How high should be a starting bank?
Is 5,000 units enough?

Well, there is no standard answer to this question. It all depends on the individual strategy.

Image: Sergey Novikov (Shutterstock)

However, what is possible, is to calculate bank fluctuations (i.e. winning and losing sequences).

With the help of knowing the best and worst case scenarios you can determine the ideal starting bank for any betting system of your choice.

At the end of the article you will find a few useful exercises to practise, with the solutions available as a free download to all of you who would like them.

Length of Winning and Losing Streaks

It stands to reason that the smaller the probability of an event occurring (i.e. higher odds), the longer the likely losing streak will be (in between winning bets).

However, the big question is how often and for how long will the losing (and winning) streaks transpire?

It is possible to mathematically calculate many things with statistics, including streaks of luck and bad luck. However, it is important to note that no matter how accurate the results may appear, they are ‘models’ (a formal representation of a theory).

In this article, we are talking about probabilities; what can we ‘predict’ about how things may develop in the future. Please bear in mind that any such hypothesis is always a “could happen” not a “will happen”.

Of course, the larger the sample size (i.e. number of bets), the more likely the prediction is to be correct. But apart from the bookmakers themselves, who else has a betting portfolio comprising thousands of bets every weekend?

Winning and Losing Streaks Formula

The longest expected losing streak (or winning streak) can be calculated using the following formula:

n = number of trials (i.e. total number of bets)
ln = natural logarithm¹
P = probability²
| .. | = absolute value or ‘modulus’

¹Suffice to say, explaining what natural logarithm is would be worthy of a series of articles. For the time being, use Excel to calculate this for you.

²For winning streak calculations use the positive value (i.e. the probability of winning). For losing streak calculations use the negative probability value. For example, if the probability to win the bet is 33% then the probability that the bet loses (negative probability) is 67%.

In practice, the formula is best applied to situations where you constantly bet repeatedly on the same probability, for example, on ‘red’ at the roulette wheel: its probability remains exactly the same with every new spin of the wheel.

For football betting the concept is much more difficult to apply as each bet is likely to have a different probability (e.g. one Over 2.5 Goals bet with a 55.3% chance, and the next with a 62.1% chance, etc.).

However, you can group bets in probability clusters – for example, bets with a 55%-60% expected hit rate, bets with a 60%-65% expected hit rate, and so on.

Longest Winning and Losing Streaks, depending on the number of bets (Examples for 50, 500 and 1,000 bets shown)

The tables above show the calculations of the expected maximum number of winning and losing streaks, depending on the expected hit rate (probability of the bet to win).

To read the tables, let’s explain the 70% line (odds in the region 1.4 and 1.45); in other words, bets with a 7 in 10 chance of winning.

The table on the left calculates the expectations of 50 tries (50 bets in a row, one after the next). You can see that the player will experience at least one streak of three lost bets in a row somewhere in the sequence.

On the other hand, he can expect at least one series of 11 winning bets in a row during the same sequence of 50 bets.

In contrast look at the 30% line (odds in the region of 3.2 to 3.4). In a series of 50 bets the bettor must expect at least one sequence of 11 consecutive losing bets, but will probably see only one set of three consecutive winning bets.

To develop a sense of probabilities and sequences, you can experiment with a dice. It has six faces; in other words, a probability of 16.67% (1 in 6 chance) of successfully landing on a chosen number.

Choose a number and count the number of throws until you succeed to roll it. Count also the number of consecutive successful rolls.

Having learned how to calculate the expected length of winning and losing streaks, the next question to ask is:

How many bets is it likely to take before I encounter ‘X’ losses in a row?

Timing of Winning and Losing Streaks

This formula is actually very simple:

= 1 divided by P, to the power of a

P = probability (expected hit rate or loss rate)
a = number of won or lost bets in a row

In the tables below you can see how many attempts (bets) it needs to experience a specific, expected length of luck or bad luck. Again, the assumption is that the bettor bets all the time on the same probability:

Expected time of occurrence of winning and losing streaks, depending on the hit rate

Looking firstly in the right-hand column at the Losing Sequences, if the expected hit rate is 45% (what you should ‘expect’ at odds of around 2.2), then it is likely that you will experience a sequence of three losing bets in a row by the time your sixth bet is settled.

