Introduction to Combinatorics and Probability Theory

This article is a step-by-step guide explaining how to compute the probability that, for example, exactly 4 out of 6 picks win, or how to calculate the likelihood that at least 4 of 6 bets win.

To help your understanding of this topic you will need to comprehend the basics of football result probability calculations, which I explained in detail in the article Calculation of Odds: Probability and Deviation.

The Basics of Probability Computation in Football Betting

The following picks table (published on 22.03.2011 in the German language Soccerwidow site), contained 6 value bets including the calculated probabilities for each bet to win:

English Football: Picks 22.3.2011

Of the 6 published picks, 4 won and made a profit of 19.9% on the 50.00 € betting bank. I will now attempt to explain the mathematics behind the above selections.

The calculation of the probability that all 6 Picks will win is relatively easy and requires no knowledge of difficult formulas. You simply multiply together the given probabilities, thus:

The result of 6.3% is the probability that all 6 picks in the portfolio win.

Of course, the other end of the scale is that all 6 picks will lose. Again, this is a straight forward calculation: simply multiply the opposing probabilities to those used in the ‘win’ scenario, thus:

The result of 0.1973% is the probability that all 6 picks lose.

Summary:

• Probability that all 6 Picks win: 6.3%
• Probability that all 6 Picks lose: 0.1973%

If you divide 6.3% by 0.1973% the result is 31.93. This means the probability in this particular portfolio that all 6 picks win is almost 32 times higher than the probability that all 6 picks lose.

Practically speaking, there is a 32 times higher chance of winning all 6 bets and cashing 40.90 € profit than losing all 6 bets together with the entire 50.00 € starting bank.

When using odds in European format (decimal) you can be forgiven for thinking that average betting odds are simply computed by building the arithmetic mean of the data to be analysed. Unfortunately, this is the wrong approach and leads to a deceptive result.

Photo: Elnur (Shutterstock)

As a reminder, European odds are calculated as the reciprocal of the statistical probabilities of each event:

and vice versa … The implied probabilities are the reciprocals of the odds:

In effect, European odds are ratios/relations representing the likelihood of an event happening in comparison to each other event (e.g. a bet priced at odds of 4.0 is half as likely to win as a bet with odds of 2.0).

If these ratios are averaged using arithmetic mean (a common error), high data points are given greater weights than low data points. (e.g. working out the arithmetic mean of a set of 20 odds, 19 of them between 2.0 and 2.4, would be skewed if the 20th figure was, say, 15.0).

The correct approach is to calculate average odds by forming the harmonic mean!

The harmonic mean is defined as the reciprocal of the arithmetic mean of the reciprocals of x₁, x₂, …, x_n (the odds):

As the reciprocals of betting odds are the implied probabilities of the events, one can calculate the harmonic mean as a reciprocal of the average probability of the respective bets:

The above equations rearranged facilitates the harmonic mean calculation by dividing n (the number of matches) by the sum of the reciprocals of the odds:

Or alternatively… dividing n (the number of matches) by the sum of their individual probabilities:

The result (harmonic mean) is the accurate average of the betting odds.

Excel users employ the following formula: =HARMEAN(number1,number2,…)

Here are some of the most popular misconceptions…

Photo: Jeff Banke (Shutterstock)

(1) Betting on low odds is not worth the effort because winnings are very little

This is absolute nonsense. Successful betting is not about the pay-out of a bet, but the main goal is to make a profit. Very often, especially in the small odds markets excellent value can be found.

In addition, low odds have the huge advantage that they stand for high probabilities and therefore losses are considerably fewer in number than with higher priced, lower probability events. Also, small odds experience shorter losing streaks and the patience of the gambler is never stretched too far.

However, what really matters is the price of the bet.

If, for example, the calculations for a match throw up an 85% chance that there will be under 3.5 goals (corresponding betting odds: 1.18), and the market is offering odds of 1.25 then this is a fantastically good value back bet opportunity.

(2) Betting on higher odds is more profitable

Again this is a very common misconception. Higher odds do not automatically hold more value and are therefore more profitable than lower odds.

Many punters believe that better returns can be achieved in the long run if playing at odds between 2.5 to 3.0. Of course, it is understandable that these bets are quite popular as the potential winnings are 1.5 to 2.0 times the stake.

