There are all kinds of explanations on the Internet about various odds types, and the majority of them distinguish between fractional, decimal, and moneyline odds.

Image: paffy (Shutterstock)

Unfortunately, this is misleading and mathematically speaking, incorrect. There are only two types of odds, which are unrelated to their displays as fractions, decimals or, as in America, whole numbers.

Just remember those long ago school days (for some of us!)… A fraction, such as 6/5, converts into a decimal 1.2, or vice versa. Both numbers are the same, only written using different formats.

Here are the two main types of odds, including their formulas…

(1) Single-tier – (Net) Return Odds:

Fractional and Hong Kong odds are actually exchangeable. The only difference is that the UK odds are presented as a fractional notation (e.g. 6/5) whilst the Hong Kong odds are decimal (e.g. 1.2). Both exhibit the net return.

The European odds also represent the potential winnings (net returns), but in addition they factor in the stake (e.g. 6/5 or 1.2 plus 1 = 2.2).

(2) Two-tiered – Moneyline Odds: highlighting the amount of money risked, or the amount of money won:

Odds commonly referred to as ‘moneyline’ are mainly US bookmakers odds and also known as American odds. Moneyline means the money wagered either to win 100 units (e.g. -400), or money which will be won from a 100 units wager (e.g. +120).

However, both Indonesian and Malaysian odds, although displayed as decimals are, strictly speaking, ‘moneyline’ odds but their basis is 1 unit and not 100. Whilst the Indonesian odds closely resemble the American moneyline odds, Malaysian odds are a kind of “inverted” Indonesian style, combined with Hong Kong odds.

Although this may all sound pretty confusing, and the odds certainly look very different at first glance (see table below), just have a closer look at the above formulas – all odds are calculated using their net returns (formula for net return: 1/probability – 1) and change their formula at 50% probability (this is what is actually understood under by term ‘money-line’).

European, British, American, Malaysian, Indonesian, and Hong Kong Odds
BETTING ODDS CONVERSION TABLE

Fractional Odds (also known as Traditional or British/UK)

Fractional odds are favoured by bookmakers in the United Kingdom and Ireland.

Fractional odds quote the net return that will be paid out to the bettor, should he win, relative to his stake. Odds of 6/5 (“six-to-five”) imply that the bettor will cash £120 from a £100 stake. Should the punter win, he always receives his original stake back, plus the winnings.

So, if the odds are 6/5 and the stake is £100, then the total return is £220 (£120 winnings plus the original £100 stake).

Odds of 1/1 are known as ‘evens’ or ‘even money’. Not all fractional odds traditionally show the lowest common denominator. Perhaps most unusually, odds of 10/3 are read as “one-hundred-to-thirty”.

Hong Kong Odds (also known as HK Odds)

Image: Scott Maxwell (Shutterstock)

Hong Kong Odds are mainly used by people living in Hong Kong. Until 1997, Hong Kong was under British rule and therefore it should not be a surprise that HK odds closely resemble traditional British odds. The only difference is that HK odds are written in a decimal notation whilst the UK odds use fractions.

Hong Kong odds, as well as their UK counterparts, both quote the net return that will be paid out to the bettor should he win, relative to his stake. Odds of 1.2 in Hong Kong are the same as 6/5 (“six-to-five”) in the UK and both imply that the bettor will earn a net profit of £120 from a £100 stake. Should the punter win, he always receives his original stake back, plus the profit.

It follows that if the HK odds are 1.2 and the stake is £100 then the total return is £220 (£120 winnings plus the original £100 stake).

Decimal Odds (also known as European or Continental)

Decimal odds are standard on betting exchanges and they are favoured in continental Europe, Australia, and Canada.

Decimal odds are equivalent to the decimal value of the fractional odds, plus one. Because decimal odds are simply 1 divided by probability (= 1/probability) they are easier to use in calculations and we therefore favour them for our articles and explanations.

Thus, ‘even odds’ 1/1 are quoted in decimal notation as 2.0; the 6/5 fractional odds (1.2 HK odds) discussed above are quoted as 2.2. Decimal odds quote the net return paid out to the bettor PLUS the original stake.

So if the odds are 2.2 and the stake is €100 then the total return is €220 (€120 winnings including the original €100 stake).