After 20 such bets it is likely that you will have seen a losing streak as long as five bets in a row.

Looking at the Winning Sequences column: you will win three times in a row at some stage during a series of 11 bets.

However, winning five in a row may only be seen once in every 54 bets.

As we mentioned before, in football betting it is extremely difficult, if not impossible, to find bets, all with the same probability of success.

However, you should at least try to understand the theory behind winning and losing streaks, as it will be easier on your nerves when you do encounter the inevitable run of bad fortune.

In particular, a thorough understanding of losing streaks is of enormous importance when setting both the size of your starting bank and stakes per bet.

Example:

A bettor prefers bets within the odds range of 2.0 to 2.5 with a hit rate between 40% and 50%. He plans to place 50 bets (e.g. two bets per round on 25 rounds of matches).

After looking at the tables, he knows that the maximum losing sequence expected is likely to be as long as six to eight lost bets in a row. Therefore, he knows that there may be at least one sequence of three or four consecutive rounds (weekends) when all bets lose.

After every 5^th to 8^th bet, he is also aware that he is likely to experience a loss of three consecutive lost bets (e.g. one weekend loses both bets, the following weekend only one loses).

He also knows that every 13 to 32 bets there will even be a streak of five losing bets in a row.

The bettor is fully aware that he has to take this into consideration and plan the starting bank accordingly to be able to ‘sit through’ these losing streaks.

Of course, he also knows that winning sequences will arrive too. In his case, with some ‘luck’, he may experience a winning sequence of five bets in a row after 32 bets. Every eight to 16 bets he will have a ‘lucky’ streak of three wins in a row.

This is certainly quite a fluctuation. When these ‘bad luck’ and ‘good luck’ streaks actually happen, nobody knows. However, what we do know is: They will happen!

Starting Bank – Rule of Thumb

A starting bank should be approximately five times the maximum expected losing streak. The reason for this is that a losing streak can happen right at the beginning, immediately followed by another bad run of luck. We are talking statistics here!

So if a bettor wants to stake 10 units per bet, the starting bank must be nine times (expected losing streak) the stake of 10 units multiplied by five = 450 units. Then he can risk 2.2% of his bank each time he bets (10 divided by 450). If losing, the stakes will remain constant at 2.2% and, if winning, raised gradually.

Questions to ask before setting the starting bank:

1. What hit rate is expected (probability to win the bets)?
2. How many bets are planned for the season?
3. How long will the longest losing streak be?
4. What is the desired stake per bet?

Calculation of the starting bank:

Exercises: Losing & Winning Streaks

Answers to the Exercises

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Optimising Your Bankroll

The factor 5 used in this article to determine the betting bank is a risk variable for risk-averse bettors. It is also the factor advisable for strategies with a 45% to 55% win probability (odds between 1.8 and 2.2).

Here is another article: How to Calculate Losing Streaks & Optimal Bankroll in which we provide a more detailed account of setting the ideal starting bank.

Risk management in sports betting is the foundation stone upon which all of your betting transactions should be built.

Risk management encompasses risk assessment, risk control and capital requirements, all of which cannot be addressed until you understand how winning and losing streaks are likely to impact upon your starting bank.

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

If I had to use just one word to describe how bookmakers think…

Image: Cartoonresource (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.

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.

The bookmakers’ priority is balancing their books

The closer to the kick-off of a game, the more ‘fluid’ the odds become, as salient information such as team news becomes public knowledge, and this then has a knock-on effect with bettors’ opinions being confirmed or changed on the outcome of the match in question. Thus, the odds tend to change more as the start of the match gets nearer and nearer and more money changes hands.

