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Calculation of Odds: Probability and Deviation


Calculating odds is a science, and statisticians/analysts, who can do the job competently, receive good annual salaries (GBP 50k-80k; €60k-95k: Quantitative Analyst).

These jobs are paid so well because the results form the backbone of each bookmaker’s business. The better an analyst understands his job, the bigger the potential profit margins for the bookmaker. In order to make long-term profits, a good understanding of odds calculation is therefore also essential for any bettor.

Man with 4 arms and juggling with calculator, abacus, note pad and penImage: alphaspirit (Shutterstock)

Probability of Football Match Results

Odds are based on the probability that a certain event occurs: for example a home win, a draw, or an away win in football games based on historical data. But what is the current probability of each of these outcomes and how are percentages computed? Also, from where does one get the data from?

The following diagram shows the distribution of results for the English Premier League for home win, draw, and away win during the last five complete seasons:

Results: English Premier League - 2005 to 2010 – Diagram

Looking at the graph, one can see that the distribution of results is rather similar for each year.

Without considering factors such as matches between strong or weak teams, teams under new management, rain affected games or anything else one can think of, fixtures in the English Premier League, according to the statistics, show on average 24.46% of all games were drawn, 27.35% ended in an away win and 48.16% were won by the home team:

Results: English Premier League - 2005 bis 2010 – Data from: www.sportpress.com

Only twice (of 15) did results fall outside of ± 2% residual from the average figures, being drawn games in 2005/2006 (-4.2%) and away wins in 2009/2010 (-3.4%):

Absolute deviation: English Premier League 2005 to 2010


Comparison of the expected results with observations

Now, it is time to compare the expected results (based on the average results of 5 years) with the observed results of the current 2010/2011 season up to 19 February 2011:

Expected vs. Observed: English Premier League 2005 to 2010

There are quite large differences from the expected values (averages or means) to the observed results (actual results for 2010/2011) in both the drawn matches and away wins categories. However, I am pretty sure that by the end of the season, the figures will adjust themselves more in line with the 5-year average figures and that the differences showing now can be explained by having compared only two-thirds of the current season with the average results of a full-year. However, this indicates that there is an uneven distribution of home wins, draws and away wins at different times over the season, which should balance out by the close.

So, if the odds of a single match are calculated, this seasonal effect must be considered, but for this article and for your general understanding, I shall not complicate matters by touching on it further.

Mean (average value), Errors and Residuals (relative and absolute deviation)

For those of you with difficulties understanding what arithmetic mean and errors/ residuals are, herewith a few additional explanations with examples:

  1. Arithmetic Mean
    The arithmetic mean, often referred to as simply the mean or average when the context is clear, is a method to derive the central tendency of a sample. In probability theory, the mean is also called expected value (or expectation, or mathematical expectation, or the first moment).

    The mean (average value) is a known value and can therefore be used as an expectancy value.

    Simply speaking, one can use the mean to predict future events (e.g. the distribution of football results) quite accurately.

    Example using the 5-Year English Premier League table above for home games:
    (50.53% + 47.89% + 46.32% + 45.29% +50.79%) divided by 5 = 48.16% (= mean/ average)

    Therefore, it “is expected” that the 2010/2011 season will produce 48.16% home wins.
  2. Errors and Residuals (relative and absolute deviation)

    Statistical errors and residuals are two closely related and easily confused measures of the deviation.

    The error of a sample (e.g. observed football results for a certain period of time) is the ‘relative’ deviation from the function value (mean), while the residual is the difference (absolute deviation) between the sample and the function value (mean).

    For example, the RESIDUAL (absolute deviation) for home wins during 2005/2006:
    50.53% (2005/2006 home wins) minus 48.16% (average value for 5-years) = 2.37% (= residual)

    For example, the ERROR (relative deviation) for home wins during 2005/2006:
    2.37% (residual) divided by 48.16% (mean) = 4.92% (= error)

    The error (relative deviation) is the proportional deviation between the observed value (the actual results) and the expectancy value (the mean/ average of all years). The error (relative deviation) puts residuals into relation to each other.

Relative deviation: English Premier League 2005 to 2010


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Last Update: 11 March 2011

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27 Responses to “Calculation of Odds: Probability and Deviation”

  1. bester
    11 March 2012 at 6:11 pm #

    the stuff you publish kicks ass. thanks!

  2. Ian538rygby
    22 March 2012 at 9:10 am #

    just found your blog……I am not usually prone to participating on forums or blogs, but I have been fascinated, reading your articles, and thought your efforts deserve to be acknowleged.

