Odds

The odds in favor of an event or a proposition are expressed as the ratio of a pair of integers, which is the ratio of the probability that an event will happen to the probability that it will not happen. For example, the odds that a randomly chosen day of the week is a Sunday are one to six, which is sometimes written 1:6, or 1/6. In probability theory and statistics, where the variable p is the probability in favor of the event, and the probability against the event is therefore 1-p, the odds of the event are the quotient of the two, or p/(1-p). That value may be regarded as the relative likelihood the event will happen, expressed as a fraction if it is less than 1, or a multiple if it is equal to or greater than one of the likelihood that the event will not happen. In the example just given, saying the odds of a Sunday are one to six or, less commonly, one-sixth means the probability of picking a Sunday randomly is one-sixth the probability of not picking a Sunday. While the mathematical probability of an event has a value in the range from zero to one, the odds in favor of that same event lie between zero and infinity. The odds against the event with probability given as p are (1-p)/p.

The odds against Sunday are 6:1 or 6/1 = 6: it is 6 times as likely that a random day is not a Sunday. Hence 'odds' are an expression of relative probabilities. Generally 'odds' are quoted in this format odds against rather than as odds in favor of, because of the possibility of confusion of the latter with the fractional probability of an event occurring. E.g., the probability of a random day of the week is a Sunday is 'one-seventh' 1/7. A bookmaker may for his own purposes use 'odds' of 'one-sixth', but the overwhelming everyday use by most people is odds of the form 6 to 1, 6-1, 6:1, or 6/1 all read as 'six-to-one' where the first figure represents the number of ways of failing to achieve the outcome and the second figure is the number of ways of achieving a favorable outcome: thus these are odds against. In other words, an event with m to n odds against would have probability n/ m + n, while an event with m to n odds on would have probability m/ m + n. Even in probability theory, odds may be more natural or more convenient than probabilities. This is in particular the case in problems of sequential decision making as for instance in problems of how to stop online on a last specific event, which is solved by the odds algorithm.

In some games of chance, using odds against is also the most convenient way to understand what winnings will be paid if the selection is successful: the winner will be paid 'six' of whatever stake unit was bet for each 'one' of the stake unit wagered. For example, a winning bet of 10 at 6/1 will win '6 × 10 = 60' with the original 10 stake also being returned. Betting odds are skewed to ensure that the bookmaker makes a profit—if true odds were offered the bookmaker would break even in the long run—so the numbers do not represent the true odds.

Odds on means that the event is more likely to happen than not. This is sometimes expressed with the smaller number first 1:2 but more often using the word on 2:1 on meaning that the event is twice as likely to happen as not.

Decimal presentation

Taking an event with a 1 in 5 probability of occurring i.e. a probability of 1/5, 0.2 or 20%, then the odds are 0.2 / 1 − 0.2 = 0.2 / 0.8 = 0.25. This figure 0.25 represents the monetary stake necessary for a person to gain one monetary unit on a successful wager when offered fair odds. This may be scaled up by any convenient factor to give whole number values. For example, if a stake of 0.25 wins 1 unit, then scaling by a factor of four means a stake of 1 wins 4 units.

Ratio presentation

Fixed odds gambling tends to represent the probability as fractional odds, and excludes the stake. For example, 0.20 is represented as 4 to 1 against written as 4-1, 4:1, or 4/1, since there are five outcomes of which four are unsuccessful. Thus, the stake returned must be added to the odds to compute the entire return of a successful bet. In craps, the payout would be represented as 5 for 1, and in money line odds as +400 representing the gain from a 100 stake.

By contrast, for an event with a 4 in 5 probability of occurring i.e. a probability of 4/5, 0.8 or 80%, then the odds are 0.8 / 1 − 0.8 = 4. If one bets 4 units at these odds and the event occurs, one receives back 1 unit plus the original unit 4 units stake. This would be presented in fractional odds of 4 to 1 on'' written as 1/4 or 1–4 , in decimal odds as 1.25 to include the returned stake, in craps as 5 for 4, and in money line odds as −400 representing the stake necessary to gain 100.

Fixed odds are not necessarily presented in the lowest possible terms; if there is a pattern of odds of 5–4, 7–4 and so on, odds which are mathematically 3–2 are more easily compared if expressed in the mathematically equivalent form 6–4. Similarly, 10–3 may be stated as 100–30.