Odds commonly referred to as ‘moneyline’ are the odds which are mainly used by US bookmakers. These odds mean the money wagered either to win 100 units, or the money which will be won from a 100 unit wager.

Please note that both Indonesian and Malaysian odds are also ‘moneyline’ odds although often wrongly referred to just as ‘decimal odds’, probably due to their decimal notation, whilst US moneyline odds are shown in whole numbers.

However, both Indonesian as well as Malaysian odds have the same underlying formulas as the US moneyline odds, only that their basis is 1 unit and not 100.

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Over/Under is a great way for people new to betting to get their feet wet…

What Does Over/Under Mean?

With over/under betting, the first thing to understand is that you are betting in relation to the number of goals scored in the game. It doesn’t matter which team scores the goals or even who wins.

You just take the number of goals scored by each of the teams and add them together to get the total goals scored during the game.

The simplicity of this bet is what makes it such a common wager.

When you place an over/under bet, you have to choose two things:

It doesn’t matter how close the result is to the goal line either, the bet pays the same whether the end result is close to the bet’s goal line or far away.

Why the “Point Five”?

Over/Under bets that end in “.5” have only two outcomes.

You either win the bet and get paid an amount equal to your stake multiplied by the betting odds or you lose the bet and your stake.

Because you can’t score half a goal, every bet is either a win or a loss.

Let’s take the most popular goal line in football: 2.5.

If you think the total goals in the game will be three or more, you would want to bet “Over 2.5”. If you think the total goals in the game will be two or less, you would choose to bet “Under 2.5”.

Over/Under bets without the “.5” or with different fractions (such as 2.25) may return part or all of your stake to you even if you don’t win.

For example, with a Total Goals bet, if you bet “Over 2” and exactly 2 goals are scored, you will not win the wager but you will get your stake refunded.

The graphic below shows which bet (over or under) will win given a certain score in the game for various goal lines.

Attention Please! Some Goals Don’t Count!

When you are betting in football, the only goals that count towards over/under bets are the ones scored during the regulation 90 minutes of each match.

If the game ends tied after 90 minutes any goals scored in extra-time or a penalty shoot-out do not count towards the over/under goals tally.

The result of the match itself does not matter; drawn matches with scorelines like 0-0, 1-1, 2-2, etc., also contribute to over-under bet results.

Over/Under 2.5: The Most Popular Bet in Football

We are using 2.5 as an example here because although there are many possible over/under bets, 2.5 happens to be the most popular goal line for football bets.

It turns out that when you look at a long history of football games, even across different leagues, the average number of total goals scored per game is very close to 2.5.

As a result, you often have a similar number of punters backing each side of the 2.5 goal line where the odds are similar no matter which choice of over or under is made.

Choose Your Bet Using Statistical Information

Bookmakers set the odds of over and under for each available goal line using statistical information on the teams who are playing each other as well as football games in general.

Demand on each side of the bet can change the bookmaker odds and knowing what the statistics say can help you determine whether a particular bet is a good or bad one to take.

The best bet is not always the bet that is closest to the actual score you expect.

In fact, it is often the bets further away from expectations that offer the greatest difference between the betting odds offered in the markets and the true statistical predictions.

Making Money at Over-Under Betting

Some punters are surprised at how important the underlying mathematics are, but long-term success at over/under betting is much more science than art!

Continue to follow our blog to learn more, or check out our book: Fundamentals of Sports Betting.

What does the Betting Odds Convertor do?

• Automatically converts odds into their implied probabilities
• Automatically switches odds between US, decimal, fractional, percentage, Hong Kong, Indonesian, and Malay formats
• Automatically converts probabilities into their corresponding odds
• Automatically displays the net return for each odd type

What Additional Information is included in the Spreadsheet?
Excel formulas, explanations and toggle facilities for the following conversions: –

• Probabilities into Odds
• Net returns into Odds
• Odds into their Implied Probability
• Odds into Net return

Why do I need to know how to convert betting odds
into probabilities and net returns, and vice versa?

In order to compare odds, you need to be able to convert them into implied probabilities and their net returns, because this is the foundation of all odds in the market. Only when you convert odds into their probabilities can you compare them.

