Odds, Statistical Probability and Gambling.
by Doug Robb (doug@ausrace.com)
This article will introduce you to some important concepts but without dwelling too much on the mathematical rigour.
What has statistical probability got to do with gambling? In a word, everything. If gamblers had even a modest understanding of probability then the casino's of the world would all be empty. In this article I will explore the basic statistical probability you will need to know before you start betting either on the horses or other forms of gambling.
Introduction - Odds ain't Odds
Most gamblers are comfortable with the concept of odds because we are very interested in what the pay-out will be for any particular wager. Many people however, fail to realise that odds are really a measure of probability and what we should really be interested in is if the odds correctly represents the statistical probability of the outcome we are about to invest in.
The words probability and odds are often used interchangeably since 'odds' is the language spoken by gamblers but always remember that when you say odds you mean probability!
To demonstrate what I mean when I say 'odds ain’t odds' consider a coin toss. Assuming the coin is without any manufacturing faults we all know that if it is tossed thousands of times the number of heads tossed will be about the same as the number of tails. The probability of heads is equal to the probability of tails.
In betting terms this is an even money bet, or a ratio of heads to tails of 1:1 and so the odds are 1/1. These odds are also called the 'true odds' because the pay-out represented by these odds correspond to the actual probability of the event happening. As a percentage the probability of tossing a head is 50%.
Therefore if you win $5 when a head is tossed and lose $5 when a tail is tossed then at best you should only hope to break even in the long run. Along the way you might get ahead for a while or get behind for a while but over time you expect to break even.
Is there a way to make money from this seemingly pointless bet? Suppose you found someone who was prepared to accept less than $5 (even money) for a correct call of the coin toss? In other words find a player who will accept a pay-out of $4 and not the $5 the 'true odds' of the bet would indicate. Of course if the player gets it wrong, you keep the full stake of $5!
Instead of breaking even over thousands of tosses you will steadily send the other player bankrupt because what you are really doing is pocketing 20 percent ($1 out of $5) of the other players money every time he wins. The longer the player bets the more money he must lose.
In betting terms you are offering odds of 5/4 ON (referred to as 'odds on') when, as you know, the 'true odds' is even money. In racing terms the punter is 'taking under the odds'. The odds offered are called the betting or gambling odds. Only the true odds represent the statistical probability of the outcome you are investing in.
The true odds are fixed for any particular bet but you can (and will) be offered any odds at all. The only predictable relationship between statistical probability and gambling odds in general is that any sensible gambler will try to offer you odds that are below the true odds dictated by statistical probability.
This is very important so one more time now and say it after me. The true odds are fixed for any particular bet but you can, and will, be offered any odds at all.
Who would be silly enough to take a bet that doesn't at least pay out the true odds you may ask? Well just wander into a casino and watch those hapless souls donating their money to the casino owners! When was the last time you brought a lotto ticket? The short answer is that we all do at one time or another. A more appropriate question however may be to ask why so many people go through their whole life betting under the odds and not know it?
In my opinion it's a scandal that in casino's people are playing games they simply cannot win, the longer they play the more they MUST lose. It is literally a licence to steal money from people unaware of the mathematical futility of their endeavour.
For you, the savvy punter make sure you know and understand the difference between ‘true odds’ and 'betting under the odds', study a few casino games if you still think you can win at the casino. If you must go to the casino then only play BackJack as this is usually the game where the house has the least advantage.
So how do we make our money?
In a casino the odds are fixed and you can either take the payout offered or have a cup of coffee or otherwise engage yourself. In horse racing however for many reasons the odds are fluctuating over the course of betting. Many of these fluctuations are totally unrelated to the statistical probability of the horse’s winning chances.
This gives you the opportunity to secure your wager for a return better than the 'true odds' would indicate.
For example continuing with the coin toss for a moment, what if someone offered us a return of $6.25 on heads for a $5 stake? A wager like this is called betting 'over the odds', an 'over' or an 'overlay'.
If you can put yourself in this position then you will win, the longer you play the more you will win. In betting terms you are getting odds of 5/4 for an event with 'true odds' of even money, or 1/1 if you prefer. The other punter is paying you a bonus of 25 percent on your stake every time you win.
Study the example I have used until you know the difference between getting 'over the odds', 'under the odds' and 'true odds' because this is the single key to the success or otherwise of your betting future.
I don’t want to introduce too many new ideas at this stage but I should point out that the 25 percent ‘bonus’ in my example is not to be confused to the percentages that punters talk about in the context of probability. My 25 percent was just a calculation based on the stake money I used ($5) and the amount of money that I would win ($6.25) . The ‘bonus’ is just $1.25/$5 or 25% of the stake money.
