Wednesday, July 17, 2024

# Champions League: Probabilities, Odds, Payouts

The Champions League final is set, featuring Borussia Dortmund versus Real Madrid. This post will cover the betting odds, payouts, and probabilities of various outcomes, as well as how to interpret and calculate them on your own. Who will you root for?

An article I read online recently stated that according to Bet365, the Borussia Dortmund vs Real Madrid match odds are as follows: Dortmund to win: 5/2; Madrid to win: 3/10. This is a format, we in the USA don’t see very often. In betting, odds are used to determine the payouts for winning bets. They reflect the probability of an event happening. The odds above are in fractional format, which is commonly used in the UK and Ireland, which makes sense as the site was based in the UK.

## Understanding odds and payouts in UK and USA

• Dortmund to win: 5/2 means that for every 2 units you stake, you will receive 5 units if Dortmund wins. So, if you bet \$2, you would get \$5 if Dortmund wins, plus your original stake back, for a total of \$7.
• Real Madrid to win: 3/10 means that for every 10 units you stake, you will receive 3 units if Madrid wins. So, if you bet \$10, you would get \$3 if Madrid wins, plus your original stake back, for a total of \$13.

The odds also give an indication of the likelihood of a particular outcome. Lower odds (like 3/10) suggest that the outcome is more likely to happen, while higher odds (like 5/2) suggest it is less likely. In this case, the odds suggest that Real Madrid is more likely to win than Borussia Dortmund.

In the USA, odds are typically expressed in a format known as “moneyline odds” or “American odds”. This format uses a baseline value of \$100. In the USA version of Bet365 site, their page shows the following:

This is the moneyline format as used in most sports betting in the USA. Here the odds are:

`Borussia Dortmund +400. Tie +333. Read Madrid -163. `

We can immediately see that likelihood of Real Madrid snatching a win is much higher than Dortmund winning according to their statistical model. Also, the odds have been updated since the original article in the UK. In this case, it means if you bet \$100, and if Dortmund wins, you’d win \$400 + the original stake with a total payout of \$500. In case of a tie, if you betted \$100 on that outcome, you’d get \$333 + \$100 payout. In the case of betting on Real Madrid winning, the negative value tells you the minimum amount of that value is needed to win \$100. So, you’ll need to bet at least \$163 in order to get \$163 back plus the \$100 winning, for a total of \$263.

## Formula to convert the fractional odds to moneyline odds

If these different formats seem confusing, that’s understandable, but there are formulas to convert from one to another. For example, to convert from UK fractional odds to US moneyline odds. Here it is:

1. If the fractional odd is greater than 1 (e.g., 5/2):
• Convert the fraction to a decimal (e.g., 5 ÷ 2 = 2.5).Multiply the decimal by 100 (e.g., 2.5 * 100 = 250).The moneyline odd is the result, prefixed with a ‘+’ sign (e.g., +250).
General formula (if odd >1):
`Moneyline Odds = Fractional Odds * 100`
2. If the fractional odd is less than 1 (e.g., 3/10):
• Convert the fraction to a decimal (e.g., 3 ÷ 10 = 0.3).Divide 100 by the decimal (e.g., 100 ÷ 0.3 = 333.33).The moneyline odd is the rounded result and prefixed with a ‘-‘ sign (e.g., -333).
General formula (if odd <1):
`Moneyline Odds = -100 / Fractional Odds`

Note that these formulas give you the profit you would make on a bet, not the total payout.The total return would include the original amount you bet + these profits.

1. If the fractional odd is 1 (or 1/1): When Odds are 1/1, it means that the event is equally likely to happen as not. In betting terms, this is known as “even money” or “evens”. Use the same formula as if the fractional odd is >1. So, 1/1 = 1. And Moneyline odds = 1 * 100 = +100 This means if you bet \$100 and the event happens, you would win \$100, plus your original stake back, for a total of \$200.
In terms of payout, for every unit you stake, you would win the same amount. So if you bet \$1, you would win \$1, plus your original stake back, for a total of \$2.

NOTE: Probability and Odds can be thought of as inversely proportional. In other words, lower probabilities (or lower likehood of an outcome: win/loss/draw for each team) result in higher odds, which generally means a higher payout. And higher probability means lower odds and lower payout.

## How are the odds determined for a game outcome?

The odds for a game outcome, such as a match between Borussia Dortmund and Real Madrid, are determined by a combination of factors:

1. Probability: At its core, betting odds are a translation of probability. A bookmaker first determines the likelihood of an event happening and then converts this probability into odds. Lower probabilities result in higher odds, which means a higher payout, reflecting a less likely event.
2. Sports-Specific Factors: These include factors such as team form, injuries, and other relevant statistics¹. For example, if a key player is injured or a team is in particularly good form, this could influence the odds¹.
3. Market Factors: Market demand can also shift betting odds. If a large amount of money is wagered on a particular outcome, bookmakers might adjust the odds to balance their books, ensuring they remain profitable regardless of the event result.
4. External Factors: These can include things like weather conditions and location (home-field advantage for example).

Bookmakers employ teams of analysts who use statistical models to predict the outcome of sports events. These probabilities are then adjusted to ensure profitability, factoring in a margin that guarantees the bookmaker earns money regardless of the event’s outcome.

While these factors can help set the initial odds, odds are dynamic and can change based on new information or changes in market conditions.

## How are the probabilities calculated?

The fundamental principle is: the probability of an event is the ratio of the number of ways that event can occur to the total number of possible outcomes.