Always remember

1. Bookmakers set odds based on a mixture of statistical probabilities and public opinion.
2. Bookmakers do not speculate (gamble). Their priority is balancing the books.

In an ideal world, bookmakers would like to see the same amount of money (risk) on both sides of a bet outcome. However, utopia is virtually unknown in the world of bookmaking and firms are rarely able to equalise their level of risk on both sides.

Therefore, you will often see a bookmaker adjusting his odds for an event over time. This fluidity aims to achieve an acceptable money line on both sides of the bet outcome.

Please note! Because it is rarely possible to “equalise” the risk on both sides, bookmakers instead look for an “acceptable” level of risk. This is the only ‘gamble’ bookmakers take.

How do Bookies Manage their Risk?

You will have certainly noticed the plethora of various betting offers used by the bookmakers to woo their customers. Unsurprisingly, these are the bets where they expect to make the highest profits (for example, pushing accumulator bets with offers such as, “If team A (usually a short priced favourite) is the one which lets down your five fold, we will return your stake!”) (how generous of them!!).

Bookmakers apply all kinds of marketing tricks to divert the sports bettor into a direction which is most profitable; for them but not for the bettors!

I risk repeating myself but the truth is that bookies’ odds never aim to predict an outcome of a match with utmost accuracy (therefore the calculated probabilities of ‘true’ odds often do not match the betting odds offered in the market). A bookmaker’s main goal is to balance the books and to do this, public opinion is taken into account.

This is the key to bookmaking success. This is the key to sports betting success.

Of course, each sport is different, but in the end bookmaking methods are always the same. Bookmakers make money with these same methods, regardless of the sport or other type of betting event.

• Their books are not perfect.
• They do not have a crystal ball.
• Bookmakers have a business plan!

The bookmakers’ mantra is very simple:

Calculate the statistical chances of the matches for a weekend and set the odds by taking into account the probabilities and public opinion. Collect enough money to pay off losing bets. Keep the profit.

Learn from the Bookmakers!

Bookmakers are not able to balance their books for each single game. To them, it is always about “acceptable” amounts of money (profits or losses) and spreading risk.

The goal of bookmakers is not to predict the outcome of a game correctly. This means that their odds often do not reflect the expected probability distribution.

Bookmakers’ odds usually reflect public opinion about a match and their primary objective is to ensure a well balanced book.

If you wish to become successful with any form of betting you must understand the way of thinking (the business plan) of the bookmakers.

Why? Because these firms survive and thrive from the money they encourage you to lose through nothing more than your own ignorance of how their ‘system’ works.

What is Bankroll Management?

Bankroll management is one of the most important pillars for success in sports betting.

Image: Alex Roz

A portfolio of sports bets placed over time can be compared to investing in the money markets on a portfolio of stocks and shares.

Indeed, the term ‘bankroll management’ comes from the financial sector and describes the use of the seed capital (i.e. in betting terminology, the initial stake).

Bankroll is the ‘starting bank’, and the intention is to manage it and increase it at the same time.

Bankroll management therefore deals with how to properly manage your starting bank.

The Continual Importance of Statistics, and Lots of Them!

The good news: It is actually possible to calculate the required starting bank mathematically.

The bad news: The calculations are naturally dependent upon statistics, and the ‘significance’ of the results relies on the amount of data used.

For example, any strategy based on one German Bundesliga team’s home games during a season produces a sample of precisely 17 sets of data, which is a very small number, statistically speaking.

The Law of Large Numbers is omnipresent so far as statistical accuracy is concerned: The larger the data sample, the more accurate the final results are likely to be, although a line has to be drawn between sample size and an acceptable level of error.

One way of coping with small data sets is to incorporate a risk discount into the equation. More about this later…

What does ‘Optimal’ Really Mean?

On face value, you might assume that calculating the necessary starting bank for a betting strategy can be derived solely from the stake multiplied by the number of bets (n).

