Mathematics of Betting Odds

Are you interested in maximizing profits and minimizing risks while betting on sports? Well, who isn’t? If you want to know whether a wager is worth the pursuit or not, you should be well versed with the mathematics of betting odds. Basically, there are three types of odds involved in this process:

  • American (Moneyline)
  • Decimal
  • Fractional

These three types of odds are essentially representatives of different formats to present probabilities. Also, each type of odd can be readily converted into another, and can also be expressed as an implied probability percentage. My main motive is to provide you with the basics of betting odds so that you can use those in sports betting.

blue and silver square ornament and dices showing The Mathematics of Betting Odds
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After you ascertain the likely probability for an outcome, you can make an informed decision whether to place a wager/bet or not. To successfully identify promising opportunities, you need to determine if the probability is greater than the implied probability seen in the odds. Since the bookmaker’s profit margin is also considered at odds, the house is always going to win.

How to convert odds to implied probabilities

Upon your first look, the calculations involved with odds seem to be rather intimidating and confusing. However, you needn’t worry because after you properly understand the details of the aforementioned three kinds of odds and the method of converting the numbers into implied probabilities, you will find the mathematical concepts much easier.

To start, let’s take a more detailed look at the three types of odds mentioned above:

  • American odds or moneyline odds are accompanied by a (+) or (-) sign, where the (+) sign is associated with the lower probability count having the better payout.
  • Decimal odds are used to represent the amount of money won for every dollar you wager. For example, if the odds for a certain candidate or team winning are 4.00, then the payout for every USD 100 wagered will be USD 400.
  • Fractional odds, also known as traditional odds or British odds, are generally expressed in the form of a ratio or fraction, as their name implies.

You can utilize certain tools to inter-convert these three types of odds among themselves. Nowadays, you will often find an option to display the odds in a format of your choice on various online betting websites. If you want, you can also convert odds manually by using certain conversion formulas. In fact, if you are mathematically inclined, converting odds to their implied probabilities can be a very exciting experience. For that purpose, you can use the formula given below:

Implied probability of an outcome is calculated by Stake divided by Total Payout.

Here, the stake is equal to the amount wagered.


  • Let’s say a bookmarker estimates that the fractional odds of team A defeating team B is 9/12. Entering this value into the above formula, you can calculate the value of the implied probability – 75%. The probability of the outcome is directly proportional to the magnitude of the number.
  • Let’s now consider an example of decimal odds. If a participant has 2.20 odds of winning a certain competition, the implied probability can be calculated to be 45.45% (dividing 1 by the odd — $\frac{1}{2.20} = 0.4545$ ).
  • Finally, here is an example of American odds. Let’s say India’s chance of winning the next ICC Cricket World Cup is -250. In that case, the implied probability can be calculated to be 71.43%.

Here's the odd conversion table along with the winning probabilities:

Odds Conversion Table

FractionDecimalAmericanImplied Probability

You should keep in mind, though, that probability estimations change with time and thus odds can change as new bets keep coming in. Also, the odds shown by various bookmakers can differ significantly, and thus are not entirely reliable. Thus, although it is tempting to simply support the regular winners every time, you should only do so if the odds accurately reflect their chance of winning. After all, it would be foolhardy to risk a large amount for half the amount of profit.

The general rule of thumb is that if the probability estimated for an outcome is higher than the implied probability that was assessed by the bookmaker, you have a solid betting opportunity in your hand.

Why does the house undoubtedly win every time?

As I mentioned above, the odds put up for viewing do not represent the real chance or probability of an event taking place or not. This is because the bookmaker always adds a sort of profit margin to these odds. Thus, the successful punter always receives a lesser payout than what he would’ve gotten if the odds had been accurate.

If the bookmaker is seeking a secure profit from a particular event, he or she should estimate the true chance or probability of an outcome. Doing this will set the odds up for viewing in such a way that the bookmaker will undoubtedly profit, without depending on the outcome of the said event. Revisiting the Cricket World Cup example above, let’s consider some data:

  • India: -250 (implied probability = 71.43%)
  • England: - +200 (implied probability = 33.33%)

The sum of these probabilities is equal to 104.76%. If you are wondering why the sum of all probabilities does not equal 100% over here, remember that the odds you see here are not fair odds. 

Basically, the additional 4.76% serves as the bookmaker’s “over-round”. This refers to the profit that the bookmaker could potentially make if they accept your bets in just the correct proportion. If you choose to bet on both India and England in the above example, it means that you are putting 104.76 dollars at stake in order to win 100 dollars. From the bookmaker’s point of view, they will be collecting 104.76 dollars from you and expect to pay you 100 dollars, with the stake included. Thus, you can calculate that they will be making an estimated profit of 4.5%. As you can clearly see, the bookie is at a clear advantage when it comes to the odds.

Experts claim that by winning more hands, a player could actually receive less money. This especially holds true for beginners. It can be explained by the fact that several wins will likely get you smaller stakes. Thus, you will be forced to play more to earn larger amounts of money, which puts you at greater risk of losses. Every player knows that occasional losses are more or less an inevitable part of betting. Over time, these losses can accumulate to a sizeable amount and set you back financially.

Behavioral Economics

Behavioral economics plays an important role in the situation described above. Basically, such a player keeps betting due to the confidence gained by a winning streak, or hoping to win a large amount and compensate for the losses suffered. In both these cases, the player isn’t compelled to keep playing by statistical or rational reasoning, but rather by the emotional euphoria that accompanies a win.

Even the owners of the casino know this. They carefully design all the elements on there – everything from the lighting and decorations to the music and drinks – to encourage you to keep playing. The games offered by them have a variable inbuilt house edge as well. Newcomers are usually poor at cognitive accounting and don’t adequately acknowledge the fact that a streak of wins doesn’t change the unpredictability of payouts. The plain truth is that frequent smaller gains are almost always overshadowed by losses, which aren’t as frequent but are definitely bigger in size.


As long as the probability estimated for an outcome is higher than the implied probability which has been estimated by the bookie, you should consider it to be a good betting opportunity. Also, you must always remember that the odds you see on display do not accurately represent the actual probability of an event occurring or not, ever. Thus, the payoff on a win is always less than what the odds seem to promise. The bookie’s profit margin is part of the odds as well, and thus the house will always win.