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Gambling odds and probabilities

(Please read this in full - it is a worthwhile investment of your time)

What has statistical probability got to do with gambling? In a word, everything. If gamblers had even a modest understanding of probability then the world's casino's 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 on 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 be more interested in is if the odds offered (the pay-out) 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 (odds on) when, as you know, the 'true odds' is even money. In racing terms the hapless punter is 'taking under the odds'. The odds offered are called the ‘betting odds’ or ‘gambling odds’. 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 pay out the 'true odds' you may ask? Well just wander into a casino and watch those hapless souls donate their money to the casino owners. How many of us can say that we have never taken 'under the odds' on a racehorse? When was the last time you brought a lotto ticket? The short answer is that we all have at one time or another. A more appropriate question to ask is why is it that so many people are quite happy to go through their whole life betting under the odds?

In my opinion it is a national scandal that in casino's people are playing games that they simply cannot win, the longer they play the more they MUST lose. It is literally a licence to steal money from those 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 'taking under the odds', study a few casino games if you still think you can win at the casino. By the way if you must go to the casino then only play Back Jack as this is the only game where the house won’t have a significant edge.

So how do we make our money?
In the casino the odds are fixed and you either bet the percentages offered or have a cup of coffee but in horse racing the odds are fluctuating over the course of betting for all sorts of reasons, many of which are totally unrelated to the statistical probability of the horse’s winning chances.

Continuing with the coin toss example what if someone offered us a win of $6.25 on heads for a $5 stake? 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' on evens, or 1/1 if you prefer. The other punter is really paying you a bonus of 25 percent 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 ($6.25) and the amount of money that I would win ($5) . The ‘bonus’ is just $1.25/$5 or 25%.

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 actual fluctuations in terms of probability between these bets is only 6 percent.

In my example you can see that if someone is offering me odds of 5/4 that I only need to win 44 percent of the time (or 44 tosses in 100) to break even, and of course I expect to really 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 manipulated by either player then one or the other will make money and the other MUST 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 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 is just as important in racing as it is 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 5/2 then he will beat you.

Eventually I saw what all 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 one is ‘you can’t beat a race but you can beat the races’. I interpret this 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 the percentages are 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’.

You will, obviously want to back the horses with the highest probability of winning but only at better odds than the ‘true odds’. The art of horse racing is being able to determine what horses are over the odds and what horses are under the odds and not simply picking winners. This of course raises the issue of how do you work out the odds (probability) of a horse in a race? A coin toss or a roulette wheel 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 probability of each horse in a race. How people do this and how well they do it is a topic for another day.

Working out the probability for a single event

Working out probability can be simple or quite difficult depending on the situation. In the simple case you need to work out just 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 the event happening is simply:

p(Success) = 1/n

A probability of one means that an event is certain to happen while a probability of zero means the event us certain not to happen. There are a couple of useful rules like:

p(Success) + p(Failure) = 1

(or in words it is certain that the even 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:

P(success) = 1 - P(failure)
P(failure) = 1 - P(success)

As an example lets work out the probability of drawing the ace of spades from a pack of cards and then convert this number to odds. 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:

1/52 = .019 or approximately 2 percent.

Thinking in terms of percentages is often useful. If this percentage was for a horse in a race you would know that for every 100 races you would only expect a horse with this probability to win twice. 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 usually express odds as odds against winning. So a 10/1 horse has 10 chances of losing and only one chance of winning and as a ratio this is 10:1. A 6/4 bet would have 6 chances of losing and 4 of winning and of course an even money bet, 1/1 has one chance of winning and one chance of losing.

Remember odds are really a ratio and should be expressed as 10:1, 6:4, 1:1. The ‘:’ (colon) is usually replaced with a ‘/’ (slash) and I can only assume that this is for the convenience of bookmakers in working out what their pay-outs will be. For the purposes or converting odds to probability the '/' does not work as the divide symbol so mentally replace it with a ':' and you will find life much easier.

Now a special case is when a horse has more chance of winning than losing, eg 4/6, 4 chances of losing a six of winning of winning. These horses are called 'odds on’ an 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 is 11.

Therefore the probability of winning is 1 chance in 11 or 1/11 = .09 or 9%. Many people get this wrong because when they see 10/1 the think that they have one chance in 10 of winning but really it is one chance in 11. Once you treat odds as a ratio you never make this mistake again.

So if odds are expressed as ‘odds/1’ then as a ratio this is ‘odds:1’ and the total number of possible outcomes, n is then ‘odds+1’.

probability = 1/n and as a percentage = (1/n)*100

Another example, odds of 4/1 (or as a ratio 4:1)

n = 4 + 1 = 5

Probability = 1/5 = .2 or 20%

Most people just add one to the quoted odds and divide this number into one. It is a simple formula and by all means use it but always remember odds are a ratio. In the real world examples understanding this will be a great help.

Converting Probability to Odds.

Again before we simply use a formula and forget about the subtleties lets work out the odds at least initially using a method that gives you some insight into what you are doing. Given the probability of drawing the ace of spades is 1/52 how do we work out the odds you would bet about doing this?

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 card 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, 51 cards 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 = 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.

If you prefer to use a simplified formula here it is:

odds = (1/prob) -1

and call the result ‘something’ :1 or ‘something/1’ whichever you prefer.

For instance suppose you have a probability of ¼ or .25.

Using .25 the odds are:

odds = (1/.25) -1 = 4-1 = 3

and so the odds are 3:1 or 3/1.

When 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. P = ¼ instead of .25, so

odds against = (1/(1/4)) -1 = 3 and odds are 3/1 as before. For most practical uses in horse racing the rounded decimal representation of probability is close enough.

Converting between odds and probability is easy once you know a few simple rules. Since as I have already stated, 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 you owe it to yourself to be able to do these calculations for yourself before embarking on any serious attempt to make money from your chosen form of gambling.

This article is copyright Doug Robb 1996. 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 in tact

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