"What Are My Odds?" - William Benter ICCM 2004

professor Yao thank you very much for inviting me here today the title of my talk is what are my odds and I think that's appropriate I think my luck is actually running rather bad today because I've been chosen to follow such a erudite and entertaining lecture by the previous speaker but hopefully I'll be able to impart a little bit of interesting information to you if I can the subject is going to be historical and modern efforts to win at games of chance and the things that are going to be discussed are gambling in a desire to win at gambling inspired the scientific study of probability I'm going to talk about three key mathematical concepts that emerged during that early study which described the majority of gambling phenomenon and then I'm going to talk a little bit about some modern contributions of mathematicians to solving present-day gambling problems everything begins with gambling or at least the the important modern history begins in 1654 and there was a well-known gambler in Paris named Chevalier de Marais and he was perplexed by a seemingly inconsistent result in some of the popular games of chance that were played at the time the illustration on the left is is not really this Chevalier but it's a person in the costume of the times that may be what he looked like his particular question that he was dealing with was a game of betting on rolls of dice and his question was why if it is profitable to wager that a six will appear within four rolls of one die then why is it not profitable to wager that double sixes will appear within 24 rolls of two dice and the reason was is because both of those represent four rolls of a dice represent two-thirds of the possible combinations on a six-sided dice and 24 rolls represents also two-thirds of the total ways that two dice could come out which is a total of 36 combinations now he thought those should be the same now Desmarais took his question to his friend Blaise Pascal as we all know mr. Pascal was one of the greats of mathematics so Desmarais was very lucky to have him as a friend stimulated by de Moraes question Pascal began a very now famous chain of correspondence with another mathematician of the time which is Pierre de Fermat of Fermat's Last Theorem who was living at that time and the two of those were friends now in their discussions and the letters have been examined by historians of mathematics for many years but it was evident that two those two that no existing theory existed that adequately described a solution to de Moraes problem what resulted was a chain of correspondence extending for some years but all of the important ideas or I should say the important ideas for a foundation of the theory of probability resulted from that correspondence the three key concepts I'm going to talk about that relate to practical problems in gambling are one probability - the concept of mathematical expectation and three what's what's known as the law of large numbers firstly probability the early work on probability dealt with games such as dice tossing games or dice rolling games or coin tossing games where the game could be broken down into a certain number of equally likely possible outcomes for instance a die has six sides and if you roll one die then there are six possible outcomes any one of the six numbers could come up and the probability of a particular side coming up is given by the number of sides or the number of events that cause that to be true so so on rolling a dice the chance of a 5 coming up for instance is one out of six because there's six possible outcomes all equally likely and one of those corresponds to winning so the problem probability in that case would be one over six a further refinement to the definition of probability I should not say a refinement but an extension is to say that the probability of any events can be limited to the or can be expressed as the limit of the number of successes or successful outcomes over the number of trials and if you see looking back at this the the form of that equation is very similar this is more of a generalization of the theory to say that it's the limit of an observed frequency can be also thought of as the probability the second very big idea that's very relevant to gambling problems is the idea of mathematical expectation this idea was first attributed to the dutch mathematician Christian Huygens who is pictured there on the right and now the definition of mathematical expectation which I'm sure you're all familiar with this concept being mathematicians is it's a weighted average of a random variable and that equation at the bottom expresses the expectation of a random variable X is simply the various values that X can assume each times its probability of occurring so P sub 1 times X sub 1 P sub 2 times X sub 2 etc some together and that would be a kind of weighted average outcome and the reason that is so important for gambling is because the result of a gamble can be thought of as a you know a random variable that can assume various values in this case usually winning or losing for instance so if we look at the probability for let's say the return on a particular bet there's a certain probability of a profit so P times the profit plus 1 minus P which would be the the opposite probability or the complimentary probability of a loss and that would equal your expectation and to make that a little bit more clear I've worked it out there in numbers at the bottom of the screen for the case of a bet on roulette now on a roulette wheel as I'm sure you're all familiar or you know what one looks like there's 38 possible numbers and if you win you get back throw a net win of 35 and if you lose you you lose the unit that you bet so a particular bed at Roulette you have a one in 38 chance of winning a 35 profit and a 37 and 38 chance of losing your $1 and that all averages out to a net return of minus 0.05 to 6 so your expectation for placing a bet on a roulette layout would be to lose 0.05 to 6 that fraction of your money basically so it's a negative expectation another property of expectation which is very very important for the gambler is that expectation is additive when you make a series of bets each one of those will have a specific expectation your overall expectation that is your overall average or mean result for a series of Gamble's or a series of plays at a particular game is going to be simply the sum of the individual expectations of the bets you made and so that leads to a rule