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# Introduction

Sports alone are one of the most ubiquitous forms of entertainment that has existed throughout all of human history. The accompaniment of various forms of sports betting adds a new component to its already enjoyable nature. However, understanding these odds can sometimes be confusing and so this article was written in the hopes of clarifying this obscurity. In addition to this, we will discuss why these odds are priced unfairly and how the bookmaker is always expected to make a profit.

# Odds Conversion Table

• To calculate the profit of a winning bet, multiply the bet size by the profit multiplier.
• To get the total payout, add the total profit to the original bet size.
• For example, if you make a \$200 bet with -500 American odds and win, you would obtain a profit of \$200*0.2=\$40 and a total payout of \$40+\$200=\$240.

# Various Odds Formats

## American Odds

• Examples: -200, +400, -1000, +2000
• Positive American Odds: If the American odds are positive, this bet represents the underdog and signifies how much money you will profit if you make a \$100 bet. If you make a \$100 bet with +400 American odds and end up winning, you will profit \$400 and have a total payout of \$500.
• Negative American Odds: If the American odds are negative, this bet represents the favorite and signifies how much money you need to wager in order to profit \$100. If you make a \$300 bet with -300 American odds and end up winning, you will profit \$100 and have a total payout of \$400.

## Decimal Odds

• Examples: 1.5, 1.2, 2.5, 4.2
• Decimal odds represent the total payout of a \$1 bet. The total payout is comprised of the profit in addition to your original bet. If you bet \$100 on 1.5 decimal odds and win, you will in total receive \$150 but only profit \$50.
• If there are only two possible outcomes for a bet, the side with the higher decimal odds represents the underdog and the side with the lower decimal odds represents the favorite.

## Fractional Odds

• Examples: 16/5, 13/10, 1/2, 2/1
• Fractional odds represent the profit of a \$1 bet. If you bet \$100 on 13/10 decimal odds and win, you will profit \$130 and receive a total payout of \$230.

# Most Common Types of Bets

## MoneyLines

This is simply a bet on which team will win the game.

A positive number indicates that this side is the underdog and needs to either win or lose by the corresponding point spread or less. A negative number indicates that this side is the favorite and needs to win by the corresponding points or more.

## Totals or Over/Under

Totals is a bet type based on whether the sum of the points scored by both teams will be over or under a certain threshold. If the total number of points is equal to the totals threshold, this is usually referred to as a push and the original bet is given back to the bettors with no gain nor loss. Sometimes the over/under total ends in 0.5 so that a push cannot happen.

## Props

Prop bets, short for proposition bets, are bets that can but do not have to be based on the outcome of the game. Almost all bets that are not moneylines, totals, or point spreads are classified as prop bets. These bets can range from which team is the first to score all the way to how many points is player X going to make.

# Why Sports Bets are NOT Fairly Priced

First, let’s look at an example and review how these odds work:

The home column shows the odds of betting on the home team whereas the away column shows the odds of betting on the away team for a specific bet. In this example, if you bet \$100 on the home team and they win, you would win \$110 (in addition to getting your original \$100 back) and would lose your \$100 if they lost. If you bet \$130 on the away team and they win, you would win \$100 (in addition to getting your original \$100 back) and would lose your \$130 if they lost. Since the returns are higher for the home team, they are less likely to win and therefore the underdog. Because of this, they are the underdog for the totals and need to either win or lose by at max 1 point whereas the away team needs to win by at least 2 points to get the payout for that bet. If the total number of points scored by both teams was above 8, a bet on the over would win whereas if the total number of points was below 8, a bet on the under would win.

Now, let’s explore a little deeper into these odds. How exactly do we convert these odds into returns (or profits) and how would we use these payouts to determine the probability of each team winning? If we look at the home odds for the moneyline, a \$100 bet would yield \$110 in returns. This is a return of \$110/\$100 = 1.1 = 110%. If we look at the away odds for the moneyline, a \$130 bet would yield \$100 in returns. This is a return of \$100/\$130 = 0.77 = 77%. Since the away team has smaller returns, they are more likely to win. But, how exactly do we quantify their chances of winning given their returns? Well, this chance is referred to as implied probability and is based on the returns of winning a made predicting a team’s success.

The relationship in Eq. (2) shows the function to map the potential returns of a bet into the implied probability of winning that bet. Implied probability is the probability that would be needed to win a bet in order to fairly justify the returns given to winning said bet. If winning a bet would yield 40% in returns, for example, the implied probability of winning that bet would be roughly 71%. This makes relationship makes intuitive sense; if a team has a higher probability of winning, a bet placed on that team should yield lower returns since they are already expected to win and vice versa. Now, let’s use this information to understand the previous example a little more clearly.

But, hold on. Why don’t these implied probabilities add up to 100%? Well, this is actually how the bookmakers make their money. The higher implied probabilities a team has, the lower payout a bet on them would be since they have a higher chance of winning. Therefore, if I’m the bookmaker and show that a team has a 65% chance to win when in reality they only have a 60% chance to win, I wouldn’t have to pay the winner as much since I am giving them an extra 5% chance to win. Therefore, the sum of the implied probabilities will ALWAYS exceed 100% . This is why sports odds are not fairly priced and the reason behind why bookmakers will always be expected to make money. The extent to which this sum exceeds 100% is known as overround, margin, or vig and varies from game to game, but always exist. This phenomenon also explains why a strategy to always bet on the favorite, or the underdog for that matter, would not have an expected return of \$0; the expected returns would be negative.

# A More Rigorous Explanation

Let’s explain this concept a little more symbolically. Imagine we are the bookmaker and wonder what our expected profit or loss would be given the returns we set for each team winning. This expected value is shown in Eq. (3). p_h and p_a are the true probabilities of the home team and away team winning respectively. In other words, if we have a model that predicts the home team will win with a probability of 60% and the away team will win with a probability of 40%, these would be p_h and p_a respectively. If the home team wins (which it will with probability p_h), we (the bookmaker) will win all of the money that was bet on the away team but lose the amount of money we have to pay out to the original bettors on the home team. The same but reverse payouts happens if the away team wins.

Side note, since p_h and p_a are the true probabilities, they must add to 100%, or 1. This will be an important step later.

Next, let’s utilize the fact that we are going to inflate the true probabilities of each team winning and use these as our implied probabilities. Remember, the idea for doing this is because the higher probability we give a team to win, the less returns we have to payout to the winner (as shown in Eq. (2)). Altering the true probabilities ONLY changes the returns we have to pay out to the winner, not the actual probabilities of each side winning. If we do this and then replace those values in Eq. (2) and Eq. (3), we will obtain:

Now, let’s determine exactly much we want to inflate the true probabilities. The amount we want to inflate these probabilities is decided by different deltas. When we express this and substitute this back into Eq. (5):

Now, let’s determine exactly much we want to inflate the true probabilities. The amount we want to inflate these probabilities is decided by different deltas. When we express this and substitute this back into Eq. (5):

The expected profit/loss for the bookmaker is the weighted sum of the money bet on the home team and the money bet on the away team, where the weights are equal to the degree to which we inflate the true probabilities. Notice that since both deltas must be positive, the bookmakers expect to see a profit. This profit multiplied by thousands of bets is how the bookmakers make money and why the odds are not fairly priced.

Thanks for reading this article! I hope you learned something and make sure to keep an eye out for future articles written — Thanks!