The Math Behind Betting Odds & Gambling

As mentioned in the earlier reading on this topic, Betting Basics: Fractional, Decimal & American (Moneyline) Odds, the three types of odds are just different formats to present the same probabilities (as estimated by the bookmakers). With that said, it becomes obvious that one type of odds can be converted into another. Although it requires seemingly complicated calculations, these are easier to understand once you get a grip on these three types of odds. There are many tools available to make these conversions, and many online betting websites offer an option to display the odds in the preferred format. If one wants to work it out by themselves, they could refer to the table below:

Here comes the more interesting part: converting the aforementioned odds to their implied probabilities. Due to the significance of this part, we will not discuss the specific formula related to each type of odds. Rather, let's remember the general rule for the conversion of (any type of) odd into an implied probability.

Rule:

As shown, divide the amount wagered (on stake) by the total payout to get the implied probability of an outcome. If the odds mentioned in the earlier examples (covered in the previous reading) are converted into percentages or implied probabilities, we get the following results:

According to a bookmaker, the implied probability for:

Man City (odds: 8/13) to win against Crystal Palace on April 07, 2015, was 61.9% [{13/(8+13)}*100].

Hillary Clinton (odds: 2.20) to win the 2016 US Presidential elections was 45.45% [(1/2.2)*100].

Australia (odds: -250) to win the 2015 ICC Cricket World Cup was 71.43% [{250/(100+250)}*100].

Remember, the odds keep changing as the bets come in, which means bookmakers’ 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. Note that it's 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 say that Man City will win against Crystal Palace, but would you be willing to risk $100 to make a profit of $61.53?

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. (For related reading, refer to Are You Investing or Gambling?).

Basically, the odds on display never reflect the true probability/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. This means that the bookmaker needs to estimate the true probability/chance of an outcome correctly in order to set the odds on display in such a way that it profits the bookmaker irrespective 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 we had discussed in the first reading on this topic.

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 percentage above 100% (i.e. 4.76%) represents the bookmaker’s over-round — 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 $104.46 to get $100 back. From the bookie’s perspective, they are taking in $104.60 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 a built-in edge here.

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. According to the research, 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 substantial losses. Here, behavioral economics comes into play. A player continues playing the lottery either in hopes of a big gain that would eventually offset the losses or the winning streak compels him/her to keep playing. In both cases, it is not rational or statistical reasoning but the person's emotions and the high of a win that lead them to play further. (For related reading, refer to The Worst Bets You Can Make at the Casino).

Consider a casino. Everything—including the game rules, music, controlled lighting effects, alcoholic beverages, the interior decor— is carefully planned and designed to the house's advantage. The house wants you to stay and continue playing. All 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 misjudge the variance of payouts when they have a streak of wins, ignoring the fact that frequent modest gains are eventually overweighed by infrequent significant losses. (For related reading, refer to Investing vs. Gambling: Where Is Your Money Safer?).

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). 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.