With the 17 matches from our example above, and a constant stake of 100 units per bet, the bank would then be: 100 x 17 = 1700 units. But is this maximal amount really needed?

Although this may be true where returns from winning bets cannot be immediately re-invested, such a bank can never be optimal because an inordinate amount of capital would be tied-up.

What you should look for is the most cost-effective bankroll where all the money you have at your disposal is working for you as efficiently as possible.

Optimal bankroll is characterized by two things:

• Cash holdings (i.e. money in reserve) is kept as low as possible
• Gambler’s ruin is avoided

Calculating the Optimal Bankroll

There are five vital criteria you will need to establish:

1. What is the size of your stake per bet?
2. How many bets does your strategy expect to be placed?
3. What is the expected hit rate of your strategy?
4. What is its expected longest losing streak?
5. Determine the risk variables and incorporate a ‘risk coefficient’.

Example Calculation

Okay, we will stick with the German Bundesliga for demonstration purposes and use a system gleaned from its latest full-time 1×2 HDAFU Simulation Table.

If you have already bought this table, you can see the full and detailed analysis of backing the underdog whenever Hamburg plays at home: This strategy has realised a yield in excess of 58% over the course of five complete seasons from 2010-11 to 2014-15.

In addition, there has been profit produced in every one of those same five seasons.

It’s an ideal candidate for incorporating into a large portfolio of other systems. (When we say ‘large’ we mean a portfolio that will generate at least 500 bets in a season.)

Remember the five criteria:

(1) Size of Stake per Bet:

This is determined by your own liquidity, and to keep this calculation simple, a Constant Stake (CS) of 100 units per bet will be used.

(2) Number of Bets:

For this mini portfolio of Hamburg home games, the Number of Bets (n) is 17 for the new season.

(3) Hit Rate:

The HDAFU Simulation Table reveals that from 85 Hamburg home games over five seasons, 32 underdogs triumphed: a Hit Rate of 38%.

The random selection of only 85 matches is a relatively small sample and the possibility of ‘random sample error’ is therefore relatively large.

To compensate, it is worth applying what is known as a ‘risk discount’ to reduce the actual hit rate experienced and to build-in an extra level of security if statistical expectations for the new season are not realised.

Taking a risk discount figure of 5%, the expected hit rate becomes: 38% – 5% = 33%.

[Have a look at this article for more information about hit rates].

(4) Longest Losing Streak Expected (LLSe):

The longest expected losing streak (or winning streak) can be calculated using the following formula:

n = number of trials (i.e. total number of bets)
ln = natural logarithm*
P = (negative) probability^†
| .. | = absolute value or ‘modulus’ (see Wikipedia if you would like to know more about these mathematical symbols)

*Suffice to say, explaining what natural logarithm is would be worthy of a series of articles. For the time being, use Excel to calculate this for you: to make life easy, the formulas to use are included in the free spreadsheet download below.

^†For this calculation, the negative probability or hit rate is used. In this case, having adjusted our hit rate down to 33% using a risk discount, the probability that the bet loses (negative probability) is 67%.

From a pool of 17 bets, you can therefore statistically expect that a maximum of seven in a row may be lost without winning one in between.

(5) Risk Coefficient (RC):

The determination of risk variables depends primarily on your risk aversion. Risk-averse bettors choose a high coefficient figure (e.g. 5), whilst gamblers who are happier taking risks choose lower coefficients (e.g. 2).

But why are we including a risk coefficient at all?

We can assume that the longest expected losing streak (in our example, seven lost bets in a row), may already start with the first bet.

Although one bet may win after that, with the gains reimbursing the loss and allowing for reinvestment, there can still be a second stroke of bad luck directly after the first bet that you have won.

Neither winning bets nor losing bets ever line up in a uniform manner; they will always appear in a random pattern, so always better to be safe than sorry.

Optimal Bankroll Formula

The formula is:

Our Bundesliga example is an underdog backing system, which by its very nature, is risky. However, as there are only a maximum of 17 bets in this mini system, we will choose a risk coefficient of 1.5: we are happy to take the risks!