However, what most gamblers do not consider (or perhaps simply do not know!) is that these betting odds represent chances of 33% to 40% of winning, meaning that on average six to seven out of 10 bets in this group will lose.

Since winning and losing bets are never evenly distributed it may well happen that, with a little luck, a few bets in a row may be won but, much more likely is that many, many consecutive bets will lose until the betting bank is depleted beyond recovery or is drained altogether.

The terms probability, expectation, likelihood, chance and hit rate are all closely related, and express more or less the same thing. The difference is that before a game one talks about ‘chance’, ‘probability’, ‘likelihood’ and ‘expectation’ but, after it has finished, these terms are replaced by ‘hit rate’.

Photo: Andrej Vodolazhskyi (Shutterstock)

The term value has no place in this relationship. The literal meaning of ‘value’ is benefit, merit, worth and price. Unfortunately, the latter meaning is probably the main reason why people find the expression ‘value’ rather confusing. Betting odds offered by bookmakers or exchanges are probabilities converted into the price of a bet. However, if the ‘value’ of a bet is discussed, this refers not to the actual price (odds) of the bet but to the merit or benefit of a particular price.

Strictly speaking, it would be more correct, instead of using the term ‘value’ to say “mathematical advantage” or “expected merit“, but these phrases are uncommon.

Probability & Expectation

In the 2012 UEFA Champions League final between Bayern Munich and Chelsea the probability (statistical expectation) for Bayern to win was 64.6% (the calculation is explained here – Sorry! You’ll have to auto-translate it if you don’t understand German! ).

Please note that the terms probability, likelihood, chance and expectation are frequently used synonymously in scripts.

Technically speaking this is not entirely correct and real mathematicians probably groan indefatigably when they read betting forums or posts.

However, for simplicity, this is allowed and even we use these terms arbitrarily in some of our articles and explanations, but we mean all the time the same thing: The expected hit rate!

An expectation, probability, likelihood (call it what you will!) of 64.6% for Bayern to win means that in the long-run, placing 100 similar bets should see 65 winning and 35 losing.

The expectation, probability, likelihood is calculated by building the mean (average) of historic match data, correcting the results by a few factors, and then taking some of the latest trends into consideration (if you would like to learn how to do this, have a glance at Soccerwidow’s Value Betting Course).

Reality Check: Hit Rate

Hit rate has no connection with the quality of predictions. High hit rates are often interpreted as a sign of a successful picking strategy. Unfortunately, this is a big and very common misconception! The only thing hit rate expresses is the number of winning bets compared with all bets. Hit rate is not a statement of any realised gains or losses.

Hit Rate is the number of winning bets in relation to all placed bets.

For example, if you bet on the full-time correct scores market, odds of 10 and over are normal. Based on such odds, any hit rate higher than 10% means profit, which in turn means if you manage to predict full-time scores with an accuracy of more than 1 correct in every 10 attempts, you will make a profit (despite a relatively small hit rate).

Another example: if you bet only on outcomes with probabilities between 60 and 70%, then the expected hit rate is between 60 and 70%. It follows that after 100 bets you should achieve 60 to 70 winning bets.

In this article, Soccerwidow puts the record straight…

Photo: Lim Yong Hian (Shutterstock)

Stake (S)

Definition: The sum of money gambled on the outcome of an event.

In the online world of gambling, stakes are electronically placed on a desired outcome with a second party that has agreed to accept your stake, whether this be a bookmaker or an anonymous person/group in a betting exchange. These ‘adversaries’ are effectively backing with their own money against your selection, hoping to make a profit of your stake if your selection in the event turns out to be wrong.

Once the outcome of the event is decided, stakes are returned to you in full if your bet has won (plus the winnings), or if you lose, the stake is lost and either retained by the bookmaker or, transferred to the winning side in the betting exchange.

Technically speaking, stakes are sureties! This means that they are short-term deposit payments to guarantee that the losing party will honour his debt obligation to the winner of the bet.

