Your Luck always starts from Scratch
Dice have no memory. Cards have no friends or enemies. How they turned up last time has nothing to do with how they’ll turn up next time because
DR. LIONEL RUBY
SIR WINSTON CHURCHILL, on his visit last summer to the U. S., was interviewed by a reporter who asked him, “If you had it all to do over, would you change anything?" . . • “A nostalgic look flitted over the great man’s face,” the reporter wrote later. “ ‘Yes,’ he said, I wish I had played the black instead of t he red at Cannes and Monte Carlo.’
That Churchill should have carried such a vivid memory so far from the tables was not unusual for anyone who gambles and, in one form or another, almost everyone does. That he should have indulged publicly in hindsight on his pH) was equally commonplace. Win or lose, the gambler has never lived who was not certain that if Lady Luck had smiled more beneficently he would have come out better. The fact is, however, that even figuratively there can be no such mystical lady as Lady Luck. Gambling is a perfectly cut-and-dried proposition, and your chances of winning and losing can be calculated by a logic of probabilities as cold and immutable as a used stat istic.
Let’s examine this logic, as well as some of the logical fallacies that afflict the thinking of many gamblers in particular an error which has been christened “the gambler’s fallacy.” Our illustrations will be drawn from some popular games of chance: bingo, lotteries, roulette, slot machines,
dice, cards, and horse racing. We shall not he concerned with the moral aspects of gambling, nor with the many deplorable consequences that, often flow from the human desire to gel something for nothing. And we won’t say that a person should not gamble, even though, as we shall sec, those who engage in public gambling such as horse racing are almost sure to be taken for a ride and not on one of the horses!
We begin with the logic of probabilities. The modern study of probability theory began about three hundred years ago, during the seventeenth
century, when the Chevalier de Mere, a famous gambler, called on his friend Blaise Pascal, a distinguished moral philosopher, and a brilliant mathematician. The Chevalier asked Pascal to work out the probabilities in games of dice for him, so that he would know how to place his bets in the most advantageous manner. Pascal became interested in the problem, and his study of the odds in gambling led to the development of an important branch of logic and mathematics known today as the “theory of permutations and combinations.”
We can illustrate the general principle with a penny. I toss it in the air. It may fall heads, or it may fall tails. There are just two possibilities, and if I desire heads then I have one chance in two to win. My chance of getting heads is thus 50-50, or one half. If I cast a die (plural: dice) my chance of getting a given number, say 5, is 1 in 6 ( 'A), since there are six faces on the die.
The simple principle illustrated in the coin and the die applies in the most complicated computations. But before we go on from here, let us note that when we speak of the probabilities of getting “heads” or a “5” on the coin or die, we make at least four assumptions:
1. We assume, first of all, that the coins or the dice are evenly balanced in weight ing, so that they do not tend to fall one way rather than another. This includes the assumption that the dice are not loaded. But crooked gamblers, as we know, use loaded dice. The logic of probabilities assumes that there are no influences of this nature.
2. We assume that the coin and the die are not manipulated or controlled by the thrower. The hand is quicker than the eye in gambling as well as in legerdemain, and we rule out such influences. Where such influences are present, the laws of chance are irrelevant. Slot machines, for example, have three wheels, each with twenty slots. If there are three jackpot symbols on one wheel and one each on the others, then the mathematical chances of hitting the jackpot would be 3 in 8,000 spins - provided that the machine is not fixed. But all slot machines are reputed to be regulated or fixed, and if they..are, then the mathematical
probabilities are inapplicable. The fixer, like the fates, controls the destinies of the wheels. Thus, when he was asked, “What would you do if you saw someone cheating at cards?” a British nobleman addicted to gambling once replied: “What
would I do? Bet on him, to be sure.”
3. We make the assumption that the desires or thoughts of the thrower have no influence on the result. Not matter how hard you may concentrate on winning, no matter how much you may need to win, no matter how many pairs of shoes your baby needs, and no matter how you may importune those familiar characters known as “Little Joe” and “Big Dick,” these considerations have no influence whatsoever on the probabilities.