If you were to consider my example in terms of percentages related to probability then what is happening is that for an even money bet you expect to win 50 percent of the time. For a bet of 5/4 you expect to win 44 percent of the time and for a bet of 4/5 (or 5/4 ON if you prefer) you expect to win 56 percent of the time. So the fluctuations in terms of probability between these bets and even money is only 6 percent.
If someone is offering betting odds of 5/4 then you need to win 44 percent of the time (or 44 tosses in 100) just to break even in the long run. With a fair coin toss I expect to win 50 tosses out of 100. This is the simple reason that I expect to win over a period of time and once you understand this concept you will never play another casino game again, ever.
If you don't feel comfortable talking in terms of odds and percentages just yet the important point to grasp is that if the pay-out when you win is less than the true odds would indicate then you will never win the game and the longer you play the more you will lose. Sure you may get ‘lucky’ and get ahead for a while but in the long run you will lose.
The probability of winning in my example is the SAME for both players but if the pay-out can be appropriately manipulated by one player then the other will lose money over a period of time.
How does this apply to horse racing?
Most people think that horse racing is about picking winners. Indeed I used to say to my percentage punting friends "you won’t go broke backing winners" and didn’t pay too much attention to the odds simply because I took the view that a winner is a winner at any price. However the flaw in my logic is that ultimately there are no good things on the race track and so the odds you take for your winners are just as important in racing as in the coin tossing example. In the long run if a bookmaker can get you to take 2/1 about a horse that should be 3/1 then he will beat you.
Eventually I realised what these ‘percentage’ players were on about. A favourite saying of these punters is "good things come and go but percentages go on forever" or another, "you can’t beat a race but you can beat the races". I interpret these statements to mean that when a horse wins it can be seen as a random event from race to race but with a probability that can be measured over many races and hence as a percentage over a period of time.
It doesn’t really matter if your next bet gets up (just as in the coin toss) as long as over a period of time you bet to percentages in your favour. If you plan to bet over hundreds of races then you must use a system that is designed to win over hundreds of races and certainly not rely on putting large amounts on this weeks ‘good thing’.
Remember even horses that start 'odds on' get beaten so there is no such thing as a certainty on the race track.
Obviously you want to back the horses with the highest probability of winning and maximise your 'strike rate' but only at better return (or odds) than the ‘true odds’ would indicate. The art of gambling on the racetrack is not simply picking winners but in being able to determine the true odds for a particular horse in a race. This of course raises the question of how this can be done - a coin toss or a roulette wheel outcome is easy - but a horse race?
Well the answer is we can't, not exactly anyway but many astute punters can analyse form to the extent of getting a good approximation of the winning probability for each horse in a race. Indeed the final starting price of winning horses is often reasonably close to their 'true odds' but unfortunately at that stage of betting a punter has usually missed the chance to get the best odds that were available during the course of betting. How good punters are able to work out these approximations is a topic for another day.
Some Background Maths - Probability of a Single Event.
Calculating probability can be simple or quite difficult depending on the situation. In the simple case you only need to work out two things - how many outcomes are possible and which of these outcomes are successful for the wager you are making. To calculate the probability of success you simply divide the total number of successful outcomes by the total number of possible outcomes.
So if an outcome has ‘n’ ways of occurring and only one outcome counts as a success then the probability of you having a win is simply:
Probability of Success = 1/n
A probability of one means that an event is certain to happen while a probability of zero means the event is certain not to happen. There are a couple of useful rules like:
Probability of Success + Probability of Failure = 1
(or in words it is certain that the event will either occur or not occur, agree?)
and so once you know either the probability of success or failure you can work the other out using the formula:
Probability of Success = 1 - Probability of Failure
Probability of Failure = 1 - Probability of Success
As an example lets work out the probability of drawing the ace of spades from a pack of cards. The total number of outcomes possible, ‘n’, is 52, since there are 52 cards in a pack. There is only one successful outcome so the probability is:
Probability of Success = 1/n =
1/52 = .019
or approximately 2 percent.
Thinking in terms of percentages is often useful. You would only expect a horse with this probability to win two in every hundred races and it would not be unusual for a run of several hundred races before a win. A long time between drinks don't you think?
Converting Odds to Probability
Now let's solve one of the great mysteries for many a punter, converting odds to probability. But before we do a word about odds. Odds are simply the ratio of the losing outcomes (or chances) to the winning outcomes.
Bookmakers express odds as odds against winning. So a 10/1 bet indicates ten chances of losing and one chance of winning, as a ratio this is 10:1. A 6/4 bet would have six chances of losing and four of winning and of course an even money bet has one chance of winning and one chance of losing.
Since you will generally only see the bookmakers style notation I will use this in expressing odds but never forget that odds are really a ratio and should be expressed as 10:1, 6:4, 1:1 on so on.
I can only assume that bookmakers replaced the ‘:’ (colon) with a forward slash (/) because this made it easier to write out tickets in the days when they did this quickly by hand.