Otherwise, how would you possibly know that 6/5 fractional odds, 1.2 HK odds, 2.2 decimal odds, +120 US moneyline, 1.2 Indonesian odds, or -0.83 Malay odds , all stand for a 45.45% chance of success and 120% net return?

Further reading:
Probability, Expectation, Hit Rate, Value, Mathematical Advantage: Explained
Understanding Betting Odds – Moneyline, Fractional Odds, Decimal Odds, Hong Kong Odds, IN Odds, MA Odds

What is the advantage of being able to switch between different betting odds formats?

As you compare prices and look for the best odds available in the market, you limit yourself if you are only at ease with one or two types of betting odds.

For example, you may wish to find the best price for a tennis match, and if you are only comfortable with European style odds, then you probably won’t even consider checking the US market or Asian bookmakers.

However, they may have far better prices on offer; you just simply don’t know because you don’t know how to convert them into probabilities and net returns.

You will never become a winner if you buy your bet at the most conveniently available price, without searching for the best price offered by the market.

Further reading:
Football Betting Odds: Study Comparison Bookmaker vs. Exchanges Odds
Why A Pinnacle Account Is Essential

Why is it important to know the formulas behind odds calculation?

If you calculate your own probabilities and need to check them against the odds offered by bookmakers to find value bets, how on earth would you work this out if you don’t know the formulas behind the odds?

The Internet is full of advice and explanations, but unfortunately the vast majority of information is either wrong or intended to lure you into betting. It is rare to find reliable information on the implied probabilities for odds offered in the markets, or the net returns from the odds.

Furthermore, only people who understand the probabilities behind the odds on offer are able to assess the potential value of a bet.

If you are not skilled in converting betting odds into probabilities you will, without exception, lose money in the long run. Keep reminding yourself: odds stand for probabilities. Bookmakers earn money from gambling, but they do NOT gamble themselves. Neither do they rely on luck. They know how to calculate probabilities and convert them into odds and vice versa. It’s as simple as this.

Therefore, if you want to become a winner, you need to start thinking like a bookmaker, and this means that you simply have to understand probabilities and odds. There is no alternative.

Further reading:
How do Bookmakers Tick? How & Why do they Set Their Odds as they do?
Calculation of Odds: Probability and Deviation

What impact does the knowledge of implied probabilities
and odds conversion have on my betting decisions?

There are two types of gamblers: successful and unsuccessful. The majority of bettors belong to the latter.

If you lose more money than you win, maybe you are basing your betting decisions more on gut feeling than mathematics. Perhaps you place your hard-earned money on a favourite team believing they are in a ‘must win’ match, thinking that they surely cannot lose against the underdog.

You may not even be aware that bookmakers adjust their odds to public opinion, and that odds offered in the market often do not represent their true value.

If your betting decisions are more gambling than anything else, relying on luck, without understanding underlying probabilities of the odds offered, then it is not really surprising that you lose more money than you win.

True understanding and lasting success is based on knowledge, not on ignorance, beliefs, or public opinion.

Only if you develop knowledge of implied probabilities from bookmaker odds, will you be able to check them against your own calculations. Then your betting decisions will be not gambling any more but a planned investment with the goal of a structured and organised increase of your bank.

Further reading:
Heart Beats Brain – Is Accurate Forecasting of Specific Results in Football Matches Possible?
How Bookmakers’ Odds Match Public Opinion

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This article is therefore a definitive guide to put the record straight…

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Stake

The sum of money gambled on the outcome of an event. The amount of money played with, or placed as a bet.

In the online world of gambling, stakes are electronically placed on a desired outcome with another 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 bet, the stake is lost and either retained by the bookmaker, or transferred to the winning side in the betting exchange.

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

Yield

The Profit/Loss* ratio^† applied to the total capital employed (total staked)

*This is Profit or Loss, NOT Profit divided by Loss

^†ratio = the quantitative relation between two amounts showing the number of times one value contains or is contained within the other.

Literally translated, the term YIELD means profit, earnings, harvest, income, revenue…

When applied to gambling, Yield measures betting efficiency compared to total turnover.

If your aversion to risk is low, you will select bets with higher probabilities. Bets with higher probabilities of winning carry lower odds. Lower odds means a smaller yield.