number one as I'll call it which is that the only way to achieve a long-term expected profit and gambling is to make a series of bets that have a net positive expectation now the last of the three big concepts I wanted to talk about is the law of large numbers and this is one way of naming this particular phenomenon but now gambling typically involves playing a series of repeated trials of a particular game that could be bets at the horse races or plays at the blackjack table in the casino or plays at the roulette wheel and that had been you know independent trials or repeated experiment repeated repetitions of an experiment had been studied quite carefully by Bernoulli Jacob Bernoulli that is of one of the Bernoulli Brothers and in fact they named the outcomes of repeated trials are often called Bernoulli trials in honor of Jacob Bernoulli and down below we see the famous formula which is the binomial distribution and that is a formula which tells us the probability of particular numbers of successes in a given string of trials now again I think is as a group of mathematicians you know much of this but the as a number of trials increases the expected ratio of successes to trials converges to Casta cailli to the expected result so in this case I'm showing a series of you know results and what's graph there is the proportion of successes and they're clustering there around the 50/50 or the 0.5 point so tying these three concepts together being able to express the chance of an event as a probability allowed the mathematical analysis of any wager and the additive property of mathematical expectation enabled the calculation of the overall expected result of a series of wagers and the law of large numbers guarantees that the actual result of a series of wagers is going to converge stochastic aliy towards the expected result now let's get down to some practical things now here's a graph or a chart showing how one might fare at a particular game of chance now what I've shown here is the solid dark line running just above the x-axis in this graph is the graph of the expectation at a particular game now this would be typical if you are playing a biased coin flipping game so let's say you're flipping a coin but instead of the coin being perfectly 50/50 it's slightly positive so that it maybe comes up heads 51 percent of the time and tails 49 percent of the time let's say you win $1 when it comes up heads and you lose $1 when it comes up tails so as you're playing along the number of trials goes along the x-axis here and your result is on the y-axis so the expected result that is the average or mean result of the series of trials you can see is creeping upwards as you move along slightly so by the end of a hundred trials you'd expect to be a couple of a couple of units ahead but what's more important here and what's kind of more interesting as a phenomenon is that the spread or the possible variation in your result is given by the blue lines first is the one standard deviation spread and the pink dotted line on the outside is the two standard deviation result so basically this chart describes what could happen to you in a hundred trials of a very even coin flipping game with an arrow advantage towards the player your mean result is to be slightly ahead of even so you would have made a little bit of money but your actual result could very easily be anywhere within those different ranges so you could be up up or down we're seeing 10 or 20 units plus or minus are quite realistic possibilities for where you could be after that period of time this is what people typically face now if you're in a casino it would be very much like this except that your expectation would be a little bit negative instead of having a an expected positive result after an hour's play or after 100 trials you'd actually have a negative expected result so this you know would be sort of a rotation of this graph around the x-axis with your probability of losing but it would basically say that you could walk into a casino and even if you had a negative or a negative expected result in one hours play or a hundred trials you could be anywhere you could be winning a lot you could be losing a lot and that's what people typically go to a casino for now what I've shown here is now the medium run that was the short run a hundred trials now I've showing the medium run of the same game so this is basically the same probabilities operating except that this time I've run it out to 10,000 trials so now what you see has happened here is that the the expectation has grown much larger we now see that we're around have an expected positive return of about a hundred units and this is shifted where the standard deviation lines fall as well now we see that most of your probable return is very much above the x-axis so you're heading to where you'd have to be at least one standard deviation below expectation that is where the the blue dashed line crosses the x-axis over around the ten thousand mark so 2/3 or most of the time you'd be winning in this particular game or I should say more like 5/6 of the time so there'd only be results more than one standard deviation below chance would result in you being a net loser so you're doing pretty good at this point and let's go to an even longer run this is the the sort of the long run this is maybe a hundred thousand plays at a particular game and what you're seeing is that your expectation that the thick black line has gone up considerably you're now expected to be a thousand units ahead and the two standard deviation line shows that even if you've had an extremely bad run of luck that is you're doing much worse than expectation you're still bound to be ahead basically so you're um you're more or less geared to be guaranteed to be winning at this point so this is the law of large numbers sort of applied to gambling in action and basically what it's saying here is a sort of a corollary to the law of large numbers that in a gambling game you know you will your real result will eventually come very close to your expected result or will become the percentage-wise will become expected so if you're playing an event advantageous game with a positive edge for the player you're definitely going to win eventually but what's you know I had actually done these particular charts up for a talk about a winning gambling system the truth of the matter is that for a typical gambler in a casino it's just the office and all of these charts would be just the reverse you know if you went in