The Mathematics of Gambling

The key to all gambling is forming an opinion and understanding how that relates to the bookmakers price.  You must base your opinion in research because myth pervades the media and other people.  To be a good gambler you need to question perceived wisdom and form your own line of knowledge.  Your ability to do this dictates your profitability.  Thus, opinion may be better expressed as knowledge which only really comes from research and judgement.

This page attempts to teach you how to take the probability of a sporting event happening and then see if the odds offered are value or not.

A good bet should never be struck solely on probability, it should be taken as a combination of probability and price.  If something is likely to happen less often than the price suggests, then clearly you would not want to back it.  This is just as true of a 1/4 shot as a 10/1 shot.

To be successful, you need to get an advantage on the bookmakers odds.  To do this you have to mercilessly seek out advantageous knowledge that allows you to make better judgements as to where you think pricing is wrong in a bookmakers market. 

My advice is always to price your own market up based on your research before considering the bookmakers prices.  This ensures that you do not follow the crowd and perceived logic but instead compare what you actually think to the prices available.  This process will condition you to using pricing and weighing up probability.  

Many gamblers act on weak premise, they might look for a price that gives a return that is acceptable to them on an outcome that they conceive as possible, without consideration to what the price represents. 

It is perverse to start with a price and then see how you feel, first you need to understand what you think the price should be and then seek better, so the advantage is with your knowledge and opinion. 

The Saturday Lesson – The Mathematics of Bookmaking

Reading from the Major Chapter 3, versus 12 -18….

To calculate your price, write down the possible outcomes that could occur in an event (3 in a football game) and then write down all of the factors you think should influence that outcome.  This is the research stage.

For football as a minimum you should consider injuries, form, squad quality and home/away form, but add in any factors you like.  

Next you need to calculate the probability of each outcome, as you see it.  The easiest way to do this is to allocate points out of a hundred as to how likely you see an outcome.  I often find a useful question at this point is ‘If this game were to play a hundred times, how often would I expect each outcome’.

Here is an example.

If I consider the Chelsea v Sunderland today, I am taking into account Chelsea’s squad troubles (I do not think all is that well in the camp), Chelsea while having a much stronger squad have won only half of their last ten home games.

Sunderland have now managed five wins in seven, revitalised under a new manager.  Chelsea are without Drogba and Kalou who go to Africa while Malouda is unlikely.  They will have Terry and Sturridge.  Sunderland are missing Brown.

On balance I score it Chelsea 50 Draw 30 Sunderland 20.

How to Calculate Odds from Probability

So how to calculate odds?  Well, you need to use the formula that the Major has adapted.  Take the probability that you worked out (expressed as a number of occurrences from one hundred) and then apply this formula..

 Odds = 100-(probability/(1-probability))

So for example, I think Chelsea will win, in todays conditions, 50 from one hundred times thus 100-(50/(1-50) = 1 or expressed as a more familiar bookmaking odd evens. 

To show this works in an easy context, consider the same market with all outcomes equally likely.  Therefore your probability (from 100) is 33.3.  In my calculator, the odds of each outcome is 2 commonly expressed as 2/1.  Thus a pound on each outcome would always get your money back in that market. 

Crucially, a bookmaker will not price this way, they need to make a profit.  Thus, they will have an extra step which will be to reduce the number by around 10-25%.  This is called their overround and is designed to ensure they make a profit.  Instead of offering 2/1 in my example, he might offer 7/4, slightly shorter on all three outcomes.  This is why it is always hard to find genuine good value in any punt.

It is useful to be able to calculate how high the overround is, it shows how greedy the market is!

Calculating Probability from Odds

The formula to calculate the probability back from the bookmaker odds then first convert the odds into a probability.  To do this, take the odds and apply the following denominator/(numerator+denominator)… so 2/1 becomes 1/3 or 33% and 4/5 becomes 5/9 or 55.6%.  Then add them together and subtract a hundred

So, in our case of three 2/1 shots, each priced now at 7/4, then we need to add three lots of 4/11, which is 36.4%, thus three times that minus a hundred is 9.1%.  To check that works, consider placing a pound on all three outcomes.  Your return is £2.75 and therefore you lost 25p from £3.  25/300 = 8%.  This is slightly short of the overround which calculates on a round basis so for example, to get £3 from 7/4 you need to stake £1.0909.. 


Calculating Probability for a First Scorer

I want to use this very simple example to show how you might apply multiple factors to reach the likelihood of an outcome.  I am going to use the football first goalscorer market for an Arsenal home game against mid table opposition.  I want to know if backing Van Persie to score first at 7/2 represents good value.

First we need to work out the probability of him scoring first, based on our own judgement.

This is a staged approach.  First we need to assess the chances of there being a goal at all.  In my game I am going to say that there would be a goal in 90% of these types of game.

Then I want to know what chance there is of Arsenal scoring it first.  The question is whether if there is a goal, Arsenal would score first, so again the % is from the full 100% – We are essentially saying, there is a goal, what chance it was scored by Arsenal.  I am going to say that chance is 70% against mid table opposition.

Finally, you need to break the Arsenal team down and ask from 100% how likely is it that the goal would be scored by each player.  You can tinker with the percentages until you are happy.  In my case I am saying that there is a 30% chance that if Arsenal scored the first goal, it would be Van Persie who gets it.

Then multiply the percentages together to reach you probability.  30% of 70% of 90% = 18.9% – This is the chance I give to Van Persie netting that first goal. 

Using the formula above, you can see that works out at a fair price of 5.3 or between 4/1 and 9/2.  Thus the market price of 7/2 is no good for me, in this example.

Calculating Value – Easy and Memorable

It is not always that you will have a spreadsheet or pen, paper and calculator to hand so I am going to give you one very easy formula to remember that can help cut through the haze and deliver an incisive view on value.

Value = (odds x probability)/100

Express  odds as a decimal and probability as a number from 1 to 100.  If the number exceeds 1 then it is a value bet, simple.

Again the probability is you own thinking so, for example, the odds for Villa to win at home are 6/4, or 2.5 (expressed as decimal odds) multiply that by your probability, mine is 50 ( In circumstances where I would believe they would win half of these encounters) so 2.5*50 = 125.  Divided by 100 = 1.25 A VALUE BET.

My Chelsea example, if priced at 2/5 would be… odds 1.4 * probability 50 = 70 DEFINITELY NOT A VALUE BET.

So if anything can be drawn from this mathematical look at odds remember to convert odds to decimals, multiply by the possibilities of the event occurring in a hundred and divide by a hundred.  Over 1, get stuck in like a lunatic, below 1, leave it for the sheep.

There, you can now price your own markets and talk more knowledgeably about why you want to back an outcome.  If someone tells you something is a ‘good price’, ask them in their view how many times from a hundred they think the outcome would occur.  Then calculate the value based on the odds and their probability and see if they are right!

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