What's the Odds?
Boondoggling in the Field of Percentages. Or Some Freaky Facts about Lady Luck 'Easy Money" 1935
BOONDOGGLING is not new, provided you belong to the political party that defines it as spending a lot of valuable time on work that isn’t worth anything. The greatest mathematicians, from Archimedes to present day astronomers, often get tired of figuring the number of light years to some distant star. So for recreation they figure out problems in mathematics that are involved, but do not concern them very much. For instance, the laws of probabilities.
But there are a lot of us who could take this branch of mathematics and learn a great deal that would save us a lot of money or, if professional gamblers, make us a lot of money. An exact knowledge of the laws of probabilities would save us the price of a lot of things. That is, if we obeyed them. After all, as you should know, when you disobey the laws of chance you lose.For instance, did you know that the minute you pick up a pair of honest dice, and start to play against a banker X number of times, you are bound to lose, no matter how honest the dice, how lucky you think you are, or anything else? Given an honest game, square dice, and plenty of money to play against the banker one unit at a time, the longer you play the more you will lose. It can be proved as a mathematical fact. Let me prove it to you.
You will handle the dice; your name is A. I will bank the game, paying you when you win, and collecting when you lose. In the problem I will be B. First, we’d better review the rules so we won’t have some local gag pulled on us as is often done in poker games, when some escaped lunatic wants to destroy all the known probability by playing the deuces wild.If A throws 7 or 11 he wins; if he throws two ones, an ace-deuce, or two sixes, he loses. But if he throws a 4, 5, 6, 8, 9 or 10 he continues throwing to duplicate the first throw. If he duplicates it he wins. If he throws 7 before throwing the number for the second time he loses.
Now, for further reference during this revelation, it is well to have before us a table of the possible points that can be thrown with a pair of dice, each of which, of course, is numbered from one to six in such a manner that the total of the numbers on each of the pair of faces opposite each other add to seven.
Incidentally, a lot of pairs of dice won’t do this, and it is only a matter of common sense for any dice shooter to “match up the dice” before playing. To do this, take one of the dice and find the point 1; place the point 1 against the point 6 on the other die, then revolve them until a pair of faces that lie side by side make seven. When that condition exists on one side of the pair of dice, it should exist on all other sides and on the two outer ends. Otherwise, you’d better remember that you have to catch a train, and excuse yourself from the game.
So, first look over the possible points, and note how many ways each can be made. (See chart.)
NOW let’s get down to the problem. Dice incidentally were supposed to have been invented by Palamedes at the siege of Troy to amuse his soldiers. You’ve been carrying the banner of ancient knowledge, but, did you ever know how to bet your money with the proper odds?
Since the odds you find in a dice game are not the proper ones, we are going to show what the proper ones are, bet that way if anybody will call the bet, and, at least, get some kind of a run for our money. Get out your slate and follow the figures.
A has the dice and is ready to shoot one unit. What chances has he of winning? It sounds like a simple problem. You are accustomed to bet even money that you win, meaning you believe that your chances of winning are exactly equal to your losing. They are not.
Let’s see. The chance of throwing 7 or 11 is two-ninths. By the chart shown in this article you will see there are six ways of making a 7, and two ways of making an 11, a total of eight ways to win out of thirty-six possible throws—eight-thirty-sixths, or two-ninths. Your chance of throwing a 2, 3, or 12, and losing, are four out of the thirty-six, or one-ninth. Added to this are your chances of throwing a 4, 5, 6, 8, 9 or 10, a total of twenty-four out of thirty-six, or two-thirds. But from the beginning, your chance of say, throwing a 4 and then repeating it before you throw a 7 are just four-ninths, of throwing a 5 and making it are twenty-two-forty- fifths, and a 6 are 52-99ths. The chances on 8, 9, 10 are the same as on 6, 5, 4.
Your total chance then as worked out in mathematics, is the formula, via a maze of figuring, equaling 722-1485ths. Which is less than fifty per cent. That would leave the odds in favor of the man banking the game as 763-1485ths. Or am I wrong?Or to put it another way, if you took 1,485 units or dollars, each owning half of them, and started to play, the banker would end up with 763 of them and you with only 722. He would take forty-one of your units every time you risked a group of 1,485 points between you; with you, risking only half of them. You can see, then, that if you kept shooting the dice against a banker, as you always do in a regular gambling game, it is only a matter of time before the bank will have all the money. They just have to let you keep shooting long enough, and the law of averages, or probability, will finally catch you without carfare.
COUNT the possible ways of making a 6. You can make it with a 1-5, 5-1, 2-4, 4-2, 3-3. The 8 can be made with an equal number of combinations of numbers. There are three groups of figures involved, and, therefore the points are called three way points. And there are three groups of figures that will make a losing seven.
But there’s a catch to it. After all, there are two dice, and therefore a six can be made with a 5-1, or again with a 1-5. Count the ways in the table and you will discover that there are five out of a possible thirty-six throws that will make a 6, or 5 that will make an 8.Then count the ways in the table to make a 7. You will find that there are six ways to make a 7.
Thus, when you throw for a 6, you have five chances of winning by making it and six ways of losing it. If you were betting five hundred units or a thousand units, or ten thousand, which you will bet in a life time, figure the law of mathematics will allow you to win five to the banker’s six. Out to happen, and the ways for it not to happen, are known, each number compared with the whole will be one of the probabilities.Example: We know how many possible numbers can be thrown with one die. It has six faces, numbered from 1 to 6. If we throw a die, one face will come up. That total of six possible faces is known as the certainty; the possible chances for any given number on the face of the die to come up is known as the probability. Since the numbers are different, each face has a probability of one out of six, or one-sixth.