4. And finally, we make one further assumption
when we say that the coin will fall heads or tails. This assumption may be illustrated by a story about the man who always found it hard to decide whether to have just one more drink before going home. He would debate the matter with himself, and finally toss a coin to decide. The coin would settle the matter in the following way: If it fell heads,
our friend would have another whisky, straight; if it fell tails he would have another whisky with soda; but if the coin landed on its edge, then he would go home. We, too, assume that the coin won’t land on its edge.
Why You Don’t Win at Dice
Let us now go on to some more complicated examples of mathematical probabilities. Probabilities are stated in arithmetical fractions. Thus, our chance of picking a spade from a deck of 52 cards is 13 in 52, or ]4, since there are 13 spades in a deck. The principle is the same as in the cases of the coin and the die, and the same principle applies in the most complex computations. The chance of a coin falling heads twice in a row is (y2 times La); the chance of getting heads 10 times in a row is 1 in 1,024.
Ina crap game it is more likely that the thrower will lose than that he will win. In 495 games the thrower will probably win 244 times and lose
251 times, the fraction 244/495 representing the probability that he will win. A mathematician with some time on his hands once figured out the probability of dealing four perfect hands in a bridge game, that is, each of the four hands having all of the cards in a single suit. The probability against this sort of thing happening is represented by a staggeringly large number, consisting of twenty-eight digits: 2,235,197,406,895,366,368,301,560,000 to 1. This is 2 octillions, 235 septillions, 197 sextillions, 406 quintillions, 895 quadrillions, 366 trillions, 368 billions, 301 millions, 560 thousand to 1 ! Another way of stating this number is that if a million bridge hands are played every day, this combination will occur once in every 3 quintillions of years, that is, once in every billion-timesthree-billion years. It is unlikely that anyone now alive will ever see such a combination of cards, but it is not impossible. It may even occur tonight.
Let us now examine an extension of our basic principle. Suppose that we toss a coin a thousand times. How many heads and tails should we expect, after completing our tosses? Obviously, we should expect five hundred heads and five hundred tails, since there is just as much chance to get tails as heads. We usually don’t get an exact division even in a much larger series of tosses, but we expect a close approximation to an equal number of each.
Now, keeping in mind this point concerning the evenness of the chances, let us examine a widespread error concerning probabilities, an error to which logicians have given the name: “the
gambler’s fallacy.” Let us assume that we toss a coin twenty times, and that it falls tails each time. There are some people who will say, “There is now more than an even chance that heads will fall on the twenty-first toss. If tails fall twenty times in a row, heads become ‘overdue.’ ” This is the gambler’s fallacy. It is sheer superstition to speak of heads being “overdue,” if by this we mean that there is more than a 50-50 chance for heads to occur on the next toss.
We noted earlier that the chance that a coin will fall heads on a single toss is 1 chance in 2. It is always that. If the coin is properly balanced, and not manipulated, Continued on page. 45
Your Luck Always Starts From Scratch
CONTINUED FROM PAGE 19
then the probability that it will fall heads on any toss, regardless of what has gone before, is 1 over 2; no more, and no less.
But the gambler is not vet convinced. He will ask: “Suppose that we throw a coin a thousand times. Is it not likely that we will get approximately five
hundred heads and five hundred tails?” Our answer is Yes. “Well, then,” he goes on, “suppose that the first twenty tosses are tails. Won’t the heads now have to make up their due proport ion, in order to come out even in the end?” To this the answer is No. When we say that in a thousand tosses we will probably get five hundred heads and five hundred tails, these are the probabilities in advance of tossing the coin. Probabilities refer to the future only. But if. in our project of tossing a coin a thousand times, we get twenty tails “right off the bat,” it now becomes
foolish to expect that there will be five hundred heads and five hundred tails when we complete our one thousand tosses. For there are now 980 tosses to go. The probability is that the remaining 980 tosses will divide evenly into 490 heads and 490 tails, and the most probable outcome now is: 510
tails and 490 heads for the full one thousand tosses. This is the logical expectation after getting twenty tails in the first twenty tosses.
Perhaps an even simpler example will be helpful here. The chance of getting two heads in a row is j4, in
advance of either toss. But suppose I throw the coin and get heads on the first toss. What are my chances now of making two in a row, including the first head? Obviously the chance is now since I need merely get a head on the next toss.
