# Implied Probability in Sports Betting: Calculation and Interpretation

In this article, Catalin Barboianu shares his thoughts and analytics on the implied probability of sports betting. Read more to find out his research and how to calculate the implied probability for various odds. The terms ‘odds’ and ‘probability’ are used in gambling in various meanings and contexts. Payout odds, true odds, odds as chances or likelihoods, implied odds, mathematical probability, and implied probability are words denoting concepts that are different, although they all have a similar nature, given their mathematical component. Many times these terms are used interchangeably in gambling speech, which can cause misunderstandings, depending on the context. In this article, we will see what implied probability means in sports betting, how to calculate it, and how we should correctly interpret this concept for our betting decisions.

## What is Implied Probability

When defining implied probability, we have to state from the beginning that, as the name suggests, it is a kind of probability, however, it is not a mathematical probability. Mathematical probability is a mathematical measure for the possibility of an event to occur, provided that the space of events has a certain mathematical structure in which some events – called elementary events – are assumed equally possible.

In sports betting, implied probability is not such a measure. It is just a form of expressing the payout odds as a percentage, which is thought to reflect likelihood (just like the mathematical probability, however, based on human intention and not on counted evidences) rather than payout. We will clarify this nature of implied probability further when we will see how to calculate and interpret it.

Whether we talk about ‘odds’ as probability or payout, there is a general formula that converts odds to percentage (usually said as “odds to probability” in gambling jargon):

probability = odds / (1 + odds), where the odds are expressed as a subunitary fraction.

Example:

For converting 3 to 2 (3 : 2) odds to a percentage, first we write the odds as the fraction 2/3 (two out of three), then replace this fraction in the above formula and do the calculation:

probability = (2/3) / (1 + 2/3) = (2/3) / (5/3) = (2/3) x (3/5) = 2/5.

The last step is to convert the result to a percentage:

Dividing the numerator to the denominator 2 : 5 = 0.40

Multiplying the result by 100 and adding the % symbol: 0.40 x 100% = 40%.

Hence 3 : 2 odds convert to a percentage of 40%. If 3 : 2 represents odds as probability, then this probability is 40% as the mathematical probability. If 3 : 2 represents a payout rate (for instance, in blackjack), then its percentage form is called implied probability (without being a mathematical probability).

The above formula is the mathematical definition of the implied probability and the core algorithm for calculating it. Therefore, implied probability is just a form of expressing the payout odds (rate) of a bet, in whatever game of chance.

In particular, in sports bets, the implied probability is calculated by an algorithm that is specific for each format in which the payout odds are expressed.

## Calculating implied probability for the various formats of odds

There are three formats for the payout odds in sports betting: fractional (British), decimal (European), and moneyline (American). We will take them each to see in detail how they convert into implied probability.

### Converting the fractional odds

The fractional odds are expressed as a fraction, namely, the ratio of the possible profit won (P) to the stake (S):

O (f) = P / S

For instance, 5/2 odds mean that you can make \$5 as a profit for a \$2 stake; this means a return of \$2 + \$5 = \$7.

For converting the fractional odds to implied probability, take the following steps:

1. Add the numerator and the denominator of the initial fraction: P + S. In our example, 2 + 5 = 7.
2. Write the fraction having as numerator the denominator of the initial fraction and as denominator the result obtained at step 1: S / (S + P). In our example, 2/7.
3. Divide the numerator by the denominator of the fraction obtained at step 2 and write the result as a decimal number rounded at four decimal places: S / (S + P) = /0.xyzt. In our example: 2 / 7 = 0.2857.
4. Multiply the result obtained at step 3 by 100 and apply the percentage to find the implied probability: /0.xyzt x 100% = / xy.zt%. In our example, 0.2857 x 100% = 28.57%.

### Converting the decimal odds

The decimal odds represent the gross payout rate of the bet, i.e. the number that your stake is multiplied by for giving your payback if you win the bet. Else said, it is the payback of a won bet with a stake of 1. It is a decimal number higher than 1 with two decimal places:

O (d) = 1.xy

For instance, 1.85 odds means that you are paid back 1.85 times your stake if you win the bet, so if you bet \$1 you will get back \$1.85 and as such make a profit of \$0.85.

For converting the decimal odds to implied probability, take the following steps:

1. Multiply the odds decimal number with 100: / 1.xy x 100 = / 1xy. In our example, 1.85 x 100 = 185.
2. Divide 100 by the number obtained in step 1 and write the result as a decimal number rounded at four decimal places: 100 : / 1xy = /0.ztvw. In our example, 100 : 185 = 0.5405.
3. Multiply the result obtained at step 2 by 100 and apply the percentage to find the implied probability: /0.ztvw x 100% = / zt.vw%. In our example, 0.5405 x 100% = 54.05%.

### Converting the moneyline

The moneyline odds do not express rates or multipliers, but amounts for either the stake or profit of a bet supposed won. The moneyline odds apply to bets on matches that can have as result a win or loss for the competitor you bet on.

