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.

Slot Machine History

The first slot machine was invented by Charles Fey of San Francisco, California, U.S., who devised a much simpler automatic mechanism. Most assert that Fey invented the machine in 1887; however some believe that he may have conceived the machine in 1895. Due to the vast number of possible wins with the original poker card based game, it proved practically impossible to come up with a way to make a machine capable of making an automatic payout for all possible winning combinations. Charles Fey devised a machine with three spinning reels containing a total of five symbols – horseshoes, diamonds, spades, hearts, and a Liberty Bell, which also gave the machine its name. By replacing ten cards with five symbols and using three reels instead of five drums, the complexity of reading a win was considerably reduced, allowing Fey to devise an effective automatic payout mechanism. Three bells in a row produced the biggest payoff, ten nickels. Liberty Bell was a huge success and spawned a thriving mechanical gaming device industry. Even when the use of these gambling devices was banned in his home state after a few years, Fey still couldn't keep up with demand for the game elsewhere. Liberty Bell machine was so popular that it was copied by many slot machine manufacturers. Thus in 1907, manufacturer Herbert Mills from Chicago produced a slot machine called the Operator Bell. By 1908 lots of bell machines were installed in most cigar stores, saloons, bowling alleys, brothels and barber stores. The original Liberty Bell slot machine can still be seen at the Liberty Belle Saloon & Restaurant in Reno, Nevada.

Sittman and Pitt of Brooklyn, New York, U.S. developed a gambling machine in 1891 which was a precursor to the modern slot machine. It contained five drums holding a total of 50 card faces and was based on poker. This machine proved extremely popular and soon many bars in the city had one or more of the machines. Players would insert a nickel and pull a lever, which would spin the drums and the cards they held, the player hoping for a good poker hand. There was no direct payout mechanism, so a pair of kings might get the player a free beer, whereas a royal flush could pay out cigars or drinks, the prizes wholly dependent on what was on offer at the local establishment. To make the odds better for the house, two cards were typically removed from the deck: the ten of spades and the jack of hearts, which doubles the odds against winning a royal flush. The drums could also be rearranged to further reduce a player's chance of winning.

Another early machine gave out winnings in the form of fruit flavored chewing gums with pictures of the flavours as symbols on the reels. The popular cherry and melon symbols derive from this machine. The BAR symbol now common in slot machines was derived from an early logo of the Bell-Fruit Gum Company. The payment of food prizes was a commonly used technique to avoid laws against gambling in a number of states, and for this reason a number of gumball and other vending machines were regarded with mistrust by the courts. The two Iowa cases of State v. Ellis6 and State v. Striggles7 are both used in classes on criminal law to illustrate the concept of reliance upon authority as it relates to the axiomatic ignorantia juris non excusat Ignorance of the law is no excuse.8 In these cases, a mint vending machine was declared to be a gambling device because by internally manufactured chance the machine would occasionally give the next user a number of tokens exchangeable for more candy. Despite the fact that the result of the next use would be displayed on the machine, both courts ruled that The inducement for each play was the chance that by that play the machine would be set to indicate that it would pay checks on the following play. The thing that attracted the player was the chance that ultimately he would receive something for nothing. The machine appealed to the player's propensity to gamble, and that is a vice.9

In 1963, Bally developed the first fully electromechanical slot machine called Money Honey, although earlier machines such as the High Hand draw poker machine by Bally had exhibited the basics of electromechanical construction as early as 1940. The electromechanical approach of the 1960s allowed Money Honey to be the first slot machine with a bottomless hopper and automatic payout, of up to 500 coins, without the help of an attendant. The popularity of this machine led to the increasing predominance of electronic games, and the side lever soon became vestigial.

The first true, video slot machine was developed in 1976 in an industrial suite in Kearney Mesa, CA by N. Cerracchio, R. Greene, W. Beckman, J. Reukes, and L. Black under the direction of Fortune Coin Co., Las Vegas, NV. This slot machine used a modified 19 Sony Trinitron color receiver for the display, and logic boards for all slot machine functions. The prototype was mounted in a full size, show-ready slot machine cabinet. The first production units went on trial in the Las Vegas Hilton Hotel. After some cheat-proofing modifications, the video slot machine was approved by the Nevada State Gaming Commission and eventually found popularity in Las Vegas Strip and downtown casinos. Fortune Coin Co. and their video slot machine technology were purchased by IGT International Gaming Technology in 1978.

The first American video slot machine to offer a second screen bonus round was Reel 'Em In developed by WMS Industries Inc. in 1996.11 This type of machine had appeared in Australia from at least 1994 with the Three Bags Full game.12 In this type of machine, the display changes to provide a different game where an additional payout may be won or accumulated.

Poker Ante

An ante is a forced bet in which all players put an equal amount of money or chips into the pot before the deal begins. Often this is either a single unit a one-value or the smallest value in play or some other small amount; a proportion such as a half or a quarter of the minimum bet is also common. An ante paid by every player ensures that a player who folds every round will lose money though slowly, thus providing all players with an incentive, however small; to play the hand rather than toss it in when the opening bet reaches them.

Antes are the most common forced bet in draw poker and stud poker but are uncommon in games featuring blind bets see next section. However, some tournament formats of games featuring blinds impose an ante to discourage extremely tight play. Antes encourage players to play more loosely by lowering the cost of staying in the hand calling relative to the current pot size, offering better pot odds. With antes, more players stay in the hand, which increases pot size and makes for more interesting play. This is considered important to ensure good ratings for televised tournament final. Most televised high stakes cash games also use both blinds and antes. Televised cash games usually have one of the players pay for everyone, normally the dealer, to accelerate play. If there are six players for example, the dealer would toss six times the ante into the pot, paying for each person.

In live cash games where the acting dealer changes each turn, it is not uncommon for the players to agree that the dealer or some other position relative to the button provides the ante for each player. This simplifies betting, but causes minor inequities if other players come and go or miss their turn to deal. During such times, the player can be given a special button indicating the need to pay an ante to the pot known as "posting"; see below upon their return. Some card rooms eliminate these inequities by always dealing all players into every hand whether they are present or not. In such cases casino staff or neighboring players under staff supervision will be expected to post antes and fold hands on behalf of absent players as necessary.