The Probability of Rolling a Yahtzee

Yahtzee is a dice game involving a combination of chance and strategy. A player begins their turn by rolling five dice. After this roll, the player may decide to re-roll any number of the dice. At most, there are a total of three rolls for each turn. Following these three rolls, the result of the dice is entered onto a score sheet. This score sheet contains different categories, such as a full house or large straight.  Each of the categories is satisfied with different combinations of dice. The most difficult category to fill-in is that of a Yahtzee. A Yahtzee occurs when a player rolls five of the same number. Just how unlikely is a Yahtzee? This is a problem that is much more complicated than finding probabilities for two or even three dice. The main reason is that there are many ways to obtain five matching dice during three rolls. We can calculate the probability of rolling a Yahtzee by using the combinatorics formula for combinations, and by breaking the problem into several mutually exclusive cases. One Roll The easiest case to consider is obtaining a Yahtzee immediately on the first roll. We will first look at the probability of rolling a particular Yahtzee of five twos, and then easily extend this to the probability of any Yahtzee. The probability of rolling a two is 1/6, and the outcome of each die is independent of the rest. Thus the probability of rolling five twos is (1/6) x (1/6) x (1/6) x (1/6) x (1/6) 1/7776. The probability of rolling five of a kind of any other number is also 1/7776. Since there are a total of six different numbers on a die, we multiply the above probability by 6. This means that the probability of a Yahtzee on the first roll is 6 x 1/7776 1/1296 0.08 percent. Two Rolls If we roll anything other than five of a kind of the first roll, we will have to re-roll some of our dice to try to get a Yahtzee. Suppose that our first roll has four of a kind. we would re-roll the one die that doesn’t match and then get a Yahtzee on this second roll. The probability of rolling a total of five twos in this way is found as follows: On the first roll, we have four twos. Since there is a probability 1/6 of rolling a two, and 5/6 of not rolling a two, we multiply (1/6) x (1/6) x (1/6) x (1/6) x (5/6) 5/7776.Any of the five dice rolled could be the non-two. We use our combination formula for C(5, 1) 5 to count how many ways we can roll four twos and something that is not a two.We multiply and see that the probability of rolling exactly four twos on the first roll is 25/7776.On the second roll, we need to calculate the probability of rolling one two. This is 1/6. Thus the probability of rolling a Yahtzee of twos in the above way is (25/7776) x (1/6) 25/46656. To find the probability of rolling any Yahtzee in this way is found by multiplying the above probability by 6 because there are six different numbers on a die. This gives a probability of 6 x 25/46656 0.32 percent. But this is not the only way to roll a Yahtzee with two rolls. All of the following probabilities are found in much the same way as above: We could roll three of a kind, and then two dice that match on our second roll. The probability of this is 6 x C(5 ,3) x (25/7776) x (1/36) 0.54 percent.We could roll a matching pair, and on our second roll three dice that match. The probability of this is 6 x C(5, 2) x (100/7776) x (1/216) 0.36 percent.We could roll five different dice, save one die from our first roll, then roll four dice that match on the second roll. The probability of this is (6!/7776) x (1/1296) 0.01 percent. The above cases are mutually exclusive. This means that to calculate the probability of rolling a Yahtzee in two rolls, we add the above probabilities together and we have is approximately 1.23 percent. Three Rolls For the most complicated situation yet, we will now examine the case where we use all three of our rolls to obtain a Yahtzee. We could do this in several ways and must account for all of them. The probabilities of these possibilities are calculated below: The probability of rolling four of a kind, then nothing, then matching the last die on the last roll is 6 x C(5, 4) x (5/7776) x (5/6) x (1/6) 0.27 percent.The probability of rolling three of a kind, then nothing, then matching with the correct pair on the last roll is 6 x C(5, 3) x (25/7776) x (25/36) x (1/36) 0.37 percent.The probability of rolling a matching pair, then nothing, then matching with the correct three of a kind on the third roll is 6 x C(5, 2) x (100/7776) x (125/216) x (1/216) 0.21 percent.The probability of rolling a single die, then nothing matching this, then matching with the correct four of a kind on the third roll is (6!