The forward slash, '/', is not the division operator so mentally replace it with a colen, ':', and you will find life much easier when doing math involving probability and odds.
A special case is when a horse has more chance of winning than losing, for example odds of 4/6 represents 4 chances of losing and six chances of winning. These horses are called 'odds on’ and usually appear in red on the bookmakers board. Just to confuse you further most people just say 6/4 ON. If you see this just convert it in your head back to 4/6, or more correctly 4:6.
As we have discussed a horse showing odds of 10/1 has 10 chances to lose and only one chance to win (remember bookies odds are odds against an event happening). Now this is where knowing that the odds are really a ratio is important. 10/1 is really 10:1 and so you have 10 chances of losing and 1 chance of winning. The total number of chances (possible outcomes) is 11.
Therefor the probability of winning is 1 chance in 11 or 1/11 = .09 or 9%. Many people get this wrong because for some reason when they see 10/1 they immediatley divide 1 by 10 rather than 11. If you treat odds as a ratio you will never make this mistake again.
So the rule is mentally convert the "/" to a ":" and hopefully this will remind you to ADD the numbers each side of the ":" to determine the total number of possible outcomes, N.
Then Probability = 1/N or as a percentage = (1/N)*100
One last example, odds of 4/1. Express this
as a ratio, 4:1, therefor N = 4 + 1 = 5
and probability becomes:
Probability of Success = 1/N = 1/5 = .2 or 20%
It is a simple formula but always remember odds are a ratio when calculating the number of possible outcomes, N. Understanding this will make it easy to work out the probablilty of success for any bet you may be offered.
Converting Probability to Odds.
Now you know that odds against a winning bet is simply the ratio of the unfavorable to the favorable outcomes. On the other hand the probability of a bet being successful is the number of successful outcomes divided by ALL the possible outcomes, both favorable and unfavorable. You can see that probability and odds are related but how?
Again before we simply use a formula it's important to work through an example to understand the process. Continuing our previous example, if the probability of drawing the ace of spades is 1/52 how do we work out the odds for this bet?
First ask yourself how many chances, or ways if you prefer, are there to win? In a horse race this will always be one and in our example this is also 1. Then ask how many ways are there to lose? In the card example this is 51 (since 1 card is the winning card the other 51 are losing cards). Now you recall that I have stated that odds against is simply the ratio of losing to winning outcomes and so:
Odds = Losing Outcomes:Winning Outcomes
Odds = 51:1 as a ratio, (51 chances to lose and only 1 to win).
or 51/1 as you would see on the bookies board.
One last example, this time for a change we'll use a dice. What are the odds of throwing a six on a dice? Well the probability of a six is 1/6, since there are six possibilities, only one of which is the number six. The odds against a six are 5:1, since there are 5 unfavorable outcomes to one favorable. So you need to get odds of better than 5/1 before you would bet the six (or any single number on the dice for that matter).
Now what if we spiced up the example and asked what odds a one OR a six? The probability of a win here is 2/6 (since now we have two favourable outcomes out of the six possible). This makes sense as we now have two numbers riding for us and so we expect to improve our chances of a win.
The odds against is the ratio of unfavourable outcomes to favourable outcomes or 4:2, which can be divided by 2 to become 2:1 or 2/1 as you would see on a bookies betting board. You will also notice the odds of a win are now down to 2/1 from 5/1 when we only had one number riding for us - as the probability gets higher the odds are getting shorter. This is your opportunity to pounce on a punter offering you 3/1 about such as result!
Of course in both examples above you could have just calculated the odds without bothering with the probability calculaton however you will find many occasions where you will need to work out one from the other.
A formula to calculate the odds about a given probablity is:
OddsAgainst = (1/Probability) - 1
and represent the answer as :1 or /1 whichever you prefer.
For instance suppose you have a probability of ¼ or .25.
Using .25 the odds are:
OddsAgainst = (1/.25) -1 = 4-1 = 3
and so the odds against become 3:1 or 3/1.
If you want to avoid rounding errors (eg. 1/52 is really 0.0192307... and not just .019) then use the 1/N representation for probability in the above calculation and not the rounded decimal probability.
ie. use P = ¼ instead of .25, so
Odds against = (1/(1/4)) - 1 = 4 - 1 = 3 and odds are 3/1 as before. For most practical purposes and especially horse racing the rounded decimal representation (eg .25) for probability is close enough and is more convenient for working with spreadsheets and the like.
Conclusion
Converting between odds and probability is easy if you know a few simple rules. The only predictable relationship between statistical probability and gambling odds is that any sensible gambler will try to offer you odds that are below the true odds dictated by statistical probability. You owe it to yourself to be able to perform these calculations before embarking on any serious attempt to make money from your chosen form of gambling.
This article is copyright ©
Doug Robb 1996-2001.
All rights reserved. May be copied
freely for personal use and yes you can put it up on your web page providing
this copyright notice stays intact.
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