In other words, coins never “make up” for past performances. On any toss, the probability is the same. But the gambler is still unconvinced. He has a new argument. Is there not a “law of averages,” he asks, which makes things come out even in the end? The answer, once more, is No. j There is no law of averages which j makes things come out one way or ! another. The law of averages, if it means anything, simply means the logical probabilities concerning future events, and in the field of mathematical ; probabilities these are determined without considering the past.
We cannot leave this topic without quoting John O’Hara’s Pal Joey. When the night club in which Joey was employed as a singer burned to the ground he felt unjustly treated: “Ten
thousand night clubs in this country but I guess they repealed the law of averages because they had to pick the one I was in to have a fire. 1 notice I never get that kind of odds when I go to the track.” But a lottery ticket holder who has one chance in a million may win, and there is no need for a Great Power to repeal the law of averages in order for Joey’s club to burn. Unlikely as it is, you may win the lottery, and your place of business may he destroyed. This is why fire insurance companies stay in business.
Dice Are Never Hot
Let us now go back to the gambler’s fallacy. In a dice game, when a player has thrown several passes in a row, some players will bet more heavily than usual against him, believing that a loss is “overdue.” This is the typical fallacy. But, curiously enough, gamblers also believe in another fallacy the exact opposite of the first one! When a player throws several winning passes in a row, the thought arises that “luck is with him,” and that it will stay with him, because the dice are now “hot.” Now, it is undoubtedly the case that I the dice have some degree of warmth ■ during a game, due to the transmission of the hand’s heat to the ivories. But the dice are never “hot” in any mystical sense. The expression “hot” here can apply only to the past, that is, the dice were “hot,” meaning that the player has won several times in succession. The same considerations apply to the notion that some people are luckier than others. All we can say is that some were luckier in the past; that is, they won; but for the future all of us are on an equal basis.
The two fallacies, then, are: (1) If a man wins several times in a row, bet against him, for a loss is “overdue”; (2) If he wins several times in a row, he is “hot” and you should bet with him. But neither notion is sensible, and amusingly enough, they cancel each other out.
The belief in a mystical kind of luck, of course, will not down, even though it is sheer superstition. When people gamble on the faith that this mystical kind of luck is in operation, and are fortunate enough to win, they are accounted as “shrewd.” On July 22, 1953, the Chicago Daily News ran one of a series of articles on “The Business” of Las Vegas, Nev. In the Desert Inn’s “deep-carpeted lobby,” it reported, “locked in a glass case and resplendent on a red velvet cushion, sits the visible symbol of the Vegas gods. It’s a pair of ordinary dice. Except that this pair, handled by a youngster in
1950, made 28 straight passes at the fnc’s dice table . . . But this youth was luckier than he was brave. He parlayed only $750 . . . though the Inc dropped $150,000 to shrewd onlookers making side bets.”
Now, were these “onlookers” shrewd to go along with this youngster? They won, but their shrewdness is determined by hindsight; they won $150,-
000 Is one shrewd if he wins $1,000 on che toss of a coin? In dice games, the odds on any pass are always the same: 251 to 244 against the thrower. The same probabilities hold on the first pass as on the twenty-seventh or nineteeioth or seventh. After the event, the non-bettor bemoans the fact that he “didn’t see it coming,” that he lacked courage, or shrewdness. But this is just the superstition of gamblers who believe in the mystical Lady Luck.
There simply are no ways of “knowing” how to bet in games of pure chacee. There are no gambling “systems” that will guarantee more than you t mathematical chances. There are no vays of beating the probabilities. Theie are no magic numbers, nor any magic combinations of numbers. Gambling when the moon is full won’t help, nor will it do you any good to touch a “lucky” person. Walking around a chair will not help, nor standing on your head, nor changing the deck in a poker game. Nor does one refute these principles by pointing out that someone did one of these things, and then won.