Yield (Y)

Definition: Profit/Loss ratio as a function of the total capital employed (total staked)

MS – money staked
ML – money lost (stakes lost)
MW – money won (purely winnings; returned stakes are ‘neutral’, not winnings)
PL – MW minus ML = net profit (or loss); equivalent to your bank growth


Formula Yield:
PL divided by MS = Yield (written as a percentage)

Example: if you staked a total of 760 € on your portfolio of bets, and your bank grew by 70 €…
70 / 760 = 0.092 = 9.2%

To simplify, the term YIELD means profit, earnings, harvest, income or revenue. It measures betting efficiency against the total turnover. The lower your aversion to risk, the higher your probabilities to win, the smaller your yield will be and vice versa.

Generally speaking and depending upon the strategy employed, a good bettor will yield between 5 and 10 percent profit in the long run. In football betting any yield over 7% is considered to be a very, very good yield.

For example, with an average yield of 7%, a starting bank of 100 €, and staking 10 € each bet: in order to win 700 € you would need to turn over your bank 1,000 times (!), or otherwise staking a total of 10,000 € to win 700 €.

Yield is NOT the same as ROI (Return on Investment)!

Of course, bookmakers are in the business of setting odds and determining prices which are offered for certain betting events.

Photo: Zadorozhnyi Viktor (Shutterstock)

When viewing odds in betting exchanges such as Betfair, Betdaq, Smarkets, or WBX, you should understand that it is neither the exchange platform or the traders using them who set the odds.

The fact is that the bookmakers are used as the market guide for traders on the betting exchanges, and it is the bookies who compile and publish their odds weeks in advance of the events in question (sometimes even months), and certainly well before the exchanges even open their markets for trading.

If you have ever calculated odds you will have noticed that the bookmakers’ offers often do not represent the ‘true’ picture, in other words, the ‘true’ mathematically calculated values (the statistically expected values).

Only occasionally (probably in less than half of all cases) are odds close to the statistical expectations of the betting event. However, in the vast majority of games, odds are either considerably higher than mathematically expected or far lower…

Why Is This So?

You have to appreciate that bookmakers do not really intend to predict an outcome (correctly). If you enjoy statistical analysis, then take a little time to do a simple calculation for any league of your choice. Simply convert bookmaker odds into probabilities and compare them to the actual distribution of the results (historical data for 22 European leagues can be found at www.football-data.co.uk).

Bookmakers have been around for thousands of years in one form or another. Their main goal is of course to make a profit. They price their odds to ensure that sufficient action is taking place on both sides of a bet.

If a bookmaker’s betting odds are not aligned to public opinion then a disproportionately large amount of money will be placed on only one side of a bet. This would be a gamble for the bookmaker. However, bookmakers are not in the business of speculating on an outcome.

The role of bookmakers is, strictly speaking, rather the function of an intermediary, similar to a stockbroker. They take money from various people on various outcomes and after the game is finished they pay out the winners.

In return for this service, the bookies take a “fee” known as the overround.

Although bookmaker odds do not always fully represent the ‘true’ (expected) probability distribution you may sometimes find it necessary to remove the overround from the market odds.

What Exactly is Overround?

The screenshot below shows a number of bookmaker odds for the match Russia v Portugal (courtesy of Oddsportal). You can clearly see the odds for Russia to win at home were between 2.50 (lowest) and 2.95 (highest):

The average odds in the market for Russia to win this game were 2.67 at the time of the screenshot
(NB. The screenshot is cropped and only shows a handful of bookmakers from a long list of over 50 different firms quoting odds for this match. The average odds are therefore based on figures from all the bookmakers, not just the small selection shown).

To convert odds into probabilities divide 1 by the odds. The following table shows odds converted into probabilities for Expekt (lowest home odds), Marathonbet (highest home odds) and William Hill (middling odds):

As you can see, when adding up the home, draw and away probabilities for each bookmaker there is not a single line which exactly totals 100%.

For example, Expekt’s probabilities add up to 108%, meaning that they were serving this market with an overround of 8%.

At Marathonbet the overround was 2.5%.

With 99% showing in the totals column for the highest odds available for the home win, draw and away win, you will understand this is a negative overround of -1% (99 minus 100). Consequently, negative overround situations present arbitrage opportunities. Betting on all three results using the highest odds available would produce a no-lose bet with a guaranteed 1% profit.

