, discover from a prophecy that Macbeth would one day rule the land of Scotland, the two did everything in power to make sure this would come true. The couple devised a plan to murder Duncan, the current king of Scotland; Macbeth carried out this plan. With this newfound immense amount of royal power, Macbeth and Lady Macbeth go through several challenges in their relationship. Throughout Shakespeare’s play, Macbeth, there are significant changes in the way the audience perceives Lady Macbeth when considering her power, brutality, and physical characteristics. Comparing the beginning to the end of the play, it is apparent that Lady Macbeth is very different when considering her power. In the beginning, it appears as if Lady Macbeth was the superior in her relationship with Macbeth. At this period of time, man being inferior to woman was found to be very unusual. In a sense, Lady Macbeth was the â€Å"man† of the relationship. While debating whether or not Macbeth should kill Duncan, Lady Macbeth says, â€Å"When you durst do it, then you were a man; and to be more than what you were, you would be so much more the man† (Shakespeare 43). This line expresses Lady Macbeth’s opinion that Macbeth will only be a true man if he follows through with the murder; until then, Lady Macbeth would question his manhood. By using this line, Lady Macbeth has total control over her husband and can basically get him to do whatever she may ask. On the other hand, Macbeth seemed to turn around and take steps toward being the â€Å"bigger man† in their relationship as the play unfolded. With his immense amount of power, Lady Macbeth had no choice but to go with his superiority. An example of Lady Macbeth’s weakness was when she was conversing with the doctor and said, â€Å"Out, damned spot, out I say! One. Two† (Shakespeare 163). This quote explains how Lady Macbeth was obsessive over washing her hands, in other words, she was trying to get rid of the guilt of Duncan’s Murder. This section of the play shows Lady Macbeth’s significant change of weakness. As a weak woman, she succumbs herself to Macbeth’s power. In sum, Lady Macbeth went from being a powerful wife to a meaningless woman within the play. Another change the audience sees in the play is Lady Macbeth’s brutality. At the beginning of the play, she was the mastermind of Macbeth’s violent endeavors (Duncan’s Murder). When the two were plotting Duncan’s murder, it was Lady Macbeth who conveyed Macbeth to follow through; he was about to back out of the plan, as he did not want to commit such crime. When Macbeth was pondering the idea of failure, Lady Macbeth was quick to respond, â€Å"We fail? But screw your courage to the sticking place and we’ll not fail† (Shakespeare 43). This quote shows that Lady Macbeth was the one with confidence in their situation at hand. Towards the end of the play, it was Macbeth who was the mastermind behind his devious plans. Macbeth develops a sense of anxiety over Banquo, as he was the only one who suspected Macbeth had anything to do with Duncan’s Murder. Macbeth makes plans to kill Banquo, â€Å"Our fears in Banquo stick deep, and in his royalty of nature reigns that which would be feared† (Shakespeare 85). Macbeth was making these plans with two murders that he hired; not lady Macbeth. With the guilt of Duncan’s murder, it is doubtful that Lady Macbeth would offer support to Macbeth’s plan, given the opportunity, but Macbeth had purposely left her out of his plans. Now that he was king, he no longer needed assistance from his â€Å"housewife†. This just goes to show that Lady Macbeth went from being a brutal organizer of murder to a wife that has no part of her Husband’s secret life. Throughout Macbeth there are several observable changes in the physical being of Lady Macbeth. Though she may appear the same when focusing on her physical appearance, there are several physical changes in her mental health. At the beginning of the play, Lady Macbeth seemed to be a typical put together woman. At the end of the play, she virtually became a psychiatric patient who was in desperate need of mental healthcare. A doctor was consulted, but even the doctor knew there was nothing that could be done to save Lady Macbeth. While the doctor observed Lady Macbeth in her sleep, Lady Macbeth exclaimed â€Å"Here’s the smell of the blood still. All the perfumes of Arabia will not sweeten this little hand†¦Wash your hands† (Shakespeare 163-5). The doctor responds to the gentlewoman, â€Å"This disease is beyond my practice†¦So, good night. My mind she has mated, and amazed my sight. I think but dare not speak† (Shakespeare 165). This line shows the astonishment the doctor faces while observing Lady Macbeth. Her illness had reached a level of severity that a doctor could not even fix. Therefore, Lady Macbeth experiences many changes in her physical health throughout Macbeth. Macbeth is a very dense read with a lot of information and different characters. However, the changes in Lady Macbeth were very significant. Though the story concluded with her tragic death, Macbeth put his wife through countless situations that no woman should have to live through. She went from being the brains behind a murder to observing her husband committing several ruthless crimes. She also went from being a strong healthy woman to a woman with severe mental health issues. Throughout Shakespeare’s play, Macbeth, there are significant changes in the way the audience perceives Lady Macbeth when considering her power, brutality, and physical characteristics. Works Cited Shakespeare, William. Macbeth. N. p. : Folger Shakespeare Library, 1992. Print.