It is not very likely that there will be two losing streaks of seven games in a row when betting 17 consecutive times. However, we are aware that it may be quite challenging for the nerves to sit through losing streaks watching the bank balance reduce before your eyes!

The optimal bankroll required to run this system for a season is as follows:

If you remember the sub-optimal bank strategy at the beginning of the article where we touched on a bankroll of 1,700 units (100 units per bet x 17), you can see we have now released 650 units for investing in another strategy elsewhere.

Calculate Your Own Longest Expected Losing Streaks & Optimal Bankroll!

With this free Excel table download, you can easily and quickly discover what the longest losing streaks are for your own strategies. Just enter your stake, number of bets, and risk coefficient figures and let it calculate everything for you!

>>> Excel Workbook – Losing Streaks <<<

 
 Click on the above button – in the new tab click on the ‘Continue Checkout’ button. Enter your name and email address to allow our automatic shopping cart to deliver the file by email to you, free of charge. The .xls file size is 93 KB. When you receive your confirmation email, just click on ‘View Purchase Online’ (in the email text) to download the file.

Today’s article discusses the question what would have happened when backing the underdog playing away from home in the German Bundesliga?

Such a match was played in this league on 23/05/2015 between Moenchengladbach and Augsburg. The best bookmaker odds for the full-time 1×2 market at kick-off were: 1.57 Home; 5.00 Draw; 7.30 Away.

Moenchengladbach were the clear favourites at 1.57; Augsburg the rank outsiders. However, the men of Augsburg won the game, 1-3, defying their long odds.

How regular do such things occur? Is it profitable to bet on outsiders?

Here’s a screenshot from the ‘Backing by Odds’ tab in the simulation table for this league:

German Bundesliga – ‘Backing by Odds’ tab – Five Seasons 2010-15

In the table above you can see that from a total of 306 matches during 2014-15, the away team won 79 times. (Click on the table to enlarge it in a new browser tab).

79 of 306 is 25.8%, and this percentage shows that the away team won, on average, slightly better than once every four matches.

Profit and Loss Sectors when Betting on the Away Team

Looking at the profit/loss (P/L) summaries in the ‘Totals’ column, adding together the first six rows of odds clusters produces a loss of -2,564 units, based on a flat stake of 100 units per bet.

Essentially this means if the away team was priced as a clear favourite or close to the home team’s prices, they won less frequently than the probabilities indicated by their odds. The last of these first six cluster groups closes at away odds of 2.90.

Look at the second row of the table. The odds cluster between 1.66 (implied probability 60.2%) and 2.00 (implied probability 50%) contains 83 matches and, if the odds had been ‘fair’, 55.1% (60.2% + 50% / 2) of the away teams priced in this group should have won.

As you can see, this was not the case! Of 83 games in five seasons only 43 were away wins (51.8%).

Therefore, punters who regularly backed away favourites in the Bundesliga during 2010-15 surrendered ‘value’ in their bets to the bookmakers. When this happens, only one side of the deal wins in the long-run; invariably it isn’t the bettors!

Okay, let’s take a look at the away underdogs…

German Bundesliga – ‘Inflection Points’ tab – Five Seasons 2010-15

This screenshot shows a steep rising curve starting at odds of 4.40 and continuing until odds of 17.0.

Over five seasons, 462 matches fell into this group (Moenchengladbach vs. Augsburg being one of them). The away underdog won 88 times = 19% hit rate!

In these odds clusters the away team won, on average, once in every five matches. The average betting odds were 6.40, representing a probability of 15.6%.

The curve shows, as well as the calculations (19%/15.6% = 121.7%), that the mathematical advantage was on the side of the gambler!

The P/L curve registered 653 units profit at the start of our selected segment and finished at 13,727 units. This is a difference of 13,074 units of profit located solely within the away odds cluster group from 4.40 to 17.0.