  3. Ian538rugby
    22 March 2012 at 9:13 am #

    just corrected username and email……..lmfao

  4. Soccerwidow
    22 March 2012 at 9:16 am #

    Ian, thank you for your very kind words 🙂

  5. 22 March 2012 at 12:08 pm #

    Thanks for a well written an easy to follow article – something thats pretty rare on the subject of calculating football odds.

  6. Soccerwidow
    23 March 2012 at 11:58 pm #

    Thank you, Rick. We try our best!

  7. Helder
    13 July 2012 at 2:51 pm #

    Hi Soccerwidow,

    Glad to see the website is up and running again. I would like to ask you a question regarding the non arithmetical calculation of odds in the “Calculation of Odds” section.

    In your example you, the calculated odds are found by using the 5-year mean of 48,16% (home wins). You therefore assume that for every game the odds to back the home team are of 2,08. You then use the found relative deviation the calculate the minimum and maximum odds.

    If we were to calculate the odds based on reality, we would have to find the probability of the home team winnning, instead of using the 5-year average mean of 48,16%. Once we find the real odds we could then calculate the maximum and minimum odds (using the 5-year relative deviation), to which we would compare the market odds given to us.

    Is this train of thought correct?

  8. Helder
    15 July 2012 at 8:55 pm #

    Thank you for your help, those articles are very good and they have helped me clarify certain aspects. But with everything, the more you more learn the more questions you have 🙂 If you don’t mind could you give me your opinion on the following question?

    Let’s say that I like to bet on certain events that have a high probability of happening (low odds), then I would have to:

    – Find something to bet on that is statistically a sure thing. In other words a high probability and a small relative deviation over time (for instance backing Team A at home).
    – Find an edge and only bet on over value home games by checking head-2-head statistics.
    – Create a staking plan.

    If we were to compare this strategy to one where we bet sistematicaly on every Team A home game, could we say that one is more reliable than the other, or more profitable in the long run, or is it just a personal preference?

    Thank you for your time.

    • Soccerwidow
      15 July 2012 at 9:20 pm #

      Without having calculations and numbers in front of me I cannot say which strategy is more reliable. Each strategy needs to be individually evaluated in detail before a statement can be made.

      However, what I can say is that in the end it really doesn’t matter. You pick any strategy according to your personal preferences, and get your teeth into it. I promise that you will be able to make it a successful strategy if you don’t give up too early, and calculate and think everything through. Though it may take a long while.

      One of the main mistakes unsuccessful bettors do is that they keep “shopping round”, trying one strategy for some time, then the next, and then again another. Your own time resources are one of the largest issues. Therefore it’s crucial to spezialise in order to succeed.

  9. A. smith
    8 April 2013 at 6:45 pm #

    How come you don’t include missing/injured players in the calculations?

    • Soccerwidow
      9 April 2013 at 6:09 am #

      Missing/insured player have a far less effect (if at all) on the final distribution of results than commonly believed. Football is a team sport. In professional football there are lots of players employed. All of them professionals. Bookmakers set their odds weeks, sometimes months, in advance. This should tell you something about the importance of injuries/ missing players for odds calculation.

      In the “normal world” it’s only very tiny and poor companies with very few employees which suffer if a major employee is missing. In larger companies a sick note from an individual will hardly affect the overall performance of a company’s results.

  10. Autumn
    20 April 2013 at 2:13 pm #

    Hello, just to ask, how do you think chaos theory would affect the results of a soccer match? Like for example, in terms of 1) movements of the ball and 2) behavioural aspects. Thank you!

    • Soccerwidow
      24 April 2013 at 6:14 am #

      Chaos theory is very, very advanced maths, and goes far beyond the simplified explanations in this blog which are aimed at readers with more basic maths skills.

      However, to answer your question: Yes, of course. If you are familiar with chaos theory and advanced statistics, then apply them to your calculations and predictions of match results! This will certainly lead to quite accurate results.

      Try also search engines for the keywords chaos theory football. This will bring up a good number of academic articles on this subject.

  11. Ineedscore
    22 November 2013 at 10:24 am #

    Hello guys! I am looking for the acceptible methods to calculate probability chances “1 x 2” on football matches. I read a lot of articles, but there are no useful information that can help me. I started to investigate probability theory myself and I freezed after some calculations.
    Let imagine we have match between Chelsea and Liverpool. I understand how calculate probabilitly for each team. I took last ten Chelsea’s home matches and last ten Liverpool’s away matches. After simple calculations(win/10; draw/10; lose/10) I received:
    P(Chelsea win at home) = 0,55;
    P(Chelsea draw at home)=0,35;
    P(Chelsea lose at home) = 0,10;

    P(Liverpool win away) = 0,20;
    P(Liverpool draw away)=0,55;
    P(Liverpool lose away) = 0,25;

    But I could now understand how to use this calculated data to predict match Chelsea – Liverpool. I tries to apply sum and multiplication theorems, but there could be incredible result, for exmaple over 1. I definetly understand that this calculation would be very approximatly and that there are a lot of parameters that affect on football result, but I think that it could be useful to predict match. Thank you for any help!