Which Bookmaker’s Odds Represent the Expected Probability Distribution?

If you look a little closer at the image above you will notice that not only the overrounds vary but the probabilities also differ substantially.

This begs the question which of this set of odds and probabilities comes closest to the true expectations?

Expekt were pricing the home win as a 40% probability; Marathonbet’s price equated to a 33.9% chance; the market average suggested a 37.5% home win likelihood.

The answer is therefore none, some, and all, and on the face of it, this may seem quite confusing!

But you must be aware that it is not the goal of bookmakers to predict the expected probability distribution as correctly as possible. Their main objective is balancing the books. Therefore, odds are adjusted to demand. Bookmakers take public opinion into account and hence, quite often their odds do not reflect the true statistical-mathematical expected distribution.

Continued…

Every bettor should try to optimise the financial return from his/her risk, which means finding absolutely the best prices available.

Of course, depending upon the betting exchange employed, profits are eroded by up to 5%, or more if you are subject to premium charges with certain firms.

The time it takes to analyse a match and formulate value bets would be wasted to some degree if you then neglected to tie in the best possible reward on offer.

With a little imagination it is fairly easy to contrive several alternatives to laying certain outcomes within a betting exchange when faced with any lay bet opportunity. The question before committing to a lay bet is whether the same outcome can perhaps be achieved using a different market on the same match?

For example, betting against a team to win a match would usually mean laying them in an exchange, but how about the ‘Double Chance‘ bookmakers’ market favouring the draw and the other team? What about an ‘Asian Handicap +0.5‘ back bet on the other team, or a ‘Dutch‘ back bet on the draw and the other team, possibly even using two different bookmakers?

As ever, the best way to guide you and explain things as simply as possible is using an example and we have chosen the German Bundesliga 1 game between Borussia Moenchengladbach and Hamburger SV (Mgladbach v Hamburg) on 26.09.2012.

Firstly, let us assume we wish to bet against a home victory:

(1) The Obvious Approach: Laying the Home Team at a Betting Exchange

Mgladbach's Lay Price 5 Hours Prior to Kick-off was 1.83 (Betfair)

Laying 10 units at a price of 1.83, provides for a loss of 8.30 units if Mgladbach win the game but 10 units of profit if they don’t win. Including the typical 5% betting exchange commission however, the net financial gain for winning the bet is 10 units minus 5% = 9.50 units.

9.50 divided by 8.30 = 1.145, which equates to a back-betting price of 2.145.

Those back odds (2.145) provide a benchmark against which to compare the various back bet alternatives.

However, it is also necessary to find the equivalent (adjusted) lay odds which take the betting exchange commission into consideration. These adjusted lay odds are the basis for calculating the stake needed to achieve a return of 10 units. This is again crucial to allow comparisons and for this some further reverse engineering is needed:

Back Odds Converted to Lay Odds

The corresponding “adjusted” lay odds to win 10 units can be calculated using three simple steps:

• 1 divided by 2.145 (see above: corresponding back odds) = 0.4662
• 1 minus 0.4662 = 0.5338
• 1 divided by 0.5338 = 1.873

Alternatively, you can also adjust the lay betting odds using a simpler formula:

Divide (Lay Odds minus 1) by (1 minus Commission) plus 1

Illustration:

(1.83-1) divided by (1-0.05) +1 = 1.873


What do these ‘adjusted’ odds mean?
To win 10 units you need to calculate your stake (risk) based on a lay price of 1.873 rather than the price on offer of 1.83.

8.73 units divided by (1.83 -1) = 10.52 units, minus 5% commission (0.526) = 10 units return.

This effectively means risking 5.18% more than the price of 1.83 suggests.

Divide (1.873 minus 1) by (1.83 minus 1) = 1.0518

Therefore, to win exactly 10 units you must risk 8.73 units at lay odds of 1.83.

Multiply 10 units by (1.873 minus 1) = 8.73 units.

Phew, are you still with me..?