If you enjoy higher risk strategies, the opposite will apply.

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 good result.

Yield Formula:

PL divided by ∑MS (written as a percentage):

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

We come across many forum threads with people talking about their betting strategies; It is also easy to find plenty of websites offering betting systems for sale.

What many of them have in common is claims of high yield results, probably intended to impress the reader.

If they are to be believed then this is an indication of high risk strategies employed.

It must be remembered that in the Yield formula, the sum of money staked (∑MS) includes all stakes, even those that have not been lost. (In other words, the refunded ‘guarantees’).

People tend not to understand this fully and as a result mistakenly overstate their yield results.

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

Return on Investment (ROI):

The ratio of money gained or lost on an investment relative to the amount of money invested. In other words, the profit/loss ratio as a function* for investment^† (capital employed).

*function = a relation or expression involving one or more variables; in this case, investment, profit, loss.

^†investment = long-term employment of tangible, financial, or other assets that are not meant for immediate gains but are intended to generate benefits (normally earnings or profits) in the future.

ROI is also known as ‘rate of profit’ or sometimes just ‘return’.

ROI Formula:

If you bet systematically, your starting capital will be turned over again and again: It is effectively the same money you are investing. (So long as you don’t lose every bet!).

The ROI formula resembles the yield formula, but here, profit/loss is related to the actual investment (starting bank) instead of the total of all stakes (turnover).

For a more accurate ROI calculation, in an ideal world, you should also factor in to the investment all other costs of ‘setting up the business’. For example, hardware/software costs (computers). However, we will leave this out of our calculations for the time being for the sake of simplicity; you can always include these costs once you have mastered the concept.

ROI is always calculated for a certain predetermined amount of time; in finances usually for one whole year, but it is also common and acceptable to calculate the ROI monthly or, in a betting sense, for only the number of bets within a specific time scale.

The return on investment index is especially suitable when the amount of capital has a strong influence on the result (e.g. with arbitrage).

However, this is probably rarely the case for the majority of punters. Therefore, it is the next formula, profitability, which is the most important one for the normal bettor.

Image: Kristo-Gothard Hunor (Shutterstock)

When people talk about ‘value’ in connection with betting there are many different schools of thought.

This article therefore attempts to shed some light on this complex concept and its terminology…

Firstly, let us begin with a few synonyms for the word ‘value’: worth; usefulness; advantage; benefit; gain; profit; amount; rate; price…

You can see straight away that there is a plethora of different connotations.

The last two meanings in particular, ‘rate’ and ‘price’, are probably the reason why the expression ‘value’, when used in the context of ‘value betting’ is often unclear.

So, what is Value in relation to Sports Betting?

In order to really understand the concept of ‘value betting’, you need to grasp several connected aspects and I will tackle each of these individually for better comprehension.

Tip: Grab yourself a caffeine shot and read every section (re-read if necessary) until you are comfortable enough to proceed to the next. What I am about to say may initially be hard for some to digest, but don’t worry, you will certainly not be alone. Headache pills may be advisable for some…

(1) Betting Odds are solely the Market Prices for Bets

Anyone interested in sports betting knows that bookmakers offer bets on a multitude of match outcomes and sometimes the list of available bets on a single event can appear endless.

For example, the 1×2 market allows you to bet on a home win, a draw, or an away win, either at half- or full-time. Will there be more than two goals in the match or not (over or under 2.5 goals), will both teams score, or will the match end goal-less? Et cetera, et cetera…

Only very few bettors fully comprehend and understand the fact that betting odds are nothing more than the market prices of bets.

Betting Odds are Prices that Few Understand

Betting odds are not only all-inclusive prices for bets but to add to the confusion they are displayed, not in simple and easy understandable monetary units, but in different formats such as decimals, fractions, or moneyline.

When buying a bottle of beer, for example, you pay one Euro, Pound or Dollar and walk home with your purchase. When buying a bet, say, at decimal odds of 1.27 (27/100 Fractional Odds; -370 U.S. Money Line), the only thing you may understand is that if the bet wins you will receive your stake money back plus 27% winnings. Of course, if the bet loses all of the stake money is forfeited.

