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.

Analysis Of At The Abbey Theater By Rory O Neill

At the Abbey Theater in Dublin, Rory O Neill, a LGBT activist and drag queen known as Panti, confessed that he was a homophobe like everyone else. Panti explained that to grow up in a society that is overwhelmingly homophobic and to escape unscathed would be miraculous.† As a black man, I am very aware of racism and how it operates whether institutionally, internally or personally mediated. As a gay man, something I am still trying to deconstruct, I have been oblivious, perhaps intentionally, to my internalized homophobia. This dissonance in self-awareness is partially due to the fact that I knew I was black from early childhood but did not realize I was gay until sixth grade and did not begin to even accept it until twelfth. Many people compare the struggles of being black to that of being gay. In my opinion, despite some similarities the struggles are completely different. One is rooted in the legacy of over 200 years of slavery and the other in the tradition in people trying to cure homosexuality. One is more of an external conflict, the other internal. When I am followed around in stores like J.Crew or Saks, I am outraged and speak up. Realizing that ignorant people need to be educated, I usually write a letter to the VP of the company or loudly claim that I am being racially harassed in front of other customers. And it works. The VP s call me back and apologize, the staff goes through a sensitivity training, and I usually get a gift card with a hefty sum. On

Renal Problems And Septic Shock Health And Social Care Essay Free Essays

string(104) " analysis and repeated freezing melt of samples was avoided in order to forestall debasement of plasma\." Systemic inflammatory response syndrome ( SIRS ) is expressed as febrility or hypothermia, tachycardia, tachypnea which may be associated with leucocytosis or leukopenia. SIRS generates broad spread inflammatory reaction in organic structure in response to external abuse which is protective for an person when its effects are restricted to pathogens, in other state of affairss inflammatory reactions are hurtful when they are directed against normal tissues in add-on to pathogens. [ 1,2 ] SIRS can ensue from legion conditions but termed as Sepsis, merely when infection sets in and morbific agent is detected. We will write a custom essay sample on Renal Problems And Septic Shock Health And Social Care Essay or any similar topic only for you Order Now Sepsis consequences from an single response to external infection, which begins with systemic redness ab initio, followed by curdling abnormalcies and eventually deranged fibrinolysis. When sepsis causes one or more organ disfunction, the syndrome is termed Multiple Organ Dysfunction Syndrome ( MODS ) or terrible sepsis. [ 3 ] Sepsis-induced hypotension which is furnace lining to fluid boluses is termed Septic daze. Hypothermia associated with infected daze indicates hapless tegument and visceral perfusion, is normally associated with hapless forecast and high mortality rates of up to 30 to 40 % . Several serum biomarkers suggested holding diagnostic or predictive value in infected daze, but a unequivocal biomarker for everyday clinical usage is yet to be identified. One such marker is serum lactate which indicates pronounced hypoperfusion and tissue hypoxia in infected daze. Similarly serum Creatinine should besides be considered as marker of lessening organ perfusion in sepsis and Acute Kidney Injury should be regarded as index of ongoing organ harm and likely possibility of oncoming of infected daze. [ 4 ] AKI is due to sudden and drastic decrease in kidney map ( within 48 hours ) characterized by absolute addition in serum Creatinine ( gt ; 50 % from baseline ) or a decrease in urine end product ( oliguria of lt ; 0.5 ml/kg/hour for gt ; 6 hours ) . Nephritic hypoperfusion and ischaemia during infected daze amendss nephritic tubules taking to acute cannular mortification ( ATN ) and have been demonstrated to be a common etiologic factor for AKI development during sepsis [ 5,6 ] . ATN was found to be a consistent histopathological determination in these patients, this would strongly propose that ischaemia and nephritic tubular cell mortification are likely an of import pathogenetic mechanism. [ 7,8 ] Acute kidney hurt have marked impact on the result of critically sick patients. Disease badness tonss such as the Acute Physiology and Chronic Health Evaluation ( APACHE II ) and Sequential Organ Failure Assessment mark ( SOFA ) both have included nephritic disfunction as forecaster of morbidity and mortality ; on the other manus liver disfunction tonss, coagulopathy, thrombocytes and other critical organ maps are non much stressed in APACHE II hiting system. To set up a unvarying definition of nephritic harm, RIFLE categorization was formulated which characterizes Risk, Injury, Failure, Loss and End-stage Kidney ( RIFLE ) . [ 9 ] An of import facet of the RIFLE categorization is that it grades the badness of acute kidney hurt on the footing of alterations in serum creatinine and urine end product from the baseline status. [ 10 ] Urine end product is an of import physiologic mark of organic structure fluid position, and unstable instability is common in critically sick patients due to extravasation of fluid into extravascular infinite or due to 3rd infinite losingss and eventually