Odds 2

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Inprobability theory, odds provide a measure of the probability of a particular outcome. Odds are commonly used in gambling and statistics. For example for an event that is 40% probable, one could say that the odds are "2 in 5", "2 to 3 in favor", or "3 to 2 against".

Odds have a simple relationship with probability. When probability is expressed as a number between 0 and 1, the relationships between probability p and odds are as follows. Note that if probability is to be expressed as a percentage these probability values should be multiplied by 100%.

The numbers for odds can be scaled. If k is any positive number then X to Y is the same as kX to kY, and similarly if "to" is replaced with "in" or "for". For example, "3 to 2 against" is the same as both "1.5 to 1 against" and "6 to 4 against".

The language of odds, such as the use of phrases like "ten to one" for intuitively estimated risks, is found in the sixteenth century, well before the development of probability theory.[1] Shakespeare wrote:

The sixteenth-century polymath Cardano demonstrated the efficacy of defining odds as the ratio of favourable to unfavourable outcomes. Implied by this definition is the fact that the probability of an event is given by the ratio of favourable outcomes to the total number of possible outcomes.[2]

In statistics, odds are an expression of relative probabilities, generally quoted as the odds in favor. The odds (in favor) of an event or a proposition is the ratio of the probability that the event will happen to the probability that the event will not happen. Mathematically, this is a Bernoulli trial, as it has exactly two outcomes. In case of a finite sample space of equally probable outcomes, this is the ratio of the number of outcomes where the event occurs to the number of outcomes where the event does not occur; these can be represented as W and L (for Wins and Losses) or S and F (for Success and Failure). For example, the odds that a randomly chosen day of the week is during a weekend are two to five (2:5), as days of the week form a sample space of seven outcomes, and the event occurs for two of the outcomes (Saturday and Sunday), and not for the other five.[3][4] Conversely, given odds as a ratio of integers, this can be represented by a probability space of a finite number of equally probable outcomes. These definitions are equivalent, since dividing both terms in the ratio by the number of outcomes yields the probabilities: 2 : 5 = ( 2 / 7 ) : ( 5 / 7 ) . \displaystyle 2:5=(2/7):(5/7). Conversely, the odds against is the opposite ratio. For example, the odds against a random day of the week being during a weekend are 5:2.

Analogously, given odds as a ratio, the probability of success p or failure q can be computed by dividing, and the probability of success and probability of failure sum to unity (one), as they are the only possible outcomes. In case of a finite number of equally probable outcomes, this can be interpreted as the number of outcomes where the event occurs divided by the total number of events:

These transforms have certain special geometric properties: the conversions between odds for and odds against (resp. probability of success with probability of failure) and between odds and probability are all Mbius transformations (fractional linear transformations). They are thus specified by three points (sharply 3-transitive). Swapping odds for and odds against swaps 0 and infinity, fixing 1, while swapping probability of success with probability of failure swaps 0 and 1, fixing .5; these are both order 2, hence circular transforms. Converting odds to probability fixes 0, sends infinity to 1, and sends 1 to .5 (even odds are 50% probable), and conversely; this is a parabolic transform.

In probability theory and statistics, odds and similar ratios may be more natural or more convenient than probabilities. In some cases the log-odds are used, which is the logit of the probability. Most simply, odds are frequently multiplied or divided, and log converts multiplication to addition and division to subtractions. This is particularly important in the logistic model, in which the log-odds of the target variable are a linear combination of the observed variables.

On a coin toss or a match race between two evenly matched horses, it is reasonable for two people to wager level stakes. However, in more variable situations, such as a multi-runner horse race or a football match between two unequally matched teams, betting "at odds" provides the possibility to take the respective likelihoods of the possible outcomes into account. The use of odds in gambling facilitates betting on events where the probabilities of different outcomes vary.

In the modern era, most fixed-odd betting takes place between a betting organisation, such as a bookmaker, and an individual, rather than between individuals. Different traditions have grown up in how to express odds to customers.

Favoured by bookmakers in the United Kingdom and Ireland, and also common in horse racing, fractional odds quote the net total that will be paid out to the bettor, should they win, relative to the stake.[8] Odds of 4/1 would imply that the bettor stands to make a 400 profit on a 100 stake. If the odds are 1/4, the bettor will make 25 on a 100 stake. In either case, having won, the bettor always receives the original stake back; so if the odds are 4/1 the bettor receives a total of 500 (400 plus the original 100). Odds of 1/1 are known as evens or even money.

The numerator and denominator of fractional odds are often integers, thus if the bookmaker's payout was to be 1.25 for every 1 stake, this would be equivalent to 5 for every 4 staked, and the odds would therefore be expressed as 5/4. However, not all fractional odds are traditionally read using the lowest common denominator. For example, given that there is a pattern of odds of 5/4, 7/4, 9/4 and so on, odds which are mathematically 3/2 are more easily compared if expressed in the equivalent form 6/4.

A variation of fractional odds is known as Hong Kong odds. Fractional and Hong Kong odds are actually exchangeable. The only difference is that the UK odds are presented as a fractional notation (e.g. 6/5) whilst the Hong Kong odds are decimal (e.g. 1.2). Both exhibit the net return.

Wholesale odds are the "real odds" or 100% probability of an event occurring. This 100% book is displayed without any bookmaker's profit margin, often referred to as a bookmaker's "overround" built in.

In gambling, the odds on display do not represent the true chances (as imagined by the bookmaker) that the event will or will not occur, but are the amount that the bookmaker will pay out on a winning bet, together with the required stake. In formulating the 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 'overround' 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:

A study on soccer betting found that the probability for the home team to win was generally about 3.4% less than the value calculated from the odds (for example, 46.6% for even odds). It was about 3.7% less for wins by the visitors, and 5.7% less for draws.[14]

To understand roulette probabilities and calculate them, you need to know the formula. You take the numbers your bet is on and divide them by the total number of numbers in roulette (depending on your version of the game). Then you multiply by 100.[15]

Making a profit 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).

Also, depending on how the betting is affected by jurisdiction, taxes may be involved for the bookmaker and/or the winning player. This may be taken into account when offering the odds and/or may reduce the amount won by a player.

In 2018 the U.S. Supreme Court gave states permission to legalize sports betting if they wished to do so. It is currently legal in 38 states plus D.C., with other states either working on legislation or not considering it.

Here we can see that the bookmaker correctly priced Biden as the favorite to win the election. The higher the total payout (i.e., the higher the decimal odds), the less probable it is that the candidate will win (in the bookmaker's opinion) and the riskier the bet is.

Negative numbers (in American moneyline odds) are reserved for the favorite on the betting line and indicate how much you need to stake to win $100. Conversely, positive numbers are attached to the underdog and refer to the amount you could win if you bet $100. You stand to make more money on positive odds, but the chances of a win are lower.

Vegas odds (aka "Las Vegas odds") are a form of American money line odds used in sportsbooks. They can include a point spread representing the expected margin of victory. This allows bettors to wager not only on the winner but over or under the bookmaker's predicted point spread.

Betting odds calculator allows you to insert your odds and automatically convert them to American, Decimal, and Fractional odds. It also calculates the implied probability of the bet and the profit if the bet wins. The default wager is set at $100.

The way sports betting odds are presented can differ between American, Fractional, and Decimal. While they all mean the same thing, understanding how they work with your wager can be tricky. Use our betting odds and moneyline calculator tool above to convert these odds and learn more about them below.

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