Combinatorics

Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size deciding. Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry, and combinatorics also has many applications in optimization, computer science, ergodic theory and statistical physics. Many combinatorial questions have historically been considered in isolation, giving an ad hoc solution to a problem arising in some mathematical context. In the later twentieth century however powerful and general theoretical methods were developed, making combinatorics into an independent branch of mathematics in its own right. One of the oldest and most accessible parts of combinatorics is graph theory, which also has numerous natural connections to other areas. Combinatorics is used frequently in computer science to obtain formulas and estimates in the analysis of algorithms. Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. In elementary number theory, study integers without use of techniques from other mathematical fields. Questions of divisibility, use of the Euclidean algorithm to compute greatest common divisors, integer factorizations into prime numbers, investigation of perfect numbers and congruences belong here. Several important discoveries of this field are Fermat's little theorem, Euler's theorem, the Chinese remainder theorem and the law of quadratic reciprocity. The properties of multiplicative functions such as the Möbius function and Euler's φ function, integer sequences, factorials, and Fibonacci numbers all also fall into this area.

Dice Game

Playing Craps

Craps is one of the most exciting casino games. It common to hear yelling and shouting at a table. played on purpose-built table two dice are used. made after very strict standards routinely inspected for any damage. As matter course, replaced with new ones about eight hours use, casinos have implemented rules in way player handles them.
The player must handle the dice with one hand only when throwing and the dice must hit the walls on the opposite end of the table. In the event that one or both dice are thrown off the table, they must be inspected (usually by the stickman) before putting them back into play. The craps table can accommodate up to about 20 players, who each get a round of throws or at 'shooting' the dice. If you don't want to throw the dice, you can bet on the thrower. Several types of bets can be made on the table action. The casino crew consist of a stickman, boxman and two dealers.

The first roll of the dice in a betting round is called the Come Out roll - a new game in Craps begins with the Come Out roll. A Come Out roll can be made only when the previous shooter fails to make a winning roll, that is, fails to make the Point or seven out.
A new game then begins with a new shooter. If the current shooter does make his Point, the dice are returned to him and he then begins the new Come Out roll. This is a continuation of that shooter's roll, although technically, the Come Out roll identifies a new game about to begin.

When the shooter fails to make his or her Point, the dice are then offered to the next player for a new Come Out roll and the game continues in the same manner. The new shooter will be the person directly next to the left of the previous shooter - so the game moves in a clockwise fashion around the craps table.
The dice are rolled across the craps table layout. The layout is divided into three areas - two side areas separated by a center one. Each side area is the mirror reflection of the other and contains the following: Pass and Don't Pass line bets, Come and Don't Come bets, Odds bet, Place bets and Field bets. The center area is shared by both side areas and contains the Proposition bets.
Pass bets win when the come out roll is 7 or 11, while pass bets lose when the come out roll is 2, 3, or 12. Don't bets lose when the come out roll is 7 or 11, and don't bets win when the come out roll is 2 or 3. Don't bets tie when the come out roll is 12 (2 in some casinos; the 'Bar' roll on the layout indicates which roll is treated as a tie).

Craps Bets

Pass Line Bet - You win if the first roll is a natural (7, 11) and lose if it is craps (2, 3, 12). If a point is rolled (4, 5, 6, 8, 9, 10) it must be repeated before a 7 is thrown in order to win. If 7 is rolled before the point you lose.
Odds on Pass Line Bet - After a point is rolled you can make this additional bet by taking odds. There are different payoffs for each point. A point of 4 or 10 will pay you 2:1; 5 or 9 pays 3:2; 6 or 8 pays 6:5. You only win if the point is rolled again before a 7.
Come Bet - It has the same rules as the Pass Line bet. The difference consists in the fact you can make this bet only after the point on the pass line has been determined. After you place your bet the first dice roll will set the come point. You win if it is a natural (7, 11) and lose if it is craps (2, 3, 12). Other rolls will make you a winner if the come point is repeated before a 7 is rolled. If a 7 is rolled first you lose.
Odds on Come Bet - Exactly the same thing as the Odds on Pass Line bet except you take odds on the Come bet not the Pass Line bet.
Don't Pass Line Bet - This is the reversed Pass Line bet. If the first roll of a dice is a natural (7, 11) you lose and if it is a 2 or a 3 you win. A dice roll of 12 means you have a tie or push with the casino. If the roll is a point (4, 5, 6, 8, 9, 10) a 7 must come out before that point is repeated to make you a winner. If the point is rolled again before the 7 you lose.
Don't Come Bet - The reversed Come Bet. After the come point has been established you win if it is a 2 or 3 and lose for 7 or 11. 12 is a tie and other dice rolls will make you win only if a 7 appears before them on the following throws.
Place Bets - This bet works only after the point has been determined. You can bet on a dice roll of 4, 5, 6, 8, 9 and 10. You win if the number you placed your bet on is rolled before a 7. Otherwise you lose. The Place Bets payoffs are different depending on the number you bet on. 4 or 10 will pay 9:5; 5 or 9 pays 7:5, and 6 or 8 pays 7:6. You can cancel this bet anytime you want to.
Field Bets - These bets are for one dice roll only. If a 2, 3, 4, 9, 10, 11, 12 is rolled you win. A 5, 6, 7 and 8 make you lose. Field Bets have the following different payoffs: 2 pays double (2:1) while 12 pays 3:1. Other winning dice rolls pays even (1:1).
Big Six, Big Eight Bets - Placed at any roll of dice these bets win if a 6 or 8 comes out before a 7 is rolled. Big Six and Big Eight are even bets and are paid at 1:1.
Proposition Bets - These bets can be made at any time and, except for the hardways, they are all one roll bets:

Any Craps: Wins if a 2, 3 or 12 is thrown. Payoff 8:1

Any Seven: Wins if a 7 is rolled. Payoff 5:1

Eleven: Wins if a 11 is thrown. Payoff 16:1

Ace Duece: Wins if a 3 is rolled. Payoff 16:1

Aces or Boxcars: Wins if a 2 or 12 is thrown. Payoff 30:1

Horn Bet: it acts as the bets on 2, 3, 11 and 12 all at once. Wins if one of these numbers is rolled. Payoff is determined according to the number rolled. The other three bets are lost.

Hardways: The bet on a hardway number wins if it's thrown hard (sum of pairs: 1-1, 3-3, 4-4...) before it's rolled easy and a 7 is thrown. Payoffs: Hard 4 and 10, 8:1; Hard 6 and 8, 10:1

Gambling

Gambling is the wagering of money or something of material value as the stakes on an event with an uncertain outcome with the primary intent of winning additional money and or material goods. Typically, the outcome of the wager is evident within a short period.

Gaming refers to instances in which the law permits the type of gambling. Gaming and gambling are not mutually exclusive. A gaming company offers legal gambling activities to the public and a gaming control board may regulate the gaming company, like the Nevada Gaming Control Board. However, the English-speaking world does not acknowledge this distinction often. For instance, in the UK, the regulator of gambling activities is called the Gambling Commission not the Gaming Commission.  In addition, the word gaming describes activities that do not involve wagering, especially online.

Gambling is also a major international commercial activity, with the legal gambling market totaling an estimated $335 billion in 2009.

Under United States federal law, gambling is legal in the country and states are free to regulate or prohibit the practice. Gambling has been legal in Nevada since 1931, forming the backbone of the economy of the state. The city of Las Vegas is perhaps the best known gambling destination in the world. In 1976, gambling became legal in Atlantic City, New Jersey and in 1990, under state law, gambling became permissible in Tunica, Mississippi; both of those cities have developed extensive casino and resort areas since then. Since a favorable United State Supreme Court decision in 1987, many American Indian tribes have built their own casinos on tribal lands as a way to provide revenue for the tribe. Because the tribes are sovereign nations, they are often exempt from state laws that restrict gambling and are instead regulated under federal law.

Gambling Disorders Studies

The beliefs of a society about a health condition can have a huge impact on the people who suffer from the disorder. Public opinion can influence public health policy, public and private harm minimization efforts, research funds and treatment support. At the individual level, negative public views of a disease and the stigma it creates can strongly discourage individuals from admitting that he or she has the problem and seeking treatment for the condition. There is little data available on public opinion of gambling disorders; however, a new study published in the Journal of Gambling Studies fills this void with a systematic examination of public opinion on gambling disorders.

Researchers conducted telephone surveys with 8,467 adults in the Toronto area and questioned people about their opinions of how to best understand gambling disorders. Researchers asked if gambling disorders should be treated as a disease or illness, a wrongdoing, a habit, not disease or an addiction similar to drug addiction. Researchers also inquired if people with gambling disorders can get well on their own or must seek treatment to improve and polled adults on whether people with gambling disorders can reduce their gambling to that of a social gambler or if they need to quit altogether. The survey also gathered information on the gambling behavior and demographics of the respondents.

The researchers found that most people viewed gambling disorders as an addiction similar to drug addiction, with one-third seeing gambling as a habit and 17 percent viewing gambling as a form of wrongdoing. Responses to whether gamblers needed treatment to recover showed a split jury, and three out of four thought that abstinence from gambling activities must happen for recovery. Examining the demographics, the researchers found that being female, married, younger and without gambling problems paralleled believing that treatment and abstinence were necessary. In addition, people who viewed gambling problems as a disease or addiction also believed that treatment and abstinence for recovery are necessary.

The researchers noted that public perceptions reported in their study mimic the results of a 2003 study that examined the views of the public on alcohol use, with 71 percent of respondents saying that abstinence must occur for recovery. This popularly held belief is also the view of much of the scientific community as reflected by the upcoming changes the American Psychiatric Association is making.

Finally, researchers concluded that people with gambling disorders were less likely to think that treatment and abstinence were necessary for recovery. This may be because many people who meet the clinical guidelines for a gambling disorder do not think they have a problem and even those who believe they do have a problem are unlikely to seek treatment.