the nephritic disfunction. This farther suggest that reduced organ perfusion in infected daze plays a cardinal function in development of AKI taking to cut down creatinine clearance and increased serum creatinine degrees. [ 11,12 ] In this survey we compared increasing serum creatinine degrees with plasma lactates and SOFA tonss to observe oncoming of sepsis and infected daze and to prove the hypothesis that ongoing acute kidney hurt can bespeak reduced organ perfusion and oncoming of infected daze in critically sick patients. Patients and Methods: This survey was carried out to happen a correlativity between lifting serum creatinine degrees and oncoming of infected daze in 115 critically sick patients admitted in ICU and were managed following Surviving Sepsis guidelines. [ 13 ] Human ethical blessing was taken by the institutional moralss commission. Written informed consents were obtained from control topics and patients or their relations. The control groups were the healthy relations attach toing the patient. Entire 90 controls were taken, among them 65 were males and 25 females with a mean ( SD ) age of 36.5 ( 8 ) old ages. Among patient group 67 were males and 48 females with mean ( SD ) age 37.5 ( 6 ) old ages. Patients included in our survey had either of the undermentioned characteristics: ( 1 ) Clinical characteristics proposing infection ; ( 2 ) Core temperature gt ; 38AÂ °C or lt ; 35AÂ °C ; ( 3 ) Heart beats gt ; 100/min ; ( 4 ) Respiratory rates gt ; 30 breaths/min or demand for supportive mechanical airing and ( 5 ) Inadequate organ map or daze within 12 hours of registration. Patients excluded were: ( 1 ) aged patients older than 75 old ages ; ( 2 ) NYHA category III or IV patients ; ( 3 ) liver inadequacy ( Child C ) ; ( 4 ) HIV, HBsAg positive serology, Cancer patients. Demographic characteristics like age, sex, primary site of infection, morbific beings and disease badness scores including Acute Physiology and Chronic Health Evaluation Scores ( APACHE II ) and Sequential Organ system Failure Assessment mark ( SOFA ) were recorded for each patient ‘s at the clip of admittance in ICU and later. The plasma of these patients was tested for serum creatinine and lactates degrees at the clip of entry in ICU, so after every 24 hours till their stay in ICU. All the samples collected and patient inside informations gathered during the survey were coded to look into for prejudice and patient confidentiality was maintained as per the guidelines for surveies of human patients. Blood sample aggregation: First blood sample was collected prior to get down of antimicrobic, steroid therapy or vasopressors. Blood samples were collected from cardinal venous line ( 9 milliliter ) into sterilized tubings incorporating 1ml trisodium citrate ( TSC ) at the clip of patient admittance I ICU and later. Plasma was separated by extractor at 10,000 revolutions per minute for 15 min. The plasma was stored at -70AÂ °C for farther analysis and repeated freezing melt of samples was avoided in order to forestall debasement of plasma. You read "Renal Problems And Septic Shock Health And Social Care Essay" in category "Essay examples" Statistical analysis: The informations were analyzed by nonparametric analysis of discrepancy ( ANOVA ) with Newman-Keuls multiple comparing post-test. The relation between serum creatinine degrees, plasma lactates and APACHE A ; SOFA mark was tested by finding the Pearson correlativity coefficient ( R ) . A P-value of less than 0.05 was considered important. All statistical analyses were performed with the Graph Pad InStat 5.0 demo plan ( Graph Pad Software, USA ) . Consequences: Among 115 patients admitted in ICU during the period April 2009 to May 2010, 45 patients were of SIRS, 39 patients were in sepsis and 31 patients in province of infected daze ( Table 1 ) . Sepsis was diagnosed on the footing of specific civilization studies from assorted possible sites of infection, including blood civilization study. Out of 115 patients studied there were 67 male patients and 48 females with a mean ( SD ) age of 37.5 ( 6 ) old ages. Mean serum creatinine degrees in healthy control group was 0.83 AÂ ± 0.26 mg/dl, runing from 0.4 to 1.6 mg/dl. In SIRS group, the average serum creatinine degrees were 3.45 AÂ ± 1.26 mg/dl, runing from 1.2 to 6.2 mg/dl ( Figure 1 ) . The APACHE II and SOFA score correlated linearly with serum creatinine degrees ( r2=0.74, P lt ; 0.001 for APACHE A ; r2=0.69, P lt ; 0.001 for SOFA ; Figure2 ) bespeaking nephritic hurt with badness of redness. In sepsis patients average degrees were 7.15 AÂ ± 1.3, runing from 4.0 to 9.3 mg/dl. Pearson ‘s coefficient showed additive correlativity of serum creatinine and sepsis badness tonss ( r2 = 0.65, P lt ; 0.001 for APACHE and r2 = 0.62, P lt ; 0.001 for SOFA ; Figure2 ) . In patients of infected daze with nephritic disfunction really high serum creatinine degrees were observed with mean of 10.31 AÂ ± 2.29 mg/dl with values runing from 6.0 to 15.2 mg/dl, these values were significantly ( P lt ; 0.01 ) higher than those with sepsis ( 7.01 AÂ ± 1.3 mg/dl ) , SIRS ( 3.49 AÂ ± 1.10 mg/dl ) and the control group ( 0.83 AÂ ± 0.26 mg/dl ) . Detection of metabolic acidosis in arterial blood gas analysis was evaluated farther by blood lactate appraisal ( Figure 3 ) . Blood lactate degrees in SIRS group were ( 5.14 AÂ ± 1.24mmol/L ) with scope from 2.2 to 7.8 mmol/L. Similarly blood lactate degrees in sepsis ( 7.73 