Friday, January 14, 2022

Implied Probabilities

Oddsmakers provide a useful prediction mechanism for many subjects of interest. Beyond sports, they host prediction markets for events in politics, entertainment, current events and other fields. There are some subtleties, however, in converting payout odds to probabilities. Note that that payout odds (also called payoff odds or house odds) are expressed in this article as fractional odds (there are several other popular formats used in casinos and on-line bookmakers, such as decimal odds and American odds: they simply indicate the same information a different way).

Payout Odds

It is important to understand the specific nature of payout odds, which are not the same as odds taught in statistics courses. Payout odds indicate the schedule of payment for win or loss at the conclusion of a wager. Typically, payout odds overpredict the probability of a specific outcome. Think about it this way: The rarer the event the player successfully predicts, the more the house will need to pay out. Hence, payout odds tend to be upwardly biased.

If the oddsmaker believes that the probability of an event is 50%, then a bet which does not favor the house nor the player would be 1/1: The house puts up $1 and the player puts up $1. To make such a situation work to the oddsmaker's advantage, the payout odds might be set by the house to, for instance, 5/6: the house puts up only $5 while the player puts up $6 . The player in this case is expected to lose an average of $0.50 dollars: Half the time, the player leaves with $11, and the other half of the time the player leaves with nothing: an average payout of $5.50 after putting up $6.

Implied Probability

Payout odds can be converted to implied probabilities by doing some quick arithmetic: the player's wager is divided by the total wager (the house wager plus the player's wager). Continuing with the hypothetical payout odds of 5/6, the implied probability is 0.54 = 6 / (5 + 6). The extra 0.04 versus the oddsmaker's prediction of 0.5 is the statistical bias of this implied probability.

Mapping to the Probability Continuum

Interestingly, it is normally the case that the sum of the implied probabilities for real payout odds exceeds 1.0. This is because the oddsmaker generally inflates each of the payout odds (to make them all in the house's favor). One common method for dealing with this is to divide each of the implied probabilities by their total.

For an illustration using real payout odds, consider an actual wager on American politics offered on Jan-14-2022 at Bovada ( ), an on-line gambling house. The listed wager is: "Which Party Will Control The Senate After The 2022 Midterm Election?", and the listed outcomes are "Republican" with payout odds of 2/5 and "Democratic" with payout odds of 37/20. The implied probability of Republican control of the Senate is 0.71 = 5 / (2 + 5). The implied probability of Democrat control is 0.35 = 20 / (37 + 20). Note, significantly, that 0.71 + 0.35 > 1. To map these implied probabilities together into the probability space, one divides by their total, 1.06. Thus, using payout odds as our model, the estimated probability of Republican control is 0.67 = 0.71 / 1.06, and the estimated probability of Democrat control is 0.33 = 0.35 / 1.06.

Important Details

A few important details are worth mentioning:

First, gambling is a product of human behavior. Betting is partially driven by emotion, and gamblers enjoy the same failings as the rest of humanity. This sometimes spoils their aim, and statistically biases predictions of the betting markets. On the positive side, the gain or loss of real money provides a powerful inducement for clear thinking, and in the long run tends to weed out poor predictors. It's one thing for an academic or pundit to casually make a prediction in a paper or on a talk show, it's quite another thing to do so when one's own money is on the line!

Second, payout odds are arguably driven by two forces: 1. the wisdom of crowds and 2. the desire of the house to minimize risk. The first force works in favor of statistically unbiased estimates, whereas the second one may bias payout odds. It has been suggested that oddsmakers will set odds to balance wagers on competing outcomes. Despite making less money per wager, this would ensure that "the losers pay the winners", avoiding situations in which the house pays out painfully on its own poor predictions.

Third, payout odds may vary from one casino or on-line house to another, but market forces (and the potential for arbitrage) tend to keep them fairly well aligned.

Last, commercial gambling houses tend to be fairly specific in the terms of their proposed wagers, regarding times, conditions, etc. In the real example given above, in the event that neither political party wins control (there is a 50/50 split), the wager will include a stipulation that the bet is cancelled, or the party of the Vice President breaks the tie or some other such specific resolution is used.

Note, too, the exact meaning of such wagers. In the real example related here, the 0.67 represents the probability that the Republican party will control (have more than 50% of the membership of) the Senate. It does not indicate the percent of Senate seats which the Republicans will fill.


Gambling markets are, at their foundation, information markets. Participants are powerfully motivated to deliver competent analysis. By aggregating such predictions, oddsmakers provide a competitive alternative to more traditional predictors, such as pollsters, forecasters and similar experts. Additionally, the events whose outcomes are anticipated by gambling houses are often important yet statistically awkward: high importance, low frequency events that often involve special, one-time circumstances which can stymie other prediction mechanisms.