and played for an hour you could be winning or could be losing you'd have a small negative expected result if you played for a hundred hours which would be roughly 10,000 trials of say blackjack over in Macau or something like this you'd be almost certainly losing but there'd be some chance again this graph would be reversed there'd be about a one in six chance that is greater than one standard deviation chance that you might be ahead but if you went to Macau and you played for a hundred thousand trials which might be something like a thousand hours of play which a heavy gambler could do in a year you'll certainly be behind basically there's almost no chance that you'll be be ahead so that's a warning to people who play games with negative expectations now I'd like to go onto something a little bit more interesting after the theoretical preliminaries and that's the the modern contributions of mathematicians to some unsolved gambling problems firstly it's true that the basic problems the important problems were all solved in the 17th century there was a particular period dating from 1650 for when Pascal and Fermat first started working on the problem within about 50 years most of the major ideas had been developed and the machinery was in place to solve most gambling problems however there are occasional new theoretical developments some of which have occurred just in the last 50 years or so but increasingly also the computer-based mathematical techniques have now been used by gamblers and researchers to try to find winning systems at games that had previously seemed you know immune to such assaults basically games that couldn't be beaten before people have now found ways to do it the first one I'll talk about is an analytic result that was made by a gentleman named JL Kelly Kelly was researcher working at the Bell Laboratories Belva of the Bell Telephone Company Fame in the 1950s and he was working on the theory of information transmission but realized that one of his findings could actually be applied to gambling and what he what he solved was the problem was what fraction of a cap of his capital should a gambler risk on each play and this was this is a very practical problem usually if you have a system and urug Ambler you also have a finite amount of capital and that's something that is not necessarily told to you by any of the other theorems that I talked about before the expectation or the law of large numbers it doesn't tell you precisely how much money you should risk on each play of the game and that's what his that's what his result told us his result has proven to be very generally generally applicable to different gambling situations and it's used very widely by professional gamblers today what Kelley did was to define something called the was called the Kelly criterion sometimes is he defined the exponential rate of growth of a gambler's capital G as being the limit of this X of n which is final capital versus starting capital as a game has played on and n being the number of trials what the derivation and the and the solution of this is a little bit involved and beyond the scope of this lecture but it gives some very simple resort so for instance in a coin flipping game that where the coin was slightly biased for instance let's say you had a coin where you are a game where you win $1 if it comes up if the coin comes up heads and you lose $1 if it comes up tails but instead of the coin being 50/50 it actually has a point 5:1 probability of heads and a point 4/9 probability of tails so that would be a slightly advantageous game and let's say you have $1,000 as your total capital well the kelie formula maximizing that function G is above as according to the fraction of your capital that you wager on each play it comes to a very simple formula which is you should wager 2 times P and P being the probability of winning 2 times P minus 1 so if P is 0.51 you would want to wager 0.02 of your capital so if you had $10,000 you'd want to risk $20 on each play or $20 on the first play I should say but then always recalculating is your capital grew or shrank depending on whether you're winning or losing you would always outlay exactly 0.02 or 2% of your capital it's a very important resultant used very widely the second big area is not a is not a theoretical advance but rather it's a practical advance in gambling I think many of you will know the game of blackjack to those who don't I'm really not going to be able to describe it here adequately but I'll assume that most of you do that's a game where the player receives two cards and the dealer receives one card which is visible to the player and another card which is hidden now the object of the game is to achieve a point total of your cards in your hand as being close to 21 as possible without exceeding that number it's a game that has both skill and chance elements because you have a choice of whether you take additional cards or you don't take cards now this has been a popular game in casinos since the late 1800s and it was solved in 1960s so it had been around for about seven years and was quite popular but there it had so far eluded solution because while the rules of the game were very well-defined and there was no theoretical challenge the fact that it was dealt from a deck of cards and that there were a large number of discrete I call them card order dependencies it made it very difficult to enumerate all the possibilities and even though it would be not theoretically complicated to simply write out on paper all of the probabilities it would resolve it it would be millions of different combinations you have to write out and test and so it wasn't able to be done manually now around 1960s the early 1960s when digital computers first became available for use in in frivolous pursuits like figuring out card games a gentleman named Edward Thorp decided he was going to use a computer to determine the optimal strategy for blackjack and he did that work in conjunction with some people at IBM and in 1962 he published a book called beat the dealer and this book was a watershed event in professional gambling or in I say the modern interest of mathematicians in gambling because for the first time it was now possible to use a mathematical strategy to achieve a positive expected return at a pocket or casino game so basically anyone who read this book and learned the strategy could walk into any casino and be playing with a positive edge against the casino and this this started something of a revolution in