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 American (moneyline).

These types are represent different formats to present probabilities, which are also used by bookmakers, and one type can be converted into another. Once the implied probability for an outcome is known, decisions can be made regarding whether or not to place a bet or wager.

### Key Takeaways

- The three types of odds are fractional, decimal, and American.
- One type of odd can be converted into another and can also be expressed as an implied probability percentage.
- A key to assessing an interesting opportunity is to determine if the probability is higher than the implied probability reflected in the odds.
- The house always wins because the bookmaker's profit margin is also factored into the odds.

## Converting Odds to Implied Probabilities

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

- Fractional odds are sometimes called British odds or traditional odds and are sometimes 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.
- American odds are sometimes called moneyline odds and 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 can help convert odds with pen and paper, for those interested in doing the calculations by hand.

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

In 2018 the Supreme Court gave U.S. states permission to legalize sports betting if they wish to do so. As of 2023, It is still fully illegal in 14 states, including California, Massachusetts, and Texas. In nine states, there is some form of pending legislation.

### Rule

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

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:

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

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%:

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

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 is to consider a betting opportunity valuable 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 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.

### $14.8 billion

The amount of gaming revenues generated by Nevada casinos in 2022, a record level.

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.

## What Is the Difference Between Odds and Probability

Odds and probability are both used to express the likelihood of an event occurring in the context of gambling. Probability is expressed as a percentage chance, while odds can be presented in a few different formats, such as a decimal, fraction, or moneyline. Odds represent the ratio of the probability of an event happening to the probability of it not happening.

## Which Casino Games Have the Best Odds for Players?

Blackjack has the most favorable odds for players (who know how to play the game properly), with a relatively low house edge. The exact house advantage for blackjack depends on a number of factors, such as the the house rules, number of decks used, the skill level of the player, and the skill of other players at the table, but it is generally in the range of 0.40% to 1%. This means that for every $100 that a player bets on blackjack, they can expect to lose only 40 cents to $1 on average. Other games that can have a relatively low house advantage include craps, baccarat, and some video poker games.

## Which Casino Games Have the Worst Odds for Players?

Some of the casino games with the highest house edge include Keno, the Big Six Wheel/Wheel of Fortune, and slot machines (which often have the highest house advantage).

## How Do You Calculate the Odds of a Casino Game?

To calculate the odds of winning a bet in a casino game, you'll need to know the number of possible outcomes that would result in a win and the total number of possible outcomes. As a very simple example, in a simple coin toss, there are two possible outcomes (heads or tails), so the odds of winning a bet on heads would be 1 in 2, or 50%.

## 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.

If you or someone you know has a gambling problem, call the National Problem Gambling Helpline at 1-800-522-4700, or visit ncpgambling.org/chat to chat with a helpline specialist.