Judging ‘value for money’ from a bottle of beer is far easier than evaluating the same in a bet. Many bettors struggle to calculate the expected return of a bet or a series of bets let alone analyse why the big win they hoped for failed to materialise.

Even if sports betting odds appear to be genuine offers, the fact is that odds are frequently distorted.

Betting Odds cannot be priced by Gut Feeling or Intuition

Very few people are born with a sense of probabilities and ratios. Statistics and financial education are rarely taught in high school and touched upon in very few university courses.

To develop a ‘feeling for price’ is impossible, even for those of us who are born with a sense of probabilities.

To make matters worse, every bet is actually a different product, even if betting on just one particular type, for example, the “over 1.5 goals” market. Think about this bet in the context of a match between two evenly matched teams as opposed to, say, the team at the top of the table playing the weakest team in the league. The name of the bet is still the same but the two match situations are totally different.

There is no way but applying maths and statistics to decide whether the price of any bet offered is “too expensive” or “too cheap”. Gut feeling is misleading and may prove fatal. Probabilities are a subject too complex to succeed with guesswork, feeling and intuition alone.

For those of you who are really serious about understanding sports betting in depth and developing a value betting strategy, there really is no alternative to learning the calculations necessary.

In order to grasp the concept of value it is of uttermost importance to comprehend that betting odds are merely prices, and not ‘true odds’.

Before I address the term ‘value’ I must first plunge into probabilities… please stay afloat!

(2) The Broken Connection between Probabilities and Betting Odds

Once you have grasped that betting odds are nothing more than the price of a bet it should now be apparent that market odds cannot possibly “represent” the true probability (likelihood) of an outcome.

Image: Dragana Gerasimoski (Shutterstock)

It should be obvious that “making a book” is the business of bookmakers and they must ensure that the advantage is on their side. Common sense dictates that betting odds (price) can have little to do with true probabilities.

You must understand that bookmakers have no option but to manipulate odds in order to balance their books. They also construct odds to match public opinion, and use odds as a marketing tool to attract customers.

It is absolutely correct that betting odds can be converted into their implied probabilities. However, these probabilities seldom represent the actual ‘real’ chance that a particular outcome will occur. Though, the implied probabilities are very important for identifying ‘value’.

To reiterate: Betting odds are the prices of bets, not their probabilities!

Here, at this point, the huge advantage bookmakers hold over the uninitiated comes into focus:

• How many people do you know personally who were born with a good sense and acute awareness of probabilities?
• Do you actually know anyone who can demonstrate a sound training or exhibit a qualification in statistics and probability?
• Is this rare person actually interested in sports betting?

On the other hand, how many people do you know who collect money-off coupons, or chase after shopping discounts at summer or winter sales?

Do you wholeheartedly think that these people truly acquire decent quality with each bargain they come home with?

Or maybe they are tempted by these opportunities to spend more money than they actually had in their household budget, and may end up with a lot of things they didn’t really need, but which were “soooooo cheap” at the time?

I shall not delve any further into the mindsets of bettors or shoppers but I hope the analogy makes sense.

So, to summarise what I have discussed so far:

1. Betting odds are the price of a bet.
2. Betting odds can be converted into implied probabilities but these have hardly anything to do with the actual expected distribution of results.

Forgive me for preaching, but these are two vital elements you must fully internalise in order to have a chance of understanding the concept of value betting.

Now we finally get to the term ‘value’ of a bet… (And about time too, Soccerwidow!)

(3) The Term ‘Value’ in Relation to Sports Betting

When professionals speak of the ‘value’ of a bet, they are referring neither to the price of a bet (betting odds), nor to the probability of winning the particular bet. Value does not even mean betting odds for a particular bet being “too high” or “too low”.

The term ‘Value’ is used when a positive return can be expected. It is about the intrinsic value of a betting decision.

Please note that although this may be contradictory to common (non-entrepreneurial) sense, a purchase may hold ‘intrinsic value’ even if computations return a negative figure.

To compute value, formulas for the calculation of Profitability can be used.

It is possible to calculate the ‘mathematical advantage’ (value of a bet). The result is (nothing more than) a mathematical expression and technical declaration. The computed value estimates the expected return on investment. This information can then be used as guidance for betting decisions.