fact that's that's how I got into the game I was caught up in this I read the book and quit what I was doing which was attending university and moved to Las Vegas and became a professional gambler so you know it worked and it still does and it really generated a lot of interest what what was interesting about this though as a you know as a technique was that it wasn't it was the use of technology to enumerate possibility so it was the use of a computer to assist in a arduous manual calculation task and it wasn't really a theoretical breakthrough but it was still an interesting use in fact the way that it's the way that it's done now in blackjack because computers have gotten so fast people don't even bother to make calculations they simply write simulators so if for instance the rules of the game of blackjack are very easy to encode into a computer and when you want to answer well what is the advantage for the player or the house or what's particular strategy is best it's easiest rather than than writing out a detailed sort of calculation by computer is to just make a simulator let it run millions and millions of times or billions of times and then tabulate the results and this will answer any question it's sort of the the brute force brute force method of calculation now the last one I wanted to talk about though and this is the one that's nearest and dearest to my heart is horse racing I would say that this particular form of gambling has inspired the most serious and sophisticated efforts in terms of applying modern mathematical techniques to a to a game of chance partly the reason is because there's so much money in horse racing it's a it's a played for very high stakes around the world especially here in Hong Kong and therefore it attracts a lot of people who are trying to win at the game now in racing the challenge that you're faced with is trying to estimate each horse each horses probability of winning now unlike a well defined or idealized game like blackjack which is really just a set of rules basically or a game with dice or cards which again is something that is not you know it can be set down as rules you know let there be a dice let it have six sides let each chance each side have an equal probability of coming up these are sort of analytical games which don't require any investigation of the real world to solve them horse racing however involves dealing with real individual horses and modeling real-world phenomenon so it's a little bit it's a departure from pure mathematics now if you wanted to to win at horse-racing you have to construct a model as we said to estimate the horses probabilities of winning now the the model you want has certain desired there are certain desired characteristics of such a model one would be that it could combine heterogeneous variables into an overall predictor of horse performance a second is that an estimate of the horses win probability is the desired output of the that model third the probability estimate should sum to one within each race that's kind of a common-sense constraint because we know there's going to be one winning horse out of each horse race and a way must exist to practically estimate the parameters of the model with real data now we could view the expected performance of a horse as being the result of a number of different variables or factors related to the horses past performance or the condition of the race I've written out one such specification here as an example and we could say that the horses expected performance equals some coefficient theta times the average finishing position I call that a V fin that's a variable which would be the average past finishing position another quote that plus another coefficient times the number of days since its last race a third coefficient times the weight carried by the horse a fourth coefficient times the wind percentage of the jockey etc you could imagine those going out to you know maybe a hundred different characteristics of a horse that all can be expressed by the the equation at the bottom which is that the the expected performance V of a horse is equal to a weighted sum of different different variables describing the horses past performance now an actual horse performance in a race even though it's its expected performance might be a deterministic thing you know the sum of a number of different predictor variables we all know from common sense that random things can happen in a horse race or there might be unknown variables that are affecting horses performance so we could say that an actual performance in an individual given race for a horse might be this deterministic component V plus some random error epsilon this is a common formulation in statistical modeling of all sorts and I'm sure many of you will be familiar with it but if we if we make the assumption that this epsilon is a normally distributed quantity and that's going to result in a performance distribution which is normally distributed around a certain mean and there would be a graph of that now this is for instance a graph of the expected running time of a particularly the a particular horse is going to run a race in an elapsed time of just under 90 seconds elapsed time being on the the x axis there but that is actual performance is going to be somewhere around that due to unknown factors that we can't control for can't predict so that's the expected performance of one horse this would be a look at what an entire race of horses might look like we might have let's say ten horses in a race or I think we've got eight in this particular example and what we see are a number of different overlapping normal distributions showing you know different potential running times for different horses now I've intentionally made these so that each horse doesn't have an identically an equally wide distribution some are some horses might be more consistent and have a sort of a narrower distribution of possible running times others might be more uncertain in their in their performance and have a wider standard deviation but this sort of mess right here is a number of overlapping predicted running times is what you really have in horse racing and this is leads to something called the probit model of horse racing and the chance of a particular one of these eight horses winning is sort of the chance that his performance will fall it will be the leftmost of all of these overlapping distributions and that's expressed by the equation on the bottom which is a rather complicated equation when if we