  12. Tsvetelin
    1 July 2015 at 9:14 am #

    hi soccerwidow

    i m getting into sportsbetting,i figured out that i might bet on soccer becouse its the sport that i know more than the others,so far so good. when i started to surch on the internet information for betting i get a lot,and i mean a looot confused,there is no precise information, everyone says diffrent shit and i m far away from descovering hot all this works
    i found your blog and strarting to read it,you talk about the analysis of the match,analysis for the odds and other things,is this the way professional sports bettors bet? for me to bet do i need this tools? if i have to,can i find this tings on the internet and learn them? when you find a site with odds,isn t it the same as you do you r own odds,saves so much time..so,how can i learn all this things on internet? and what are the really main thing i have to learn in order for me to bet? please help me cuz i m a lot confused and i see you are truly an expert in this so thanks if you answer my questions

  13. scott
    29 May 2016 at 10:58 am #

    The minimum odds are computed as follows:
    Home win: 2.08 calculated odds multiplied by (1 minus ‘error’ 4.14%) = 1.99

    The maximum odds are computed as follows:
    Home win: 2.08 calculated odds multiplied by (1 plus ‘error’ 4.14%) = 2.17

    am i just being really stupid beacuse i cant seem to get these same answers when i do this sum??

    many thank!
    scott

    • Soccerwidow
      30 May 2016 at 7:29 am #

      Hi Scott, here’s the calculation a little bit more in detail. Hope it helps.

      2.08 calculated odds multiplied by (1 minus ‘error’ 4.14%) = 1.99
      1 minus ‘error’ 4.14% >> 1 – 0.0414 = 0.9586
      2.08 × 0.9586 = 1.9939 (rounded: 1.99)

      2.08 calculated odds multiplied by (1 plus ‘error’ 4.14%) = 2.17
      1 plus ‘error’ 4.14% >> 1 + 0.0414 = 1.0414
      2.08 × 1.0414 = 2.1661 (rounded: 2.17)

      Best wishes,
      Soccerwidow

  14. Jimbo
    20 October 2016 at 5:03 pm #

    Hey,

    I just came across an app that offers the chance to win 50,000 to correctly predict the odds of two entire leagues.

    As an example it says if I can correctly predict 100% premier league results this weekend (10 games) + (10 games of the Spanish League)

    What is the average chance per game or the chance winning the 50,000?

    • Soccerwidow
      28 October 2016 at 6:32 am #

      Do you have to predict the expected odds (antepost), or predict the actual results which will be played? These are two completely different things.

  15. erez
    7 December 2016 at 10:59 am #

    if the home team have advantage in the odds ?

    for example odds are 2 but this home team get 1.75 ?

    • Soccerwidow
      8 December 2016 at 7:58 am #

      Hi erez, I don’t understand your question. Sorry!

      Generally speaking, home favourites are mainly overpriced, meaning that bookmakers price them at a higher chance to win than their true chances are. For example, odds of 1.75 stand for a 57.1% chance to win, whilst odds of 2.0 for a 50% chance. If bookmakers offer 1.75 odds for a team which should be actually priced at 2.0 then they are ensuring that they have the mathematical advantage on their side.

      You may be interested into our HDA Simulationtables. Just have a look.

  16. Kelvin mulwale
    15 July 2017 at 12:09 pm #

    If odds are 2.0 and 3.12 what is possible outcome

  17. Erny
    4 October 2017 at 11:31 pm #

    Hello
    I have been using some data like
    1. Average Home Team goals,
    2. Avereage Home team conceeds
    3. Average scored by Away Team
    4. Average Conceedes by away Team

    I also use:
    1. Attacking rating
    2. Defensive Rating

    How can I combine these statistics to get me an idea of expected result.

    I am aware that per chance things can change

    Erny

    • Soccerwidow
      5 October 2017 at 10:17 pm #

      Hi Erny,

      I’ll have to think about that and add your question to the 1×2 course I have been planning to write for a long while, especially the attacking/ defensive rating… No idea, to be honest, because I don’t even know where I could get enough data from to analyse it properly.

      Generally speaking, from a statistical perspective, football matches do not occur frequently enough. For example, looking at a single league such as the German Bundesliga with only 306 matches per season, a relevant sample size is never going to be large.

      The plain truth is that any football league is simply not large enough to generate a significant amount of completed match statistics per season. This means that the standard deviation (margin of error) is always going to be relatively large…. and attacking/ defensive rating… it’s only the last few matches, isn’t it?

      I probably wouldn’t burn my fingers with it.

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