There are two different ‘Values’:

• Value I: Mathematical advantage of a bet. (Expected return = profitability)
  This ratio indicates if the offered price in the market seems too low or too high. It also computes how much investment increase can be expected if the same transaction (bet) is repeated over and over again.
• Value II: Relative deviation, being the ratio of the implied probability (market odds) relative to the mathematical probability expectation (true odds).
  This ‘value’ is used for further evaluation and statistical computations; for example, it is very helpful for correlation analysis, cluster grouping and portfolio construction.

Now comes the fun… Caution! It’s getting mathematical!

The Effects of a Break

After the break between seasons there are usually many changes observed in teams. The manager may have changed; the playing staff may have shuffled; the team could even be playing in a different league, etc.

Undoubtedly, three months’ ‘holiday’ must have an effect on super-fit professional footballers – After all, even millionaires are humans!

Just reflect on yourself…

• Are you usually in full working order when you return to work after a long break?
• How long does it take you to get back into your routines at work?
• How often have you experienced changes in staffing, organisational structure, etc., after you returned to work?

Unexpected Results and Deviations at the Start of Season

It does not matter which betting system or value betting approach is monitored, there are usually many matches observed at the start of a season finishing with unexpected results and not within statistical expectations.

It usually takes approximately six weeks to two months, until teams “settle” statistically and begin following averages from previous seasons.

Here is a screenshot from the Lay the Draw system explained in my article 1×2 Betting System: Analysis of HDA Data and Strategy Development – LAY THE DRAW…

Lay the Draw – 5 seasons – fixed stake
1×2 Betting System: Analysis of HDA Data and Strategy Development

The image above shows a deficit at the start of the 2010-11 season, which persisted until mid October.

The other four seasons finished with profits around the 100 unit mark, but to gain this level of profit 50 bets were needed, translating to an average loss/win of between minus two and plus two units per bet.

The remaining 120 bets (from mid October to mid May) produced profits between 300 and 500 units which averages a win of between 2.5 and 4.2 units per match. This is not only more efficient but also easier on the nerves.

To bet and win consistently depends on the portfolio structure and betting strategy. Some systems may stabilise after the first six weeks, others may need six rounds of home matches, for example. Each strategy needs to be assessed individually.

Bookmaker Primary Aim – Diversify Risk and Balance the Book

It may appear that bookmakers don’t seem to worry too much about the start of a new football season. They invariably publish their odds far in advance of the games.

Indeed, how would the market react if bookies offered no bets during the first few weeks of a season?

To counter the unpredictability of results and the deviation from statistical expectations at the start of a season bookmakers ensure they offer odds which balance their books even if it is more of a juggling act than usual.

In summary, to bet and win consistently it is necessary to have a reliable portfolio, synchronised with the most advantageous betting time for optimum performance. Fluctuations within a season need to be taken into consideration.

It is crucial to construct a football betting portfolio which is large enough to diversify risk.

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 contains 6 value bets including the calculated probabilities for each bet to win:

English Premier League - Example 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.

Accumulated Betting Odds

• To win all 6 picks: 15.9 (1 divided by 6.3%)
• To lose all 6 picks: 506.7 (1 divided by 0.1973%)

These odds express that on average all 6 selected bets should win once in every 16 rounds and only once every 507th round should a total loss of the portfolio occur.

A single season’s football league betting will usually comprise approximately 80 rounds of matches (midweek and weekend betting). This means that statistically a total loss may happen once every 6.3 years betting on a similar portfolio to the example above each time. Of course, it could happen more often as wins and losses have a nasty habit of not lining up as cleanly as statistical theory says they should. For example, 2 total losses could occur in the first 2.6 years and then no more for another 10 years.

What is the probability that exactly ‘X’ picks win or lose?

Further interesting questions include what are the probabilities that exactly 5 of the selected 6 picks win, or at least 4 of the picks win, and following this, it is natural to ask whether it is viable to make long-term profit on this type of portfolio and if so, how much?