There is ont* exception to the rule that there are no systems, or rather, there i.» a system that •would be unbeatable if it could be applied. This is known as the “Martingale.” 11 operates as follows: Let us say that
1 bet $1 on the toss of a coin, or on red or black in roulette. If I win, I put aside the dollar I won and bet $1 again. If I lose, I bet $2 on the next toss. If I win on the second toss, I am $1 ahead on my two bets. If I lose twice in a row, I now bet $4. I have now invested $7: $1 on the first toss, $2 on the second and $4 on the third. If I win now, I collect $8 and am again $1 ahead. If I continue to double up in this way, I will always be $1 ahead when I win after any series of losses. But if a long series runs against me then I am in real trouble. For if a series of twenty-seven passes, or tosses, or reds or blacks comes up against me, that is, if I lose twenty-seven times in a row in craps, or flipping a coin, or roulette, then I must be prepared to back the twentyeighth toss with $134,217,727. If I can cover the next bet then I must have had an initial capital of $268,435,455, and twice this if I lose on the thirtieth toss! If I lose three more times I will require a capital outlay of four billion dollars! All this to win that elusive single dollar! Anyone with this amount of money available would probably do better to invest it in tax-exempt bonds.
But, of course, one cannot use this system in any actual gambling house, for each house sets a maximum limit
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on bets. In Las Vegas’ Desert Inn,
according to the story noted above, the limit is $500.
There is, however, a more promising method for outwitting the house. This new method is based, not on a mystical belief in magic, but on ordinary physical facts. Several years ago newspapers reported that two college students were “breaking the bank” in Reno, Nev. They had no gambling system, socalled; they simply kept tab on the numbers that fell on a roulette wheel. They found that a certain number, let us say 19, was being hit more regularly than it should have been according to the mathematical probabilities. They inferred from this that the machine was imperfectly constructed. The laws of chance, you will recall, assume a perfectly balanced machine. These students then bet exclusively on number 19 and were apparently correct about the imperfection of the machine, for they were very successful until the management found out what they were doing and the machines were then replaced.
These students, be it noted, acted on a principle quite different from the theory underlying the gambler’s fallacy. The fallacy tolls us to avoid number 19 on the ground that it has already had more than its “share” of hits. The students acted on the basis of scientific or logical reasoning. Since no machine is perfectly constructed, the ball must fall more frequently in some slots than others, though perhaps not always to a significant degree. Their reasoning was sound in that they bet on the number that fell most frequently. The gambler in the fallacy says: “Bet on the numbers that have fallen less frequently than they should have in accordance with the laws of probability, for these numbers are ‘overdue.’ ” The students, who were good reasoners, said, “It is a legitimate scientific principle that what has happened in a long series of runs on a particular machine is likely to happen again.”
Let us now turn to a consideration of one’s chances of winning in public gambling. The most important factor here is what is known as the “percentages.” At a race track, for example, a percentage of the money wagered is retained by the track for itself and the State, so that the return to the players is the total amount bet minus this percentage. The odds are thus against the player by a certain percentage. Gambling houses also operate on percentages. A gambling house arranges the odds in its favor, and in the long run the percentage by which the odds favor it establishes the “return” for the gambling house on the total amount wagered. The percentages are the basic source of the fabulous profits of organized gambling. (There are a number of other factors—such as the house’s larger capital—which also make it likely that it will win.) Thus, for the individual player, gambling is not only a form of risk-taking, but it is also a battle against the percentages. And the higher the percentage in favor of the house, the higher is the average rate of loss by the individual player.
The principle may be illustrated by a simple example. If you toss a coin and agree to pay $2 every time it falls heads, and you receive $1 whenever it falls tails, you would soon lose all of your money. You might win, in the sense that this is theoretically possible, but it is very unlikely.
Now, it is obviously foolish to play games of skill for money when you play with players who are more skilful than you are, for you are almost certain to lose your money. You don’t have an even chance. You may be temporarily
lucky, and win, but in the long run your lack of skill, like the percentages, counts against you. Gambling houses do not rely on luck. And there are many forms of gambling in which the percentages against the player are so high that it is practically impossible for the player to win in the long run. I refer to bingo games, lotteries, the Irish Sweepstakes, and horse racing in general. Ina bingo game, for example, the house may collect $100 on each set of cards, and pay out $50 in prizes. This means that on the average each player will lose one half of his total bets during an evening of play. Some players will come out ahead, but this means that others will lose more than half of their play. The odds against winning are formidably high.
This principle may also be illustrated in its application to a form of gambling established in the United States in the year 1664 by Richard N ¡colls, first Governor of New York. Nicolls said that he wished to stage horse races, “not so much for the divertissement of youth as for encouraging the betterment of the breed of horses, which, through neglect, has been impaired.” Horse racing also offers other inducements besides the opportunity to improve the breed. When a man visits a race track, he enjoys an outing in a pleasant countrylike atmosphere; he can admiringly contemplate those things of beauty, the sleek thoroughbreds; he sees the graceful jockeys perched high on their mounts; and the silken colors of the stables make lovely patterns. And there is always the greatest of all inducements, an opportunity lo lose one’s money.
