The Math Behind Betting Odds and Gambling

The math underlying odds and gambling can help determine whether a wager is worth pursuing. The first thing to understand is that there are three distinct types of odds: fractional, decimal, and moneyline (also known as "American" odds).

The different types of odds are simply varying formats for presenting probabilities. They're commonly used by bookmakers, and one type can be converted into another. Once the implied probability for an outcome is known—as represented by odds—decisions can be made regarding whether or not to place a bet or wager.

Converting Different Types of Odds

Although odds require seemingly complicated calculations, the underlying concept is easier to understand once you fully grasp the three types of odds and how to convert them into implied probabilities.

• Fractional odds are sometimes called British odds or traditional odds. They are written as a fraction, such as 6/1, or expressed as a ratio, like six-to-one.
• Decimal odds represent the amount that is won for every $1 that is wagered. For instance, if the odds are 3.00 that a certain horse wins, the payout is $300 for every $100 wagered.
• Moneyline odds are sometimes called American odds. They are accompanied by a plus (+) or minus (-) sign, with the plus sign assigned to the lower probability event with the higher payout.

There are tools available to make conversions between the three types of odds. Many online betting websites offer an option to display the odds in the preferred format. The table below demonstrates the step-by-step calculations behind conversions, for those interested in doing them by hand.

Converting Odds into Probabilities

The general rule for the conversion of any type of odds into an implied probability can be expressed as a formula:

$\begin{aligned} &\text{Implied Probability Of An Outcome} = \frac{ \text{Stake} }{ \text{Total Payout} } \\ &\textbf{where:} \\ &\text{Stake} = \text{Amount wagered} \\ \end{aligned}$​Implied Probability Of An Outcome=Total PayoutStake​where:Stake=Amount wagered​

As shown, the formula divides the stake (amount wagered) by the total payout to get the implied probability of an outcome.

For example, a bookmaker has the fractional odds of Man City defeating Crystal Palace at 8/13. Plug the numbers into the formula, which is a simple matter of dividing 8 by 13 in this example, and the implied probability equals 61.5%. The higher the number, the greater the probability of the outcome.

Note that you will also receive back your initial wager if you make a winning bet. For instance, in the above example, you would win $61.50 and receive back the initial $100 wager.

Using an example of decimal odds, a candidate has 2.20 odds to win the next election. If so, the implied probability is 45.45%, or:

$\begin{aligned} &\left ( \frac{ 1 }{ 2.2 } \times 100 \right ). \\ \end{aligned}$​(2.21​×100).​

Lastly, using the American methodology, Australia's odds to win the 2015 ICC Cricket World Cup is -250. Therefore, the implied probability equals 71.43%:

$\begin{aligned} &\left ( \frac{ 250 }{ 100 + 250 } \times 100 \right ). \\ \end{aligned}$​(100+250250​×100).​

Remember, odds change as the bets come in, which means probability estimations vary with time. Moreover, the odds displayed by different bookmakers can vary significantly, meaning that the odds displayed by a bookmaker are not always correct.

It is not only important to back winners, but one must do so when the odds accurately reflect the chance of winning. It is relatively easy to predict that Man City will win against Crystal Palace, but would you be willing to risk $100 to make a profit of $61.50? The key to assessing whether a betting opportunity is valuable is when the probability assessed for an outcome is higher than the implied probability estimated by the bookmaker. 

The odds on display never reflect the true probability or chance of an event occurring (or not occurring). There is always a profit margin added by the bookmaker in these odds, which means that the payout to the successful punter (or bettor) is always less than what they should have received if the odds had reflected the true chances.

The bookmaker needs to estimate the true probability or chance of an outcome correctly in order to set the odds on display in such a way that it profits the bookmaker regardless of an event outcome. To support this statement, let’s look at the implied probabilities for each outcome of the 2015 ICC Cricket World Cup example.

• Australia: -250 (implied probability = 71.43%)
• New Zealand: +200 (implied probability = 33.33%)

If you notice, the total of these probabilities is 104.76% (71.43% + 33.33%). Doesn't that conflict with the fact that the sum of all probabilities must equal 100%? This is because the odds on display are not fair odds.

The amount above 100%, the extra 4.76%, represents the bookmaker’s "over-round," which is the bookmaker’s potential profit if the bookie accepts the bets in the right proportion. If you bet on both the teams, you are actually risking $104.76 to get $100 back. From the bookie’s perspective, they are taking in $104.76 and expect to pay out $100 (including the stake), giving them an expected profit of 4.5% (4.76/104.76), no matter which team wins. The bookie has an edge built into the odds.

Psychological Considerations

According to a study published in the Journal of Gambling Studies, the more hands a player wins, the less money they are likely to collect, especially with respect to novice players. That is because multiple wins are likely to yield small stakes, for which you need to play more, and the more you play, the more likely you will eventually bear the brunt of occasional and substantial losses.

Behavioral economics comes into play here. A player continues playing the lottery, either in hopes of a big gain that would eventually offset the losses or the winning streak compels the player to keep playing. In both cases, it is not rational or statistical reasoning but the emotional high of a win that motivates them to play further.

Consider a casino. All of the details—including the game rules, music, controlled lighting effects, alcoholic beverages, and the interior decor—are carefully planned and designed to the house's advantage. The house wants you to stay and continue playing. Naturally, the games offered by the casino have a built-in house edge, although the house advantage varies with the game.

Moreover, novices find it particularly difficult to do cognitive accounting and people often misjudge the variance of payouts when they have a streak of wins, ignoring the fact that frequent modest gains are eventually erased by losses, which are often less frequent and larger in size.

The Bottom Line

A betting opportunity should be considered valuable if the probability assessed for an outcome is higher than the implied probability estimated by the bookmaker. Furthermore, the odds on display never reflect the true probability of an event occurring (or not occurring). The payoff on a win is always less than what one should have received if the odds had reflected the true chances. This is because the bookmaker’s profit margin is included in the odds, which is why the house always wins.