An easy starting point for assessing whether a portfolio is ‘worthwhile’ is by calculating the ‘expectancy’, in other words, how many of the picks are likely to win. This is simply the average of the win probabilities of the selected picks:

This value means that by betting on the above portfolio a success rate of 63.55% is ‘expected’, which would correspond to a hit rate of 4 from 6 picks (i.e. 6 [picks] times 63.55% = 3.81 [roughly 4 picks]). This means that on average this portfolio should usually bring around 4 successful picks. However, it is obviously necessary to check if the combination of 4 successful picks and 2 failed ones will produce a profit:

Profit Calculation: Exactly 4 out of 6 Selections Win

The above illustration shows that every combination of 4 picks from our 6-match portfolio would have returned a profit of between 7.02 € and 16.71 € depending upon the combination.

Important Note

Please note that the average value (expectancy value) does not mean a 63.55% probability that exactly 4 picks will win every betting round. The average value indicates that if you bet on this type of 6-match portfolio often enough, an ‘average’ of 4 hits can be expected.

In other words, the maxim of gambler’s ruin is that if you play long enough you will eventually go bankrupt and have to quit the game prematurely.

Collage of Shutterstock images; Foreground: wacpan, Background: Lisa S.

The World of Sports Betting

The truth is that in the world of sports betting, the common gambler has far less money than a bookmaker or casino, and there will inevitably be a time when he will simply be unable to continue playing and, of course, the house will not be giving credit.

“Long enough” may be a very long time though. It mainly depends on how much money the gambler starts with, how much he bets, and the odds of the game. Even with better than even odds, the gambler will eventually go bankrupt. But, this may take a very long time indeed.

Please note that we are talking here about a “fair” game; e.g. each bet with zero value. The practice of bookmakers and betting sites to offer odds with an overround in their favour makes this outcome just much quicker.

Fair Coin Flipping

To make the dilemma of gambler’s ruin a little easier to understand imagine coin flipping with a friend. You each have a finite number of pennies (n₁ for yourself and n₂ for your friend).

Now, flip one of the pennies (either player). Each player has a 50% probability of winning (head or tail). If it’s a head you win a penny and if it’s a tail you surrender a penny to your friend. Repeat the process until one of you has all the pennies.

If this process is repeated indefinitely, the probability that one of you will eventually lose all his pennies is 100%. In fact, the chances P₁ and P₂ that players one and two, respectively, will be rendered penniless are:

Now let’s populate these equations with real numbers:

The above example is based on both players starting with the same amount of pennies (100 each). In other words, you and your friend have both an exact probability of 50% to end up with all of the pennies after many, many coin flips. This means that after an unknown number of coin flips either you or your friend will finish banking all the pennies. At the start, your chances are equal, and it is impossible to say who may win.

However, if one of you has many more pennies than the other, say you start with 100, and your friend with 10,000, then your chance of finishing with all of the pennies (yours as well as your friend’s) is as low as 1%, whilst your friend’s chances are 99% to win this unequal match.

Bankruptcy Probability Table – Gambler’s Ruin

To visualize the gambler’s ruin problem further, here is an overview of the probabilities of finishing with N amount of pennies.

Player 1 starts with 5 pennies. Player 2 has an infinite amount of pennies.

The top row shows the number of flips. The left hand column shows player 1’s current amount of money. The numbers in the table are probabilities (click on the image to enlarge; opens in a new tab):

Overview of the probabilities to finish with an N amount of pennies after X flips

Reading the table (examples):

After the first flip player 1 has a 50/50 chance of ending up either with 4 pennies (i.e. he lost the first coin flip), or with 6 pennies (i.e. he won the first coin flip).

In 10.4% of the trials player 1 will be broke (penniless) after the tenth flip of the coin. This means that every 10th experiment of this nature player 1 will have been forced to give up after the 10th flip of the coin due to a run of “bad luck” whilst player 2 is not affected by “bad luck” purely because he has plenty of coins to sit through and survive any such spell.

82.04% of the players will still be in the game after coin flip 15. However, 17.96% of the gamblers will already have retired due to exhausted funds.

Of course, you will now probably surmise that player 1 started with only 5 pennies, and by staking 1 penny each bet he was risking 20% of his starting bank on each coin flip, which is way too much. Player 1 should ideally have started with a much larger pile of pennies, and risked a far smaller percentage of his bank with each coin flip.

Anyway, eventually the same thing will always happen, albeit just much more slowly. Player 1 will still go broke sooner or later, if player 2 has an infinite amount of pennies. It’s just for the sake of the above table and illustration that we choose to show the calculations with a starting bank of 5 pennies only.