AÂ ± 1.4mmol/L ) and infected daze group ( 9.53 AÂ ± 1.2mmol/L ) were significantly high ( 95 % CI in sepsis 7.25 to 8.21 ; daze 8.9 to 10.16 mmol/L ) . In control group mean values were 0.93 AÂ ± 0.39mmol/L ( 95 % CI 0.84 to 1.0 mmol/L ) . In order to detect a relation between extent of nephritic hurt and oncoming of infected daze, creatinine clearance ( taken as step of nephritic map ) was so compared with blood lactate degrees ( taken as index of anaerobiotic metamorphosis and daze ) , Pearson ‘s coefficient showed reverse relation between lifting lactate degrees and Creatinine clearance in patients with terrible sepsis and infected daze ( Figure 4 ; r2=0.48, P lt ; 0.001in sepsis ; r2=0.56, P lt ; 0.001in infected daze ) . The patients included in this survey were non given any nephrotoxic drugs and drug dose were modified as per criterion chronic nephritic disease guideline to forestall farther nephritic hurt [ 14 ] . Discussion: Patients come oning to infected daze during their stay in ICU may hold altered sensorium, deranged liver map, malabsorption syndrome or may hold respiratory trouble. It was observed in our survey that all such patients with infected daze have one common characteristic of crazed kidney map with AKI. The pathophysiology of AKI in infected daze is ill understood due to miss of histopathologic information, which in bend requires nephritic biopsy to observe any parenchymal or vascular alterations in nephritic tissue ; such biopsies are non routinely performed in most ICU set up. [ 15,16 ] In absence of such information, we performed an indirect appraisal of nephritic map during oncoming of infected daze. These observations were based on serum creatinine degrees and creatinine clearance which was so correlated with disease badness tonss ( APACHE II and SOFA ) and serum lactate degrees ( index of anaerobiotic metamorphosis and daze ) . Assorted carnal surveies of AKI have been done which helps in more complex and invasive measurings of nephritic maps. [ 17 ] Most of these carnal surveies were based on either ischemia-reperfusion hurt or drug induced hurt and informations generated from such surveies may assist us in understanding the pathophysiology in a infected patient with AKI. A major paradigm developed from these observations in animate beings and worlds with daze is that AKI is due to nephritic hypoperfusion and ischaemia. This fact supported our position of utilizing AKI as index for decreased organ perfusion and oncoming of infected daze. [ 18 ] Under basal conditions blood flow to kidney is 360 ml/min/100gm of tissue where as blood flow to encephalon ( 50ml/min/100gm ) , bosom ( 70ml/min/100gm ) , lungs ( 25ml/min/100gm ) and liver ( 95ml/min/100gm ) . [ 19 ] Among all the variety meats, kidneys have really high flow rates, approximately three times higher than other critical variety meats, therefore metab olic maps of kidneys were more likely to be affected earliest by decrease in blood flow during infected daze, this may ensue non merely in a decrease in glomerular filtration but besides, in metabolic impairment perchance doing cell decease, acute cannular mortification and terrible AKI. [ 20 ] Glomerular filtration rate ( GFR ) is governed by glomerular filtration force per unit area, which in bend is determined by the relationship between the sensory nerve and motorial arteriolas. When the sensory nerve arteriola constricts due to hypoxia in infected daze, glomerular filtration force per unit area will fall and urine end product and GFR will besides diminish, ensuing reduced creatinine clearance and increased serum creatinine. [ 21 ] Arterial lactate concentration correlates with badness of infected daze and reflects metabolic changes associated with hemodynamic via media. Therefore, the association between high lactate degree and diminishing creatinine clearance are built-in to the badness of the daze and of multi-organ failure. [ 22 ] In add-on, epinephrine extract during infected daze besides enhances lactate production by itself ; the higher adrenaline dosage required in more terrible patients may besides hold contributed to this association. Following the natural history of infected daze in most ICU, it is impossible to happen patients deceasing of infected daze but without nephritic failure, the huge bulk of patients deceasing of multiorgan failure has a outstanding nephritic constituent and requires nephritic replacing therapy during their class in ICU. Renal biopsy during infected daze is non performed routinely because of important hazards owing to azotemic or infected coagulopathy and patient instability. However, this restriction does non impair the relevancy of our observations sing the pathophysiology of infected daze, as kidneys have highest blood flow per gm of tissue and AKI is changeless characteristic of terminal phase multiorgan failure so any via media in nephritic map in critically sick patients should be considered as indicant of decreased organ map and likely oncoming of infected daze. Decision: Kidneies are invariably involved in multiorgan failure of infected daze. Nephritic lesions associated with AKI in infected daze are more complex than the simple ague cannular hurt, so reduced creatinine clearance should be regarded as index of underlying mechanism of decreased organ perfusion and likely possibility of oncoming of infected daze in critically sick patients. 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