For example, flipping a coin can result in only 2 total possible outcomes: head (H) or tail (T). In a fair coin, each outcome of either H or T is equally likely (same probability). Next, the probability of an event P(E) is calculated as the number of ways that a particular event can occur divided by the total number of outcomes.

So, for a coin toss, probability of getting heads P(H) is 1 since there’s only one way to get heads; and then we divide it by 2 (the total number of outcomes), which equals 0.5 or 50%. P(H) = 50%
Same for P(T) as there’s only one way to get tails, and the total number of outcomes is still 2.

Probabilities are always between 0 and 1 (or 0% and 100%), where 0 means the event will not happen, 1 means the event is certain to happen, and values in between reflect varying degrees of certainty.

This is a basic example with equally likely outcomes. In more complex situations, such as a soccer match between two teams, determining the exact probabilities would require more detailed information and analysis. However, the fundamental principle remains the same.

## Back to real-world soccer match probabilities

Let’s now apply the above general theory of probability to the Champions League final game. If we take their Head-to-head history and results, we see: Borussia Dortmund and Real Madrid played 14 games in history against each other. And Borussia Dortmund won 3, and Real Madrid won 6 of the matches. The remaining 5 matches were draws (or ties).

So, let’s compute the 3-way probability: a) Borussia Dortmund wins, b) Real Madrid wins, c) Games ends in a tie

Given this historical data, we can calculate the probability of each outcome as follows:

1. Borussia Dortmund wins: The probability of Borussia Dortmund winning is the number of times Dortmund has won divided by the total number of games. So, Dortmund has won 3 out of 14 games, the probability P(D) :

P(D) = Number of Dortmund wins / Total number of games = 3/14 = 0.214 or 21.4%

2. Real Madrid wins: Similarly, the probability of Real Madrid winning P(R) is the number of times Madrid has won divided by the total number of games. If Madrid has won 6 out of 14 games, the probability would be:

P(R) = 6/14 = 0.429 or 42.9%

3. Game ends in a tie: The probability of a tie P(T) is the number of times the game has ended in a tie divided by the total number of games. If there have been 5 ties out of 14 games, the probability would be:

P(T) = 5/14 = 0.357 or 35.7%

## Probabilities as Odds

So, next logical question would be: How can we express the above probabilities as odds?

The odds of an event are a way to represent the probability of the event as a ratio. i.e. The event’s likelihood of happening : it not happening.

So, the odds formula is:` O(E) = P(E) / (1- P(E)`
Where O(E) is the odds of an event happening; P(E) is the probability of the event occurring; therefore, 1-P(E) is the probability of the event not happening.

NOTE: Probability is expressed as a percentage (or a fraction 0 to 1); while odds are expressed as a fraction first and then either in UK (fractional) or USA format (moneyline or American Odds).

Let’s now convert the match probabilities we calculated to odds:

1. Borussia Dortmund wins: The odds of Dortmund winning O(D) would be the probability of Dortmund winning divided by the probability of Dortmund NOT winning. As we found that the probability of Dortmund winning is 0.214 (or 21.4%), the odds would then be: O(D) = P(D) / (1-P(D)) ​= 0.214/ (1-0.214) = 0.273 approx.

This means the odds are approximately 0.273 to 1, or can be expressed as 273 to 1000 in favor of Dortmund winning. *

1. Real Madrid wins: Similarly, the odds of Real Madrid winning O(R) would be the probability of Madrid winning divided by the probability of Madrid not winning. If the probability of Madrid winning is 0.429 (or 42.9%), the odds would be: O(R) = P(R) / (1- P(R)) ​= 0.429/ (1-0.429​) = 0.753 approx.

This means the odds are approximately 0.753 to 1, or can be expressed as 753 to 1000 in favor of Madrid winning. *

1. Game ends in a tie: The odds of a tie (O(T)) would be the probability of a tie divided by the probability of not a tie. If the probability of a tie is 0.357 (or 35.7%), the odds would be: O(T) = P(T) / (1−P(T)​ = 0.357/ (1-0.357) = 0.556 approx.

This means the odds are approximately 0.556 to 1, or can be expressed as 556 to 1000 for a tie. *

• NOTE: These odds we calculated here are odds of an event happening. These are not expressed as Betting Odds as we saw earlier from Bet365. Betting odds are typically expressed as fractions or decimals.
• Here’s how we can express the probabilities we calculated earlier as betting odds:
1. Borussia Dortmund wins: The probability of Dortmund winning was approximately 0.214 or 21.4%.
And as before, the odds of Dortmund winning O(D) we calculated above = 0.273, or 0.27 rounded.
This is expressed as Betting Odds as 27 to 100 (after multiplying both sides by 100 and rounding to the nearest whole number for simplicity). Betting Odds for O(D) = 27 to 100.
2. Real Madrid wins: Similarly, the probability of Madrid winning was approximately 0.429 or 42.9%. The odds of Madrid winning O(R) we calculated above = 0.753, or 0.75 rounded. This can be simplified to a fractional odd of approximately 75 to 100. Betting Odds for O(R) = 75 to 100.
3. Game ends in a tie: The probability of a tie was approximately 0.556 or 0.56 rounded. This can be simplified to a fractional odd of approximately 56 to 100. Betting Odds for O(T) = 56 to 100.

So, the betting odds would be approximately:

• Borussia Dortmund wins: 27/100
In payout speak: If we bet \$100 on Dortmund, we win \$27 + get the \$100 initial bet back if they win. Total: \$127