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.

Betting arbitrage, miracle bets, sure bets, sports arbitraging is a particular case of arbitrage arising on betting markets due to bookmakers’ different opinions on either event outcomes or plain errors. By placing one bet per each outcome with different betting companies, the bettor can make a profit. In the bettors' slang an arbitrage is often referred to as an arb; people who use arbitrage are called arbers. A typical arb is around 2 percent, often less; however 4-5 percent are occasionally seen and during some special events they might reach 20 percent. Arbitrage betting involves relatively large sums of money stakes are bigger than in normal betting.

Bookmakers generally disapprove of arbers, and restrict or close the accounts of those who they suspect of engaging in arbitrage betting. Although arbitrage betting has existed since the beginnings of bookmaking, the rise of the Internet, odds-comparison websites and betting exchanges have enabled the practice to be easier to perform. On the other hand, these changes also made it easier for bookmakers to keep their odds in line with the market.

The best way of generating profit, which has been established in Britain via sports arbitrage, consists of 'key men' employing others to place bets on their behalf, so as to avoid detection and increase accessibility to bookmakers. This allows the financiers or key arbers to stay at a computer to keep track of market movement.

While often claimed to be risk-free, this is only true if an arb is successfully completed; in reality, there are several threats to this:

Arbs in online sports markets have a median lifetime of around 15 minutes, after which the difference in odds underpinning them vanishes through betting activity. Without rapid alerting and action, it is possible to fail to make all the legs of the arb before it vanish, thus transforming it from a risk-free arb into a bet. High street bookmakers however, offer their odds days in advance and rarely change them once they have been set. These Arbs can have a lifetime of several hours.

Making errors: In the excitement of the action and due to the high number of bets placed, it is not uncommon to make a mistake like traders on financial markets. For example the appropriate stakes may be incorrectly calculated, or be placed on the wrong legs of the arb, locking in a loss, or there may be inadequate funds in one of the accounts to complete the arb. Those errors might temporarily have an important impact. In the long term, the benefit will depend on the odds. For example one could actually make more money by placing the wrong bet where the outcome happens to be beneficial, though not justified by the arbitrage calculation. However, this stroke of luck being repeated is unlikely, assuming the bookies have calculated the odds so they make a profit.

Bet cancellation: If a bettor places bets so as to make an arbitrage and one bookmaker cancels a bet, the bettor could find himself in a bad position because he is actually betting with all the risks implied. The bettor can repeat the bet that has been cancelled so as minimize the risk, but if he cannot get the same odds he had before he may be forced to take a loss. In some cases the situation arises when there are very high potential payouts by the bookie, perhaps due to an unintentional error made while quoting odds. Many jurisdictions allow bookmakers to cancel bets in the event of such a palpable [obvious] error in the quoted odds This is often loosely defined as an obvious mistake, but whether a palp in fact has been made is often the sole discretion of the bookmaker.

Other problems: Bookmakers who suspect arbing can set very low maximum stake limits, making arbing insufficiently profitable. Capital diffusion is serious; many bookmakers make it very easy to deposit funds and difficult to withdraw them. Making a return involves many bets spread over typically many bookmakers so keeping track is a considerable challenge, and requires excellent record-keeping.

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Shuffling

Shuffling is a procedure used to randomize a deck of playing cards to provide an element of chance in card games. Shuffling is often followed by a cut, to help ensure that the shuffler has not manipulated the outcome. A common shuffling technique is called the riffle or dovetail shuffle, in which half of the deck is held in each hand with the thumbs inward, then cards are released by the thumbs so that they fall to the table interleaved. Many also lift the cards up after a riffle, forming what is called a bridge which puts the cards back into place. This can also be done by placing the halves flat on the table with their rear corners touching, then lifting the back edges with the thumbs while pushing the halves together. While this method is more difficult, it is often used in casinos because it minimizes the risk of exposing cards during the shuffle. There are two types of perfect riffle shuffles: if the top card moves to be 2nd from the top then it is an in shuffle, otherwise it is known as an out shuffle which preserves both the top and bottom cards.

Riffle shuffling does, however, carry a risk of damaging cards from excessive bending. Casinos often replace their playing cards to prevent cheating from players that detect deformations in the cards. However, collectible card game cards are considerably less replaceable than playing cards, and CCG cards can be damaged from riffle shuffling, even when protected with card sleeves.

Because standard shuffling techniques are seen as weak, and in order to avoid inside jobs where employees collaborate with gamblers by performing inadequate shuffles, many casinos employ automatic shuffling machines. They also save time that would otherwise be spent shuffling, allowing several more hands per hour to be played and increasing the profitability of the table. These machines are also used to lessen repetitive motion stress injuries to a dealer. Note that the shuffling machines have to be carefully designed, as they can generate biased shuffles otherwise: the most recent shuffling machines are computer-controlled, though they have not yet fully been integrated into gaming.