Gambling odds versus probabilities

In gambling, the odds on display do not represent the true chances that the event will occur, but are the amounts that the bookmaker will pay out on winning bets. In formulating his odds to display the bookmaker will have included a profit margin which effectively means that the payout to a successful bettor is less than that represented by the true chance of the event occurring. This profit is known as the 'over-round' on the 'book' the 'book' refers to the old-fashioned ledger in which wagers were recorded, and is the derivation of the term 'bookmaker' and relates to the sum of the 'odds' in the following way:

In a 3-horse race, for example, the true probabilities of each of the horses winning based on their relative abilities may be 50%, 40% and 10%. These are simply the bookmaker's 'odds' multiplied by 100% for convenience. The total of these three percentages is 100%, thus representing a fair 'book'. The true odds against winning for each of the three horses are 1-1, 3-2 and 9-1 respectively. In order to generate a profit on the wagers accepted by the bookmaker he may decide to increase the values to 60%, 50% and 20% for the three horses, representing odds against of 4-6, 1-1 and 4-1. These values now total 130%, meaning that the book has an over round of 30 130 − 100. This value of 30 represents the amount of profit for the bookmaker if he accepts bets in the correct proportions on each of the horses. The art of bookmaking is that he will take in, for example, $130 in wagers and only pay $100 back including stakes no matter which horse wins.

Profiting in gambling involves predicting the relationship of the true probabilities to the payout odds. Sports information services are often used by professional and semi-professional sports bettors to help achieve this goal.

The odds or amounts the bookmaker will pay are determined by the total amount that has been bet on all of the possible events. They reflect the balance of wagers on either side of the event, and include the deduction of a bookmaker’s brokerage fee vig or vigorish.

Chicago Poker Card Game

The poker-related card game called Chicago is one of the most popular card games in Sweden today. Relying on the keeping of score instead of the placing of bets, it is suitable even for environments such as schools, where gambling is often prohibited. The game exists in countless versions, so here a somewhat arbitrarily chosen basic game will be followed by a number of possible variations.

Hand scores

The backbone of the game is that each poker hand has its own point value, as given in this table:

One pair - 1 point.
Two pair - 2 points.
Three of a kind - 3 points.
Straight - 4 points.
Flush - 5 points.
Full House - 6 points.
Four of a kind - 7 points
Straight flush - 8 points

Basic rules

Chicago is played with a standard 52-card deck. Each player is dealt five cards. The objective is to reach 52 points.
Exchanges and hand scoring

The players are allowed to exchange any number of their cards. If a player chooses to exchange one card only, he may choose "one up", meaning that he is dealt one card faced up, which he can either accept, or instead take the next card unseen. After the exchanges, the player with the best hand and only one player gets points for his hand. Then follows another round of exchanges, but no hand scoring.

Now, the first player begins by playing one card. Ordinary whist rules apply, but the players keep their cards collected by themselves. The player who wins the last trick gets 5 points. Also, the player with the best hand whether it is the same player or not gets points for his hand. Please Note: After achieving 42 points a player is no longer permitted to trade cards as they normally would. Instead, they are dealt 6 cards at the beginning of the game and must discard one before the first scoring round. No further exchanges are permitted.
Chicago

After the second exchange, any player can choose to play Chicago. In this case, he pledges himself to win all the tricks of the game. If he does, he is awarded 15 points, but if he fails, the penalty is just as harsh: -15 points.
Blind Chicago

Sometimes, a player given five cards below ten either inclusive or exclusive - must be decides before game starts is allowed to replace them before the exchanges begin.
Some play with 3 exchanges instead of 2. Then of course, scoring for hands will be made after both the first and the second exchange.
Some do not use the "one up" rule.
Often, a game will require that a player declare "Chicago" before they can win the game. The declaration is accepted regardless of whether one wins or loses the 5 tricks.
Often, one wants to give higher rewards than 7 or 8 points for four of a kind and Straight flush respectively. There are several ways to achieve this, most notably by elevating the player immediately to 52 points, or lowering either all players or one player of the holder's choice to 0 points, or a combination of these. Some also separate the Royal Flush from the Straight Flush, awarding 9 points for a Royal Flush. Holding a Royal flush usually means immediate victory.
The confusion is great as to what scores are appointed in the case of Chicago. Some will argue that no player will get any points at all besides the +15 or -15, whilst others will allow other points to be awarded. The +5 for the game, however, can never be stacked with the +15 for Chicago. Yet another variation is to award +13/-13 points for Chicago and the declaring player gets to go first. In that variation it is forbidden to declare Chicago unless the player has reached 13 points, ruling out the possibility of a negative score.
Some prescribe that any player with 45 points or more is not allowed to replace any cards.
Some require that after and not in the same hand as a player reaches 52 points, he must win the game once more before he actually wins. This handles the possibility that more than one player reach 52 points in the same hand.
Some award 10 points instead of 5 if the last trick is taken with a deuce. If this variant is employed, 30 points must also be awarded for a Chicago hand successfully ended with a deuce.

Wagering is 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.