/7776) x (625/1296) x (1/1296) 0.003 percent.The probability of rolling three of a kind, matching an additional die on the next roll, followed by matching the fifth die on the third roll is 6 x C(5, 3) x (25/7776) x C(2, 1) x (5/36) x (1/6) 0.89 percent.The probability of rolling a pair, matching an additional pair on the next roll, followe d by matching the fifth die on the third roll is 6 x C(5, 2) x (100/7776) x C(3, 2) x (5/216) x (1/6) 0.89 percent.The probability of rolling a pair, matching an additional die on the next roll, followed by matching the last two dice on the third roll is 6 x C(5, 2) x (100/7776) x C(3, 1) x (25/216) x (1/36) 0.74 percent.The probability of rolling one of a kind, another die to match it on the second roll, and then a three of a kind on the third roll is (6!/7776) x C(4, 1) x (100/1296) x (1/216) 0.01 percent.The probability of rolling one of a kind, a three of a kind to match on the second roll, followed by a match on the third roll is (6!/7776) x C(4, 3) x (5/1296) x (1/6) 0.02 percent.The probability of rolling one of a kind, a pair to match it on the second roll, and then another pair to match on the third roll is (6!/7776) x C(4, 2) x (25/1296) x (1/36) 0.03 percent. We add all of the above probabilities together to determine the probability of rolling a Yahtzee in three rolls of the dice. This probability is 3.43 percent. Total Probability The probability of a Yahtzee in one roll is 0.08 percent, the probability of a Yahtzee in two rolls is 1.23 percent and the probability of a Yahtzee in three rolls is 3.43 percent. Since each of these are mutually exclusive, we add the probabilities together. This means that the probability of obtaining a Yahtzee in a given turn is approximately 4.74 percent. To put this into perspective, since 1/21 is approximately 4.74 percent, by chance alone a player should expect a Yahtzee once every 21 turns. In practice, it may take longer as an initial pair may be discarded to roll for something else, such as a straight.

How Parenting Styles Affect Childhood Development

How Different Parenting Styles Affect Childhood Development Parenting is one of the most challenging yet rewarding experiences in an adult’s life. Parents can greatly affect their children’s behavior and development. Children are like sponges, they soak up everything they see a parent do and model what they see into their own lives and actions. It is important that parents are good examples and set high standards for their children. Negative examples can be harmful to a child’s development and will usually lead to bad behavior. Baumrind, who studied parenting styles during the early 1960s, concluded that there is four important areas: parental nurturance, discipline strategy, communication skills, and maturity. Incorporating these areas, she posted four types of parenting styles: authoritarian, authoritative, permissive, and neglectful. Authoritarian Parenting In this style of parenting, children are given arbitrary rules and expected to give unquestioning obedience. Failure to follow instructions will result in severe or unnecessary punishment. Authoritarian parents tend to fail to explain the reasons for the rules, causing the child to see the rules as useless and meaningless to them. For example, the parent might reply to a child’s request with â€Å"No, because I said so.† Little to no respect is given to the child choices or opinions. These parents have extremely high standards, but are not emotionally connected with their child. The parents can be very demanding,Show MoreRelatedInfancy and Early Childhood Development1399 Words   |  6 Pagesand Early Childhood Development Cynthia Miranda PSY 375 July 2, 2012 Dr. Aneta Bhojwani Infancy and Early Childhood Development Every human being begins developing in one way or another, from the moment they come into this world. Development and all its aspects begin in infancy and continue throughout one’s entire lifetime. It is a common belief that infants and young children are influenced by the environment that surrounds them. Cognitive and social aspects of development are shapedRead MoreInfancy and Early Childhood1150 Words   |  5 PagesInfancy and Early Childhood Development Paper Beverly Mahone PSY/375 July 11, 2011 Andrew Rodriguez Infancy and Early Childhood Development Paper The immature years of life and the stage in which the most development occurs in a child are referred to as infancy and early childhood. 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The Gold Rush Writers free essay sample