A Boy and A Bobcat free essay sample

Most people who ask a little kid what they want to be when they grow up receive the stereotypical answers such as wanting to be a firefighter, policeman, teacher, astronaut, or anything else that is considered a usual response. My answer, however, was a little different than most. When someone would ask me what I wanted to be when I grew up I would proudly say, â€Å"I want to be a boy.† Before I go on, I feel it is important for you to know that I am a girl and have been my whole life. But wait, it gets better. When they suddenly froze up and had a look of shock on their face, clearly not sure of the appropriate response, I would takeit one step further by informing them of another aspiration; I wanted my first car to be a Bobcat bulldozer. Even as a child, I had no limits on my imagination and had no issues sharing these dreams with anyone and everyone that I met. We will write a custom essay sample on A Boy and A Bobcat or any similar topic specifically for you Do Not WasteYour Time HIRE WRITER Only 13.90 / page Looking back, I am in disbelief that I made friends as a child – especially after making such bold announcements. Needless to say, I grew out of the stage of that particular ambition; however, I am still the same big dreamer I was as a child. As a child, I believed that anything could happen and often found myself wishing for things that would be considered impossible or over-the-top. I still carry that with me today and never let myself deem any aspiration impossible. No matter how senseless or ridiculous I must have sounded as a little girl wishing to be a boy, it never fazed me or caused me to change from what I thought to be possible.I never changed my answer to a more practical career, such as a veterinarian or something that is normal and expected.Instead, I would look the person straight in the eye, smiling from ear to ear, and proudly say that I was going to be a boy. Even from a young age, I have not been one to screw a cap onto the bottle of possibilities or tie my self down to keep from failing. No adventure was considered too dangerous; no obstacle was seen as unconquerable. I still see myself today as that little girl, jumping and twirling around, telling anyone and everyone she met about her ideas. This girl was not just an average, go-with-the-flow kind of person. This girl knows who she is and knows that she can accomplish the toughest of tasks. . She never placed limits on what she thought she couldachieve; even more, she never let others place a limit on her.. The child who was not afraid to speak her opinions or develop high expectations has become immutable. Although the hope for my future may have altered a bit, particularly the boy and bobcat part, my dreams have not grown smaller or become suppressed. They will forever shape who I am and what I will become. They cannot be concealed or put down, just like a wild child that dreamed of driving a Bobcat to school.

Dances With Wolves- Lakota Sioux essays

Dances With Wolves- Lakota Sioux essays Dances With Wolves, an academe award-winning movie, is about a solder that is united with a Sioux tribe. This movie however does not go into much detail about the Sioux tribes life. The Tribe has a very orderly system within its government, how their women are incorporated within the daily tribes duties, and how the tribes uses its land and its surroundings. As stated above, the Sioux tribe has a very stable government system. The tribe believes in the separation of power within the group, almost comparable to how the United States is ran on the checks and balances system. The tribe even has an attorney general who provides the enforcement of the codes of ethics, codes of the land, and the codes of commerce. If there is an appeal to the codes or anything to that nature, then they can appeal it to the Supreme Court level group. At the head or center of the tribe are the Board Members. The members are usually the elders of the tribe and are chosen by their wisdom, integrity, and knowledge of their culture. This is seen in the movie when Kevin Costners character is first taken to the tribe and he gets to be in the meeting. Women can also be involved in the government system, though it was not shown very good in the film. The women in the tribe are not excluded form the government as like it has been in the western civilizations. The main reason that women are involved in decision-making is because the tribe believes that input of all views are necessary for the stability of the tribe. Some of the jobs that women held or could do within the government were being able to choose or fire a chief and they also helped negotiate treaties. In some rare cases, women could even be chief. At the very lest, women could influence their husbands. Which was commonly done. This was portrayed in the movie when the wife asked the husband if their daughter was done mourning. The Sioux also had many uses with the land. As showed in ...

Development of Phoneme Perception Essay Example | Topics and Well Written Essays - 1250 words