Why does this advantage exist? How does it happen?

Backing Low Odds Favourites – Downfall of any Betting System

Most bettors prefer betting on the more popular and ‘emotionally safer’ shorter-priced favourites, but please ask yourself the following two questions:

• How does a profit-oriented company (i.e. bookmaker) set its prices?
• Should the prices (odds) for favourites rise or drop?

Both common sense and business acumen prevail in this situation:

The market dynamics are the following: The more bets expected to be placed on a particular outcome, the more bookmakers reduce their odds. Reducing odds mean that the bettor must risk more money (stake more) to achieve the same financial outcome. The punter therefore pays a ‘higher price’ (gets lower odds) for the same product:

Odds 2.0 → stake 50 = win 50
Odds 1.5 → stake 100 = win 50
Odds 1.25 → stake 200 = win 50

Falling odds means:

⇒ Rising stakes
⇒ Potential to lose more money
⇒ Lower percentage returns should the bet win!

Although this relationship may seem paradoxical, falling odds means rising prices!

Bookies adjust Favourite & Underdog Odds to Public Expectations

To reiterate: Falling odds for an outcome is a clear indicator that this is a favourite. Warning! Dropping odds do not indicate that the statistical probability for the favourite winning the game is improving; purely the fact that the outcome is becoming more and more favoured by bettors. This is a betting fundamental, which many gamblers are totally unaware of.

Falling odds mean bookmakers are effectively raising the price for the product! The product itself does not change in the slightest (i.e. betting on the favourite), but it becomes more expensive to buy. The bettor has to risk more money in order to win the same amount. In this case, you do not get ‘more for your money’, but considerably less!

Let’s use a different example. A confectionery company launches a new chocolate bar, which becomes an instant success. Demand increases; the company naturally takes advantage of the situation by raising the price. You can certainly make the statement that if the price of the chocolate increases it is a ‘favourite’, but the product itself never changes – it’s still a 100g chocolate bar!

The last word here is that since the books have to be ‘balanced’ (i.e. the payout of all three 1×2 bets combined needs to add up to around 100%), whilst the ‘prices’ for favourites are lowered to take advantage of the demand, on the opposite side, the odds for the underdogs rise.

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,…)

Since the growth of online poker caused the average IQ of poker players to increase dramatically it has become even more important to understand the numbers behind the game.

Indeed, while it was once possible for players to make money with a touch of experience and a stoic demeanour, it’s now the statistically minded that rake in vast sums of money at the table.

So, does this mean you need to be like John Nash (the famous math professor portrayed by Russell Crowe in A Beautiful Mind) to make money? Fortunately, the answer is “no” because online tools such as the CardsChat poker odds calculator are on hand to unravel the poker’s numerical matrix.

Why Do You Need to Use a Poker Odds Calculator?

Although it can be an advantage to know how to perform complex equations without the aid of a calculator, it’s certainly not a necessity at the poker table.

In fact, given the time you have to make decisions at the felt it’s virtually impossible to accurately work out the exact odds in each scenario. Thankfully, however, the ability to use calculators and other poker tools can help you build up a general appreciation of the numbers involved in various situations.

In the same way you have a kinaesthetic awareness of where your limbs are without having to look at them, using a poker odds calculator often enough allows you to know the right move without having to look at the numbers.

Breaking Down the Numbers

With that said let’s break down the basics of the CardsChat calculator and then put it to use in a selection of scenarios.

The main purpose of the calculator is to work out how often a particular hand will win, tie or lose against another, and with this particular piece of software you can compare up to six hands at once.

Once you’ve set the number of hands you want to assess (you must choose at least two) you need to highlight the empty boxes and choose the two cards that will make up each hand.

After setting each player’s hand you can hit the “calculate odds” button and the software will tell you how the hands stack-up against each other pre-flop.

Example 1: Big Slick

Looking at the scenario in the image below you’ll see that when comparing Ad Kd against Jd 9c the former will be a 65.73% favorite if all the money went in before the flop, and the five community cards were laid out.