Go to the next page, to see some more examples and illustrations…

Introduction

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’.

Image: Daniilantiq (Shutterstock)

However, although often referred to, 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 the market’s expectations (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 not usually connected with the mindset of football bettors.

Probability & Expectation

To give you an example, 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.

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).

More examples… 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.

If you want to achieve a hit rate of over 80%, you must only bet on probabilities of 80% and above. These are back bets with odds below 1.25 and lays above 5.0.

If you prefer betting on odds between 1.8 and 2.2, a hit rate of around 50% can be expected and if you can achieve a hit rate of over 55% in this bracket, profit will be made.

At these odds if you expect to achieve an 80% hit rate from your own strategy or gut-feeling, or rely on a picking service to deliver, then you will be sorely disappointed as it is unrealistic.

A further example: If your strategy is to lay correct scores at odds between 7 and 12, then a hit rate of around 90% is realistic.

Hit Rate in Soccerwidow’s Context

Soccerwidow’s own régime relies on portfolio betting to spread risk and choose ‘value’ bets from the entire spectrum of probability groups (clusters).

Image: Kaonos (Shutterstock)

For example, up to the end of May 2012, the monthly evaluations of Soccerwidow’s match previews showed an average hit rate of 57% (Soccerwidow’s Match Preview Results: 6 Months, 207 Bets, Dec 2011 – May 2012).

However, as already mentioned, this number tells you nothing about the quality of the predictions, because to link these criteria would be comparing apples with pears.

Nevertheless, what it does say is that 57% of the recommended bets won. Nothing more. Again, this is not a statement of a good or a bad hit rate.

Soccerwidow’s recommendations encompassed all probability clusters (0-10%, 10-20%, etc.), and thus all possible expectations of concrete hit rates. Each cluster needs to be evaluated separately, since the average hit rate has no meaning whatsoever.

Remember: The hit rate, no matter how high, says absolutely nothing about potential and/or realised gains or losses. This leads us now to the concept of VALUE.

Value of a Bet

Betting profits can only be achieved in the long run if you ‘lay’ or ‘back’ at odds that do NOT exactly represent the average odds (expected outcome).

In other words, you should lay when the odds are lower than the expectation and back when the odds are higher.

To reiterate, a bet is called a Value Bet if the market’s back odds are higher than the expected odds, or the lay odds are lower than expected.

The ‘value’ of a bet is its mathematical advantage (‘edge’), being the expected profit from the betting transaction.

The concept of value is often mixed-up with the term hit rate. However, these two terms have no relation to each other.

Back to our example, Bayern Munich v. Chelsea: The chance (= probability, expectation) of a Bayern victory was 64.6%, which, translated into odds, is 1.55 (1 divided by 64.6%).

The betting odds in the market that day were 1.81, representing a probability of 55%. This meant that Bayern were 16.7% over-priced (1.81 divided by 1.55 minus 1), and therefore a back bet on Bayern held ‘value’ (= a mathematical advantage).

As it turned out, Bayern lost the match to the great disappointment of many German fans (Boo!), but to the professional gambler a loss such as this makes no difference because he/she knows that from 100 similar bets, ultimately 65 will win and 35 will lose. The profit from these transactions will be around 17% of the total capital employed (= Yield).

A professional bettor appreciates that not every bet will win. A 64.6% probability means a 64.6% expected hit rate; no more and no less! The value lies in the odds (price), not in the hit rate on the day.

Summary

The terms ‘hit rate’, ‘probability’, ‘likelihood’, ‘chance’ and ‘expectation’ are, more or less, synonymous. Ahead of the game one speaks of chance, probability, likelihood and expectation (future events), and after the match in the evaluation process it is then referred to as the hit rate (past events).

The value of a bet is its mathematical advantage or ‘edge’ over the market price (odds), and it is also the expected profit from the betting transactions. The term ‘value’ must not be confused with ‘hit rate’, ‘probability’ and ‘expectation’…

If you would like to learn how to calculate true probabilities and convert them into odds, identify cluster groups for betting, and understand value, then why not have a glance at Soccerwidow’s Fundamentals of Sports Betting Course).

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.

Image: 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,…)