Summarising the results:

• Adjusted back-betting price: 2.145 - used to compare the lay price equivalent with the bookmakers’ back bets
• Adjusted lay odds: 1.873 - used to calculate the lay stake to win exactly 10 units
• Necessary stake to win 10 units at lay odds of 1.83: 8.73 units

International Club Competitions

Price: £29.90
WITH Asian Handicap



Price: £19.90
WITHOUT Asian Handicap


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The instructional video concentrates on the Home-Draw-Away odds section explaining the calculation of expected odds, and comparing them to market prices. We also show you where to find records of the opening bookmaker odds and their changes over time, and how to interpret this information.

Later in the video we identify a method of easily identifying matches to analyse with the spreadsheet, those with possible strong discrepancies in their odds, by looking at the money matched and waiting to be matched for bets in the betting exchanges.

This is an educational video, aimed at those who wish to further understand odds calculation, and would like to know how market prices are set.

This video is about: Calculation of 1X2 (Home-Draw-Away) True Odds in UEFA Champions League games & How Market Prices are set

The examples used in the video are the UEFA Champions League matches Real Madrid vs. Manchester City, Dortmund vs. Ajax and Manchester United vs. Galatasaray.

The spreadsheet can be used to calculate the ‘true’ odds for any match of your choice for international tournaments between individual clubs. The accuracy of the calculations improves if the teams in question have a similar recent history in the tournament under analysis (for example, both team’s last 10 international club competition games have all been in the Champions League).

All you need to do is fill-in basic data:

1. the best odds you can find for the various betting options dealt with by the spreadsheet
2. the home team’s last 30 home game half-time and full-time scores in their domestic league and all other competitions (no friendlies)
3. the away team’s last 30 away game half-time and full-time scores in their domestic league and all other competitions (no friendlies)
4. the home team’s last 10 international club tournament home game half-time and full-time scores (or if not 10, as many within the last 5 years as can be gathered)
5. the away team’s last 10 international club tournament away game half-time and full-time scores (or if not 10, as many within the last 5 years as can be gathered)

The team stats can easily be found in sites such as www.soccerway.com or www.oddsportal.com.

Use a combination of an exchange site such as www.betfair.com and www.oddschecker.com for the best odds.

For each of the following events, the expected mathematical ‘value’ is automatically flagged up together with the expected probability (statistical mean) of winning the bet:

• Match Result (home-draw-away) at both half-time and full-time
• Draw No Bet
• Win to Nil
• Clean Sheet
• Both Teams to Score
• Over/Under ‘X’ Goals
• Half-time/Full-time
• Half-time and Full-time Correct Scores
• The full realm of Asian Handicap options including all the half and quarter point thresholds

The spreadsheet calculates ‘true’ odds (expected probabilities), and compares market prices with the statistical probability of winning the bet to show you whether there is any ‘value’ in the bet and therefore whether it is financially worthwhile undertaking.

It should be noted that the spreadsheet is not a tool for automated betting. Although the tables provide vital information to improve the accuracy of predictions, the figures do not substitute for reading salient news and applying good judgment in addition.

Required Acumen

Ideally, you should already possess a basic understanding of odds and how they are calculated. The spreadsheet is a tool to identify potential bet candidates.

However, you need to understand what is meant by the term ‘value’ and also have a good idea how to spread risk with a well-structured betting portfolio. The spreadsheet is intermediate level and you should become entirely familiar with its mechanics before committing money to any bet recommendations suggested by it.

Good Excel skills would also be an advantage.

For beginners we offer a starter course: Soccerwidow’s Football Betting Course – How to Calculate Odds: Betting on Over/Under ‘X’ Goals, which is a step-by-step guide including exercises and tutorials on how to understand odds calculation. Although the example used is the popular over/under ‘X’ goals market, the principles can be just as easily applied to any market of your choosing.

Full support for all of Soccerwidow’s products is always provided should you require pre- or after-sales assistance or clarification. Feel free to ask any question via the Comment section below.

International Qualifiers

Price: £29.90
WITH Asian Handicap




Price: £19.90
WITHOUT Asian Handicap



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View Your Shopping Cart

Take…

…the last 30 competitive international (no friendlies) matches of the two teams involved (the last 30 home games for the home team and the last 30 away games for the away team)

…Obtain the half-time and full-time scores for each team’s 30 games plus the historical odds for these games (www.oddsportal.com is a good site for this)

…Filter these results to produce the 10 closest matches against comparable opponents (in other words, similar strength to the match in question)

…Run these data through a series of formulas

…And finally, compute the expected probabilities and corresponding odds

Sounds simple enough, doesn’t it?