Beyond the ability to analyze pre-flop situations you can also assess the winning/losing ratios of hands on the flop and turn. The reason this is useful is because the majority of Hold’em hands don’t start and end before the flop.

Thus, it’s important to understand how the changing dynamics of the board affect will alter the win/ lose percentages.

Image: Sergey Novikov (Shutterstock)

This requirement does seem to divide opinions, and our betting friends find it difficult to see that results of games from the past 10 years have any significance at all when predicting the outcome of the next game.

Categorically speaking, their thoughts are not entirely wrong – A game that was played, for example, six years ago, cannot have any real influence on the result of an upcoming game.

However, that’s not the point when determining betting odds and identifying value!

Betting success is not about predicting a result of any forthcoming game with any great certainty; it is about calculating probabilities and distributions, and then getting the right prices for bets.

The sports bettor always needs to know the answer for one question, whether the offered market price [odds] for the bet holds value, or not.

Each betting decision must be based on the expected probability, not on gut feelings, personal assessments, or hopes. We present the following classic example…

Heart of Midlothian vs. Aberdeen on 24.08.2013 – Scottish Premier League

We processed this match with the Value Bet Detector and found that Hearts had a 54% chance of winning [‘True’ odds column], which translates into odds of 1.85 [one divided by 54%]:

Hearts vs Aberdeen – Scottish Premier League match 24.08.2013 – True Odds Calculation 1×2

Sure, a 54% chance never means guaranteed victory, but in comparison, Aberdeen’s chance of winning was calculated at just 26%. It was immediately apparent that Hearts had more than twice the statistical likelihood of winning than Aberdeen.

However, these statistical calculations contradicted the emotional expectations of the public. The sports press at the time was full of information that Hearts were experiencing financial instability and had lost many good players before the start of the season, whilst Aberdeen were developing more positively and had been celebrating their comeback.

In addition, Hearts finished 10th last season [2012-13], two places behind Aberdeen, in a league of only 12 teams.

Although none of this information was of any statistical value for determining probabilities and odds, it was of immense importance to the supporters of both clubs, and certainly influenced punters’ betting decisions.

Bookmakers Match their Betting Odds to Public Opinion!

We now come to the approach of bookmakers… Remember, the primary aim of bookmakers is to balance their books and produce a portfolio that generates profits for them every week. Bookmakers set their odds to ensure betting action on both sides of an event in order to win, regardless of the result.

In this example, bookmakers were neither able to offer high odds for Aberdeen to even remotely reflect their low statistical chance of winning; nor could they offer low odds for Hearts, who were not favoured by the public – The sports world, comprised mostly of non-statisticians, would never have understood.

Therefore, the bookmakers simply reversed the probabilities corresponding to public opinion and expectations:

Instead of following the true probabilities and offering odds for Hearts in the region of 54% probability (i.e. 1.85), bookmakers chose to offer odds of 3.90, as show in the Value Bet Detector screenshot.

For Aberdeen it was exactly the opposite. Instead of the real 26% ‘true’ probability of winning the match (approximately once every four meetings), bookmakers priced their odds at 2.24, representing a 45% chance to win (nearly once every two matches).

The game ended in a 2-1 home win for Hearts. If you had looked up the H2H history of these two teams from the last 10 years, you would have noticed that something was amiss with the market odds, because Hearts won 60% of these games against Aberdeen, not just 26%.

I hope that it is now a little clearer why the Value Bet Detector for League Games with H2H History factors in the H2H’s between both teams in order to calculate the true odds.

On a similar note, there is another article, written some time ago (indeed, it was one of my first articles), where I analysed a weekend of matches in February 2011 and dispelled the conspiracy theories which were used to explain the observed deviation from a fan’s expectations.

It really isn’t easy to convince a committed football fan about the role and importance of statistics when it comes to pure enjoyment of the game, or betting on the matches!