The math behind betting odds and gambling

The odds and the mathematical basis of gambling can help determine whether the bet is worth it. The first thing to understand is that there are three different types of odds: decimal odds, decimal odds, and American odds (win or lose). The various types represent different formats to express probability. Bookmakers also use these formats and can convert one type to another. Once you know the implied probability of the outcome, you can decide whether to place a bet or place a bet.

Key points

  • The three types of odds are fractions, decimals and American.
  • One type of odd number can be converted to another, or it can be expressed as an implied probability percentage.
  • The key to evaluating interesting opportunities is to determine whether the probability is higher than the implied probability reflected in the odds.
  • The dealer always wins because the profit margin of the bookmaker is also included in the odds.

Convert odds into implied probabilities

Although odds require seemingly complex calculations, once you have a complete grasp of the three types of odds and how to convert numbers into implied probabilities, the concept will be easier to understand.

  • Fractional odds are sometimes called British odds or traditional odds, and are sometimes written as fractions, such as 6/1, or expressed as ratios, such as six to one.
  • Decimal odds represent the amount of money won for every dollar bet. For example, if the odds for a horse to win are 3.00, the payout for every $100 bet is $300.
  • American odds are sometimes called winning odds, with a plus (+) or minus (-) sign, which is assigned to events with higher odds and lower probability.

There are some tools that can be used to convert between the three types of odds. Many online betting sites offer the option of displaying odds in the preferred format. The following table can help those interested in manual calculations to convert the odds with pen and paper.

Converting the odds to their implied probabilities is probably the most interesting part. The general rule for converting odds (of any type) into implied probabilities can be expressed as a formula:


Implied probability of result = total bet payout where: bet = amount of bet begin{aligned} &text{implied probability of result} = frac{ text{Stake} }{ text{Total Payout}} &textbf{where:} \ &text{Stake} = text{stake amount} \ end{aligned}

Implied Probability of Outcome=total expensesbetWhere:bet=Bet amount

As shown in the figure, the formula divides the bet (bet amount) by the total payout to obtain the implied probability of the result.

For example, the bookmaker has (decimal) odds for Manchester City to beat Crystal Palace 8/13. Substituting numbers into the formula, in this case is a simple problem of dividing 8 by 13, with an implicit probability of 61.5%. The higher the number, the greater the probability of the result.

Taking the decimal odds for example, the candidate has an odds of 2.20 to win the next election. If yes, the implied probability is 45.45%, or:

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


Finally, using the US method, Australia’s odds to win the 2015 ICC Cricket World Cup are -250. Therefore, the implied probability is equal to 71.43%:

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


Remember that the odds change with the arrival of bets, which means that the probability estimates will change over time. In addition, the odds displayed by different bookmakers may vary greatly, which means that the odds displayed by the bookmaker are not always correct.

Not only is it important to support the winner, but it must be done when the odds accurately reflect the chance of winning. It is relatively easy to predict that Manchester City will defeat Crystal Palace, but are you willing to risk $100 to make a profit of $61.50? When the probability of the evaluation result is higher than the implied probability estimated by the bookmaker, the key is to consider a valuable betting opportunity.

Please note that if you win the bet, you will also receive the initial bet. For example, in the example above, you would win $61.50 and withdraw your initial bet of $100.

Why does the House of Representatives always win?

The odds displayed will never reflect the true probability or chance of an event happening (or not happening). Bookmakers always increase their profit margins in these odds, which means that if the odds reflect the real chance, the successful bettor will always pay less than they should be paid.

The bookmaker needs to correctly estimate the true probability or probability of the outcome in order to set the displayed odds, regardless of the outcome of the event, which can benefit the bookmaker. To support this statement, let’s look at the implied probability of 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 sum of these probabilities is 104.76% (71.43% + 33.33%). Does this not conflict with the fact that the sum of all probabilities must equal 100%? This is because the odds shown are not fair odds.

The amount exceeding 100%, that is, an additional 4.76%, represents the “super round” of the bookmaker, that is, the bookmaker’s potential profit after the bookmaker accepts the correct proportion of bets. If you bet on both teams, you actually have to risk $104.76 to get back $100. From the bookmaker’s point of view, they get 104.76 US dollars and expect to pay 100 US dollars (including principal), no matter which team wins, their expected profit is 4.5% (4.76/104.76). Bookmakers have an advantage in the odds.

According to the publication in ” Journal of Gambling Research, The more hands a player wins, the less money they collect, especially for novice players. This is because multiple wins may result in smaller bets, for which you need to play more, and the more you play, the more likely you are to bear the brunt of the occasional heavy losses in the end.

Behavioral economics comes into play here. Players continue to play the lottery, either hoping to get a big profit to offset the loss, or consecutive wins force the player to continue playing. In both cases, it is not rational or statistical reasoning but a high mood of victory that motivates them to perform further.

$12 billion

The amount of revenue generated by Las Vegas casinos in 2018.

Consider a casino. All details-including game rules, music, controlled lighting effects, alcoholic beverages and interior decoration-have been carefully planned and designed to take advantage of the house. The house hopes you stay and continue playing. Of course, the games offered by the casino have a built-in banker advantage, although the banker advantage varies from game to game.

In addition, novices find it particularly difficult to perform cognitive accounting. People often misjudge the difference in expenditure during consecutive wins, ignoring the fact that frequent and moderate gains will eventually be offset by losses, and losses are usually not so frequent, and Larger in scale.

Bottom line

If the probability evaluated for the result is higher than the implied probability estimated by the bookmaker, then the betting opportunity should be considered valuable. In addition, the odds displayed will never reflect the true probability of the event occurring (or not occurring). If the odds reflect a real chance, the reward for winning is always lower than the reward people should get. This is because the profit margin of the bookmaker is included in the odds, which is why the casino always wins.


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