Of course, this is a long-winded task on paper, and even for moderately competent Excel users this may prove more than a challenge.

Soccerwidow has produced the ultimate tool to solve these problems leaving you just to enter the half-time and full-time scores of each team’s 30 games plus the best odds you can find for each betting option to be analysed. Our Excel spreadsheet will then automatically calculate the expected (= true) betting odds and compare them with the market prices providing you with ‘value’ bet indications and the statistical probability of success for each bet.

The spreadsheet offered for sale comes pre-filled with data from the World Cup qualifier Moldova v. England scheduled for 7/9/2012. Afterwards, just delete or over-type the prerequisite data to analyse any match of your choosing.

For each of the following events, the expected mathematical ‘value’ is automatically flagged up together with the statistical probability of winning the bet:

• Match Result (home-draw-away) at both half-time and full-time
• Draw No Bet
• Win to Nil
• Clean Sheet
• Both Teams to Score
• Over/Under ‘X’ Goals
• Half-time/Full-time
• Half-time and Full-time Correct Scores
• The full realm of Asian Handicap options including all the half and quarter point thresholds

The spreadsheet calculates ‘true’ odds (expected probabilities), and compares market prices with the statistical probability of winning the bet to show you whether there is any ‘value’ in the bet and therefore whether it is financially worthwhile undertaking.

It should be noted that the spreadsheet is not a tool for automated betting. Although the tables provide vital information to improve the accuracy of predictions, the figures do not substitute for reading salient news and applying good judgment in addition.

Read some sample match previews applying the calculations in the German Betfair blog:

• England – San Marino: Starker Tor-Hagel in London zu erwarten
• Russland – Portugal: Eroberungszüge schlagen oftmals fehl in Russland

Frequently Asked Questions:

Can I use the spreadsheet for international friendlies, league games and domestic cup games, or the final tournament stages of international competitions on neutral ground?

Unfortunately, not.

The calculations are developed for international competitive qualifiers, such as the World Cup tournament, European Championships, African Cup of Nations, etc.

The data for both teams must come from similar competitive situations and therefore offer a direct comparison.

However, in some circumstances, the calculations may be applicable for national cup games, but only if there is enough data for the participating teams from competitive knock-out tournaments (no league games!). If there are 30 matches for each of the participating teams, the spreadsheet can be used. Please note that the data should not be older than 10 years and strictly without gaps in the past two seasons.

How long does it take to enter historical data into the spreadsheet?

This is the only manual procedure involved and using a site like www.statto.com or www.soccerway.com to obtain both team’s match results will probably take 15 minutes per team. Inputting the best odds will probably take another 15 minutes. Finding and inputting historical odds is another 15 minutes’ work. Of course, these figures are very arbitrary and depend upon your speed and keyboard proficiency.

The process will obviously become faster the more practiced you become.

Are the formulas for the spreadsheet calculations visible?

Yes, none of the formulas are hidden.

Exactly which historical records do I have to input?

The full-time and half-time scores for both team’s last 30 competitive international matches (home games only for the home team, away games only for the away team), plus the historical odds for these games.

What knowledge is required?

You should already have a basic understanding of odds and how they are calculated. The spreadsheet is a tool to identify potential bet candidates.

However, you need to understand what is meant by the term ‘value’ and also have a good idea how to spread risk with a well-structured betting portfolio. The spreadsheet is intermediate level and you should become entirely familiar with its mechanics before committing money to any bet recommendations suggested by it.

Good Excel skills would also be an advantage as it is necessary to sort data and filter information to identify matches of comparable strength.

For beginners we offer a starter course: Soccerwidow’s Football Betting Course – How to Calculate Odds: Betting on Over/Under ‘X’ Goals, which is a step-by-step guide including exercises and tutorials on how to understand odds calculation. Although the example used is the popular over/under ‘X’ goals market, the principles can be just as easily applied to any market of your choosing.

Full support for all of Soccerwidow’s products is always provided should you require pre- or after-sales assistance or clarification. Feel free to ask any question via the Comment section below.