Development of Phoneme Perception - Essay Example It is possible that there could be listeners whose discrimination patterns nearly manifest the phonetic system of their corresponding native language(s). In relation to this, Maye, Werker and Gerken (2002) demonstrated that there are possibilities by which infants may lose their ability to distinguish certain foreign language contrasts according to their sensitivity to the dispersal of sounds in the language input. When a child becomes aware that spoken words are comprised with sounds, he or she is described to have achieved "phonological awareness." The phonological awareness can exist in the form of awareness of rhyme, awareness of syllables, awareness of the onsets of words, etc. Phoneme awareness is essential to the process of learning to read; and explicitly teaching phoneme awareness facilitates reading acquisition, in the later stage of childhood. Research have shown that reading failure is linked to deficiency of phoneme awareness. It is important therefore, that early childhood teachers should be active in phoneme manipulation; demonstrate knowledge on the spoken words as it made up of phonemes which can be rearranged to make different words. Added cognitive benefits for young children that are at the same time exposed to more than one language is that they may have greater neural activity as well as denser tissue in the fragile areas of the brain which are related to many functions like memory, association, attention, and language than those who are monolingual learners. These variables are connected with long-term and positive cognitive outcomes (Bialystok 2001, Kovelman, Baker, Mechelli et al., 2004; & Petitto, 2006). Investigation on Performance on Native-languageStudies on the performance on native language is not new. Kuhl, et al. (2004) have conducted experiments on this topic. They compared and contrasted the 6-8 and 10-12 months Mandarin learning infants and American infants' listening to their respective native-language. The participants of the study were the 17 American infants whose mean age is 6-8 months (ten boys; seven girls) and 19 American infants whose mean age is 10-12 months (ten boys; nine girls). The two groups were recruited via the database of an Infant Studies Subject Pool ISSP based in University of Washington. There was same inclusion and exclusion criteria in the selection of the subjects. As part of the procedures, computer synthesized tokens in English were utilized and created using a male voice. The voice were matched in all acoustic details aside from the temporal features in the event of the initial portion of the consonants. The amplitude elevation time was 30 ms lo wer than the frication period to generate more natural-sounding conversation for English speakers. Also, the duration of the vowel reached 245 Your Name 3 minutes. The procedure and apparatus were just the same to that used to investigate perceptual development among infants' speech differentiation on native and non-native contrasts. Results/Findings The English and Mandarin-learning infants manifested

Lean concept in manufacturing field Literature review

Lean concept in manufacturing field - Literature review Example With the failure of MRP to offer enhanced system performance, the shift has focused to Just-in-time production systems which are driven by customer demand Hence, the jobs are â€Å"pulled through the system† with linkage between stages in the production process (Deleersnyder, et al., 1989). The Japanese system of Kanban is required in order to physically implement the pull production system. Kanban is often used in conjunction with Just-in-Time production whereby the right quantity must be manufactured at the appropriate level and at the right time. Literally translated as â€Å"a card†, the Kanban signal, which is generated through the master production schedule (MPS) or customer demand, triggers JIT as it works its way backwards through each work centre. Generally, a Kanban is tied to each container of work-in-progress (WIP) which contains specifications pertaining to that WIP such as the lot size, card number, due date etc. Research has demonstrated various benefits associated with the use of Kanban. JIT allows most companies to achieve the benefits of shorter lead times, enhanced quality and low inventory buffer (Cimorelli, 2013). However, choosing the Kanban size often requires tradeoffs. For instance, a large size of Kanban will often result in higher level of stock albeit with a shorter lead time and less time for setting up machines frequently. Furthermore, Kanban acts as a means of communication from usage points to the prior operation as well as serve the purpose of visual signage (Wang, 2011). As far as the types of Kanban are concerned, there are generally two types of Kanban systems: single card and dual card systems. This entails separating the storage of output of a particular stage from the storage of input in the succeeding stage along with the use of extra cards named as â€Å"withdrawal† Kanban (Krieg, 2005). This Kanban is defined as one which accompanies the containers that are responsible for storage at the input stage . This is followed by removal of the withdrawal Kanban and its subsequent storage in a â€Å"collection box† when the production system uses a container (Krieg, 2005). Subsequently, the withdrawal Kanban is further removed from the collection box by a carrier and moved into the storage for output from the previous stage. On the other hand, the single-card Kanban is more efficient for manufacturing processes that contain high â€Å"changeover time† owing to batch production (Basu & Wright, 2005). The major difference between a single-card and dual-card Kanban is that the former lacks a â€Å"production† Kanban and specific inventory points (Basu & Wright, 2005). To conclude, the use of Kanban in the Just-in-time production system is fairly old. However, their importance has steadily increased owing to the paradigm shift towards demand-pull manufacturing systems as opposed to push-manufacturing systems. Nevertheless it must be noted that although the Kanban system lead to efficient levels of inventory, shorter lead times and better flow of communication across the production system, it may not always fulfill all order qualifiers at the same time as tradeoffs may be involved. 2. Push/pull systems Lean manufacturing systems are usually split into pull and push production systems. Push systems are based on scheduling work such that it is released on the basis of customer demand (Kimura & Terada, 1981). This is because their aim is to maximize the usage of production capacity. These systems are driven by the system of due-dates. The release date is controlled and the level of Work in Progress is then observed. By using this system, companies are able increase the volume of their production which in turn reduces the cost of production per unit. The push strategy has been used by companies such as Dunkin Donuts for producing their donuts. Customer demand for each of the donuts (such as Glazed, chocolate, cream etc.) is arrived