The odds for the favorites are in the form of a negative integer (accompanied by the minus sign) and represent the amount you need to stake to make a profit of \$100:

O (mf) = -n

The odds for the underdogs are in the form of a positive integer (accompanied by the plus sign) and represent the profit made for \$100 staked:

O (mo) = +m

For instance, assume a match between Team A and Team B, with the following moneyline:

• Team A: –130
• Team B: +110

We take the steps for converting these odds into implied probability for each competitor:

Team A (the odds with minus):

1. Remove the minus sign from the odds (–n) to have a positive number (n) and add 100 to it: n + 100. In our example, 130 + 100 = 230.
2. Divide the odds as a positive number (n) to the number obtained at step 1 and write the result as a decimal number rounded at four decimal places: n : (n + 100) = /0.xyzt. In our example, 130 : 230 = 0.5652.
3. Multiply the result obtained at step 2 by 100 and apply the percentage to find the implied probability: /0.xyzt x 100% = / xy.zt%. In our example, 0.5652 x 100% = 56.52%.

Team B (the odds with plus):

1. Add 100 to the odds: m + 100. In our example, 110 + 100 = 210.
2. Divide 100 by the number obtained at step 1 and write the result as a decimal number rounded at four decimal places: m : (m + 100) = /0.vwqr. In our example, 100 : 210 = 0.4761.
3. Multiply the result obtained at step 2 by 100 and apply the percentage to find the implied probability: /0.vwqr x 100% = / xw.qr%. In our example, 0.4761 x 100% = 47.61%.

Now you know how to convert any payout odds into implied probability. But how can this format be of any help and how should we interpret it?

## Using implied probability

We saw that implied probability is just the payout odds in a percentage form, so as the information it is something that we already knew when we saw the bet we intend to play, in the sportsbook’s list. However, in their percentage form, the payout odds reflect likelihood, a kind of degree of belief in the possibility of the desired event occurring. This is also why it is called a ‘probability’. That degree is relative to 100%, which is associated with the degree of belief in the occurrence of a sure event. As likelihood, the implied probability is then useful for evaluating the chances you have to win the bet with respect to the odds offered as payout by the bookie, but also to the other bettor’s beliefs since payout odds of a bet are also adjusted continuously with the incoming bets on that event.

This likelihood is better visualized and perceived as the degree of belief written as an implied probability rather than in the odds format, especially for the fractional and moneyline formats.

Used as likelihood, the implied probability is helpful not only in the individual assessment of chance for your bets but also in the organization of your betting activity in a given competition and market. Converting odds to implied probability and keeping track of the results over a definite period will help you come to a sharper categorization of the competitors into heavy favorites, clear favorites, slight favorites, potential toss-ups, and underdogs. It is a tool that makes you better equipped to spot value opportunities in sports betting odds.

Besides using it concretely as an analytical tool, implied probability can improve your understanding of sportsbook odds and give you a broader perspective on the strategic dimension of betting.

## Interpreting implied probability

Many gamblers tend to take implied probability to be the mathematical probability of the event that they bet on. We already said that it is not a mathematical probability, just because sports events cannot be represented in the required mathematical structure. When we roll a die and assign the probability 1/6 to the occurrence of a number we do that because such a structure exists and in that structure any outcome is equally possible for us, regardless of the physical factors influencing it. In other words, randomness allows us to do that. Instead, sports events are influenced by multiple factors that cannot be quantified mathematically for a probability to be assigned to them. Their nature is much more deterministic than stochastic.

Take a previous example when converting moneyline to implied probability. The implied probability for Team A to win was 56.52% and for Team B to win was 47.61%. If these measurements were mathematical probabilities, then their sum would be 1; but this is not the case: 56.52% + 47.61% = 104.13%. The difference from 100% actually reflects the bookie’s vigorish. Like in any casino game, the difference between the payout odds and true odds stands for the house’s advantage. If implied probability were true odds, then the bookie would make no profit from that bet. This gives us another interpretation for implied probability: If a mathematical probability were possible for a sports event, the implied probability would be that value of it for which a bet on that event would have zero expected value (such a bet is called a fair bet).

Still, the possibilities in sports events are assessed by bettors and experts and implied probability is such a tool of assessment. Implied probability measures the overall bettors’ tendency – since it expresses the payout odds as bettors’ intentions. With this description, the implied probability is a kind of subjective probability (a theoretical probability defined as a degree of belief in terms of betting intention). When the bookie first assigns the payout odds to an event, they take into account the past statistics of that competitor and of their past matches. Then, the initial payout odds are adjusted with the next coming bets, but at their core, they reflect that initial statistical analysis. As such, implied probability is also a kind of frequentist probability (a theoretical probability defined as the relative frequency of occurrence of an event).

In whatever interpretation, remember that implied probability is just a form of expressing the payout odds of a bet. It is the “probability” that the bookie offers you for winning the bet, which you can refuse as an assessment tool. In fact, sports betting is about exploiting assessment and using information from outside the sportsbook for your own assessment.

## Conclusion

Implied probability is a way of expressing the payout odds in a percentage form, looking like a probability. Such a form is useful for a better perspective on the chances of winning a bet, especially when the payout odds are given in fractional or moneyline form, and for a better organization of your betting activity. Implied probability should not be meant as an absolute measure for the possibility of the event you bet on. It essentially reflects the bookies and the bettors’ predictions of that event.

## More Researches and Articles by Catalin To Read Next

Author
Catalin BarboianuCatalin Barboianu is an experienced iGaming writer and gaming mathematician. He has written various books and articles about problem gambling and gaming probabilities.