A brief historical review of the California gold rush and the writers that came to record, illustrating the lives, times and seasons that brought hordes of people out West in search of gold. This paper looks at the styles of the many Gold Rush writers that came out west in search of writing the great American story. The writers shows that while often that didnt happen, these writers each had their own descriptive analysis of the way the gold rush affected the society. I was out prospecting on the Stanislaus, tramping all day long with pick and pan and horn, and washing a hat full of dirt here and there, always expecting to make a rich strike, and never doing it. It was a lovely reason, woodsy, balmy, delicious, and had once been populous, long years before, but now the people had vanished and the charming paradise was a solitude? (Twain 14). We will write a custom essay sample on The Gold Rush Writers or any similar topic specifically for you Do Not WasteYour Time HIRE WRITER Only 13.90 / page This quote from The Californians Tale shows the affect of the gold rush upon the California landscape. Once the land was charming and full of people, but now it is barren. While most people think of the gold rush era of American history as a time where miners grabbed their shovels and headed west to dig in the clean air and peaceful surroundings of the California mountains, the gold rush was anything but peaceful. Mining was difficult and miners had little affinity for the land; they only cared about excavating the land for the gold. These miners brought a carefree attitude to the west, not weighing the effects of their actions upon the California landscape. However, the miners were not alone in their actions. The gold rush writers, the most popular of whom was Mark Twain, also brought a something-for-nothing attitude to California. He was surprisingly unconcerned about the impact of so many people on so fragile a land. The most salient of which concerned romanticizing the gold rush, exploiting the resources of the California landscape, and the destruction of the land.

How much sympathy does the reader feel towards Curleys wife free essay sample

Curley’s wife is a character in the novel â€Å"Of mice and men† set in California and written in 1937 by John Steinbeck. She is the only woman on a ranch of itinerant working men, and because of this she gets treated by each man in a different manner. Most of the men treat her in a negative way, therefore causing different degrees of sympathy from the reader. Sympathy implies that the reader feels an emotional connection towards the character. Her unhappy marriage to the boss’s son causes her great loneliness and unhappiness as she tries, in vain, to find someone to talk to on the ranch. The reader may feel certain degrees of sympathy for Curleys wife due to the fact that the ranch workers are always talking behind her back. This is shown when Candy is talking about her to George and Lennie in the bunkhouse when they arrive. We will write a custom essay sample on How much sympathy does the reader feel towards Curleys wife? or any similar topic specifically for you Do Not WasteYour Time HIRE WRITER Only 13.90 / page He states that he thinks â€Å"Curley’s married a tart†. This quotation could cause very strong feelings against Curley’s wife, as she seems to have a reputation amongst the men for being flirtatious. The noun â€Å"tart† might infer that Curley’s wife is a promiscuous character who flirts with all of the ranch workers. It could also imply that she is always ‘tarted up’ and making an effort to look attractive in front of the men. Candy’s opinion seems harsh, which could portray the ranch worker’s extremely negative feelings towards Curley’s wife, as they assume she is a flirtatious attention seeker. However, the reader may feel sympathetic towards Curley’s wife because she is labelled as a â€Å"tart† and so the ranch workers don’t look past her appearance to get to know her personality. In this quotation, Candy pauses before completing the phrase, which could suggest that he is apprehensive towards voicing his opinion about Curley’s wife. This would be understandable, as most of the ranch workers seem intimidated by Curley and insulting his wife may result in a fight or losing their job. Some might think that this quotation reveals that Curley’s wife is incredibly lonely, as the ranch workers are prejudiced against her from the minute they get to the ranch, so this implies that it must be hard for her to make friends. She is the only woman on the ranch and seems to have trouble fitting in. She may be seen as very glamorous and she could like to get dressed up for fun, as it would make her feel like a movie star, which is her dream. Overall, I think this quotation causes a high level of sympathy from the reader towards Curley’s wife, because the ranch workers judge her and don’t try to get to know her personality. The judgement as to whether Curley’s wife is malicious or kind lies with the reader’s perception of her character’s description