Horse Players Always Lose
The percentages against winning in horse racing are very high. The typical percentage (or “take”) today is about fifteen percent. Of every dollar bet the provincial government takes a fraction (it’s seven percent now in Ontario, although it has varied from province to province and from year to year), the track owners take a fraction (nine percent in Canada), and there is also what is called “breakage”: in
making payments, the track keeps all pennies down to the nearest dime. In winning a bet, for example, your share of the pari-mutuel pool might be $5.08; under the “breakage” you would receive just $5 while the track took your eight, cents.
For convenience let’s say that the total of track and provincial government “takes” is fifteen percent. This means that of every dollar collected by the track in bets only eighty-five cents is returned to the customers in pay-offs. As a result every player will lose, on the average, fifteen percent of whatever he bets.
Let us apply these figures to a typical situation. On a holiday, a big race track draws, let us say, forty thousand fans, and they bet $4 millions. A fifteen percent take means that the track deducts $600,000 from the total
amount bet. The customers, in other words, go home $600,000 poorer than they came, if they are taken as a group, and they are taken! Divided among the forty thousand fans, we find that, on the average, each individual loses $15. Now, some individuals will break even, and some will win, and this means that there are others who must lose a great deal more than $15. If one man breaks even, another must lose $30 if one wins $15 another must lose $45 to make up for the average loss of $15 each, and the $15 that was won. On this basis, the average bettor, over a season of 150 days, will lose $2,250 a year on the take alone. Remember that on a single day all of us together are $600,000 poorer than when we came.
Some Jockeys are Better
If you are fifteen percent better than the average in your skill in picking horses, or fifteen percent luckier than the average on a given day, then you will break even. Fifteen percent is a very sizeable amount, so this is really a rather remarkable accomplishment. Its unusual character gives point to the well-known story about the worriedlooking gambler who was on his way to the race track. He met a friend who asked why he looked so worried. “I have to break even today,” he answered, “I need the money.”
Now, there is a certain amount of judgment involved in betting on horses, for a horse race is not a matter of pure chance in the mathematical sense. One may appraise the horses and the skill and experience of the jockeys. There are other factors not subject to our judgment which we cannot appraise, such as the possibility that a horst* may deliberately be held back to ensure better odds in a later race. And then there are the notorious deviations from rectitude which may be hidden from us in a particular race. And then, of course, there are the hand ¡cappers, who seek to eliminate the element of judgment: the best horses carry the
heaviest weights. The best horses also carry the shortest odds, so that the chances of winning are equalized, and the value of good judgment greatly discounted.
The experience of some California race-news reporters at the Santa Anita track some years ago illustrates how hard it is to profit from even the best judgment. These reporters made their bets at special windows, and a record was kept of their total bets. They won three percent on their total “investment.” Now, these men were experts and the results prove it, for their ability to judge horses and odds was eighteen percent better than the average. They were three percent ahead; the average bettor loses fifteen percent. If any one of these reporters bet $100 then he won $3, on the average.
But the ordinary bettor, whose luck is average or not at all bad, should expect to lose $15 on every $100 he wagers. These percentages are likely to prevail in the long run.
Such are some of the gloomy prospects confronting the person who indulges in public gambling. The odds are against us. But it would not be sensible to expect anyone to be influenced by anything that has been said about the foolishness of certain forms of gambling. Fond hopes are too ineradicably fixed in the human heart for us to be influenced by anything so inconsequential as a logical argument.
I am reminded of the story about the gambler who met a friend on the street of a small town. His friend asked him where he was going. “To Joe’s gambling parlors,” he answered, “to try my luck at the roulette wheel.” “What!” cried his friend, “are you crazy? Don’t you know that the wheel at Joe’s place is fixed, so that you can’t win?”
“Yes,” sighed the gambler, “I know all that, but it’s the only wheel m town.” ★
This article is an excerpt from the The Art of Making Sense, to be pub by the J. B. Lippincott Company.