assume that that is the model of a particular race then you can create a likelihood function which you can associate with a past series of race and this was basically the likelihood of the observed winners in a series of let's say a past 10,000 races is going to be this product of the winning probabilities assigned to the horse that actually won in each of those particular races and this results in a very difficult to solve problem it has to be solved by a very complex iterative method there's a gentleman actually here in Hong Kong who's pioneered one of the ways to do that a professor booming Gow at Chinese University and we've personally have been using his methodology for solving this particular problem in practice well that's I showed you that to give you really a flavor of what kind of work is going on I've included this graph just because I'm proud of it this is the actual record of betting using the aforementioned horse racing system or a methodology very similar to that and this is a consecutive series of about six thousand races which was about eleven years of hong kong racing and the graph is of a sort of an accumulated net profit that we were experienced betting on this particular system and in fact to those of you who are mathematicians which i assume as most everyone the y axis on this graph is actually a logarithmic scale because the the actual profit chart from using this system was you know very much a very very steep so to make it fit on a particular page we had to use a logarithmic graph so is that you know something I'm something I'm proud of here's a quote from a gentleman who wrote probably one of the best books on the study of mathematics and gambling and his conclusion was that the gamblers can rightfully claim to be the Godfather's of probability theory since they are responsible for provoking the stimulating interplay of gambling and mathematics that the impetus to the study of probability now I think that's I think that's a true statement on and I think that in the past certainly it was true in the in the 1700s or the 1600s when all of this started these days it seems that rather than being the the you know the inspiration for innovations in mathematics Gamblers these days are more or less the beneficiaries it's other factors that are driving the development of mathematics now but keen gamblers are making use of the fruits of mathematics that was oftentimes developed for other areas of endeavor and applying these same mathematical techniques back to gambling and oftentimes - great - great - great profit and there's a few references thank you great we entertain one or two questions sorry questions sorry could you say that again oh yeah the question was can these strategies be applied to the stock market yes in a matter of fact I think horse-racing is probably the closest analogy because in the stock market you're oftentimes one way of profiting is to be able to predict how a certain company shares are going to fare or a particular currency and oftentimes this involves real-world modeling of the particular underlying instruments so you would have some model that would predict the future share price based on some past data in the same way that you would predict a racehorses performance so yes these things can be done but horse racing is a particularly good field for applying this type of modeling because the stock market often times you have unique situations occurring where as horse racing is very stable and you've got the sort of same sort of event occurring thousands of times which makes it very much amenable to modeling and statistical analysis the book is using the same no yeah yeah the question was are the bookies that is are the the bookmakers or the people who accept bets on horse racing using the same type of models and the answer is no basically usually bookmakers or people like this act as kind of intermediaries for taking bets and they usually act in a in a responsive manner whereby they try to set the odds based on how the people are betting such that they always keep a certain edge or advantage over the they act as kind of a toll taker really they oftentimes try to limit their risk and specifically in in Hong Kong I think people may be unclear what you mean by bookmakers because here we don't have bookmakers that is individuals who accept bets on racing it's all done at the race track and the race track operates on the paramutual system where it withdraws a fixed percentage from the betting pools and therefore is taking no risk basically so no but it's it's the gamblers using the models and not that not the casinos or the race tracks or the bookmakers well I want to answer the whole house raising and Hong Kong officer I think that US is a little big change before the race and after you get the probability and was however the Aussie chain then how to get there is with expectation value and before the race and before the bad thing there was the optimal time to be there may be okay yes I understand that's that's a very good question a very astute question I should say that calculating the overall expectation for horse-racing once you know the probability of a horse winning the other thing you have to know is what the payoff is going to be because it's you know that the probability of winning times the amount you get if you win plus the probability of losing times the amount you lose if you lose that's what adds up to give you the expectation and if the payoff is unknown that is if the payoff is you suggest could change at the last minute that introduces an uncertainty which might mean that your expectation is not what you thought it was and a positive expectation if the odds drop at the last minute becomes a negative expectation this is a problem the idea is to try to bet at the very last minute if possible so you know the expect or you know the actual payoff odds as closely as possible and mostly people have to build in a certain cushion on the assumption that the payoff odds can and usually do drop at the last minute on horses that are likely to win so it's it's something you just have to deal with and yes it does hurt your profits but and the best time did that always is the very last minute Christmas okay thank you very much appeals so you you

Channel: Betfair Pro Trader

Views: 77,340

Keywords: Gambling (Interest), Sports Betting (Literature Subject), Horse Racing (Interest), Betfair (Business Operation), Odds, Football (Interest)

Id: YOVrZrJ-wtc

Length: 36min 57sec (2217 seconds)

Published: Wed Apr 01 2015