and actions in the novella. When Curley’s wife first appears in the novel in the bunkhouse scene, the reader is presented with an in-depth description of her appearance with multiple references to the colour red. This is expressed through quotations like â€Å"she had full rouged lips† and â€Å"her fingernails were painted red†. These declarative phrases could imply that she is an evil temptress who is not to be trusted and attracts lots of attention, which could be either wanted or unwanted on her part. Contrary to this, the colour â€Å"red† is a passionate, loving colour which could present her as a kind, affectionate character. The adjective â€Å"red† is however sometimes associated with the devil and malice, which links to the bible, as evil was introduced into the world when Eve was tempted by the snake, so by dressing in red Curley’s wife could be presenting herself as a temptation to the male ranch workers throughout the novella. Also, Curley’s wife has some aspects of her personality that imply softness, for example â€Å"little bouquets of red ostrich feathers†. These â€Å"feathers† are a significant part of Curley’s wife’s appearance, because the reader knows of Lennie’s fascination and admiration of soft objects. This feature of her appearance could portray a more attractive personality because the â€Å"ostrich feathers† create a gentle, toned-down persona for her character, which may cause a higher level of sympathy from the reader because the ranch workers treat her as if she is repulsive and manipulative, which could be a prejudice based on her appearance. Towards the end of her initial introduction, Curleys wife addresses George â€Å"playfully†, with could portray her as an intentionally promiscuous character. She is known around the ranch for teasing the male ranch workers and being apparently disloyal to Curley. The adjective â€Å"playfully† could cause the reader to feel unsympathetic towards Curleys wife, because she may give the men false hope by flirting and acting in a promiscuous way when she is around them. On the other hand, Curleys wife flirting could be interpreted as unintentional, harmless conversations and friendships between her and the ranchworkers. This would be understandable because as she mentions later in the novella in Crooks’ room, she gets â€Å"awful lonely† on the ranch. Curleys wife is seen by everyone on the ranch as a sexual object, due to her makeup and clothes. Whenever she tries to interact with any of the workers, they assume she is flirting and spread rumours about her, however she may just be a lonely woman on a ranch full of men. Curley regards her as an item that he possesses ownership over, and so he is always looking for her to make sure she isn’t with any of the other men. This shows that Curley doesn’t trust his wife, and so their relationship and marriage isn’t a happy one. In chapter four, the ranch workers go out to a brothel in the local town, leaving Lennie, Crooks, Candy and Curley’s wife at the ranch. In Crooks’ room, where the men are talking, Curley’s wife comes in and states that â€Å"They left all the weak ones here†, which shows the hierarchy of the ranch and highlights the fact that nobody wants to be with the â€Å"weak ones† at the bottom of the hierarchy. The reader may feel highly unsympathetic towards Curley’s wife because she says cruel, cutting things when she can. She says that â€Å"they left all the weak ones here†, but it isn’t clear whether she is including herself in that group or not. If she meant to include herself, the reader may feel sympathetic towards her because she feels that she is a weak person at the bottom of the hierarchy. This reflects the way she is treated by the male population on the ranch because she is constantly put down and distrusted when she just wants to make friends. On the other hand, if she was leaving herself out of â€Å"the weak ones† group, this may cause the reader to feel unsympathetic towards her because she is offending Candy, Crooks and Lennie, when they haven’t purposely done anything to offend or anger her. This quotation could also cause the reader to feel unsympathetic towards Curleys wife because she has two different personalities. When she is around Curley and other people who are higher in the hierarchy, she lets them disrespect her and treat her in a negative manner, however when she is around people who are lower in the hierarchy than her, she brings out a side to her that is evil and cutting, and so would make the reader feel unsympathetic towards her. However, this may make the reader feel sympathetic towards her because she never gets the chance to look down on anyone, she is always treated like she is the lowest of the low, therefore she makes the most of being higher while she can.