Now when you want a number that can be made by the use of two dice, it must be composed of one probability and one certainty on each of them. You use the probability for the numerator of a fraction and the certainty for the denominator. Then, to get the chances of any number showing on two dice, you multiply the top figures, or numerators to get the new numerator, and the denominators to get a new denominator. That new fraction will give of eleven throws, you get the small end, out of eleven thousand you would get five thousand and the banker six thousand; in a lifetime you’ll lose plenty on what you believed were even chances.
That’s not all of the unfair odds in dice. But before showing you any more, let me show you a simple rule of arithmetic to get the laws of probability in any even chance game, like matching coins, or anything else where there is a definitely known number of ways a thing can happen, and where the chances of any part of it happening are constant.
The first rule is:
In every event where the number of ways for it not to happen are known, each number compared to the whole will be one of the probabilities.
Example: We know how many possible numbers can be thrown with one die. It has six faces, numbered from 1 to 6. If we throw a die, one face will come up. That total of six possible faces is known as the certainty; the possible chances for any given number on the face of the die to come up is known as the probability. Since the numbers are different, each face has a probability of one out of six, or one-sixth.
Now when you want a number that can be made by the use of two dice, it must be composed of one probability and one certainty on each of them. You use the probability for the numerator of a fraction and the certainty for the denominator. Then, to get the chances of any number showing on two dice, you multiply the top figures, or numerators to get the new numerator, and the denominators to get a new denominator. That new fraction will give you the chance or probability of any number coming up on the two dice with one throw.
You must bear in mind that the six spot on one die and the six spot on the other are two separate dice faces and that, although it is true the totals on more than one pair of faces will add up to the same thing, it is equally true that the chance of throwing up a given face on a given die is still just one out of six.
Very well. To get the chance of throwing a given pair of faces out of a known possibility, you have to do some multiplying.
Say, to throw a 12, you learned a minute ago that the chance of a six on one die is 1-6, and the same would apply to the other. The chance of a 12 showing, then, is 1-6 times 1-6, which equals 1-36th. (The numerators 1 times 1 equal 1 for the new probability, and the certainty being six times six or thirty-six, the total of possibilities.)
Now you ought to be able to figure some odds for yourself, and be ready to know whether you are making a good bet or not. Cut out the chance table further back, and paste it in your hat alongside the new baseball schedule. And while you’re doing that let’s examine the little point known, affectionately, behind the barn, and over the green tables, as 4, or “Little Josie,” and what we find will also apply to her strapping brother 10, known to the sports writers as “Big Dick from Boston.”
SUPPOSE we throw a 4. It’s admittedly a hard point to make. By looking at our little schedule we see that there are only three combinations of dice faces that will make it for us and collect the dough. They are 1-3, 3-1, and 2-2. Three out of thirty-six.
The thirty-six, however, is no longer important to us, because it doesn’t matter what we throw, so long as it is a 4, or is not a 7.But how about that 7? We lift up the kelly and take a peek at the schedule, and we find that there are six ways of making a 7 and having to walk home. In other words, out of nine combinations of dice faces that will affect our fortunes, those that will whip us are two to one. For every chance of winning there are two chances of losing!
But when you are in a crap game, do you ever get those odds? Not by a whole lot. People will bet you three to two that you can’t make it. They should bet two to one against you. You’re giving them a lot of odds. To be exact, you are betting $6.66 against their $10 when you should be betting $5 to their $10. All you have to do is give these unfair odds long enough, and no matter how good your luck is, they will take your money.
Here's something maybe you didn’t know. If you ever worried about making a 5 or a 9 it is certain that you didn’t know it. Because those two points, once you are betting on making them, are the fairest on the dice.
Number Ways of Chances of of point throwing it throwing it
2 1-1 1 3 1-2,2-1 2 4 1-3,3-1,2-2 35 1-4,4-1,3-2,2-3 46 1-5, 5-1, 2-4, 4-2, 3-3 57 1-6, 6-1,2-5, 5-2, 3-4, 4-3 68 2-6,6-2, 3-5, 5-3,4-4 59 3-6, 6-3,4-5,5-4 410 4-6,6-4,5-5 311 5-6,6-5 212 6-6 1
Possible Throws 36
So, our little friend, little Colonel! His philosophy would fill several books. He sold everything from collar buttons to pianos, and sold them well. A liar? Certainly. A petty racketeer? Just as certainly. But a bon vivant, a hail-fellow-well- met, a raconteur of rare tales—more certainly than ever!
The little Colonel has gone the way of all flesh. His tripod is folded, his pitch-case closed. No more will he do his faster-than-the-eye stuff from the back of a buckboard, no more will he sing the praises of nostrums and cure-alls beneath the glare of his sputtering torches.
But—take a bet from one who was his companion—I’ll wager my last chance for paradise that somewhere up in Heaven he’s pitching Colonel Stratton’s Surefire Halo Polish! Or, if things went wrong. Colonel Stratton’s Positive Cure for Blisters and Prickly Heat.
Fiva in the South, and her big sister, Nina with her hair down, aren’t such bad gals, after all. Look up your table again.
They usually give you three to two you can’t make either of those points. For once, the odds are just right. There are four ways to make either of the points and six ways to lose. Or, for every two ways to win there are three ways to lose. The three to two odds are all right.
And now we have swung back to that happy couple, six and eight, Jimmy Hix and Ada from Decatur. And here, as soon as we look at our table, we begin smelling a mouse, a whole nest of mouse, for that matter.
Old even points are no longer even on our slate, and the schedule will tell us why. It is true that they are called three-way points, and that there are three groups of figures appearing in their makeup. But—Count up the possible 6s, and the possible 7s, and you will be surprised to learn that there are only five ways of making a 6, five ways of making an 8, and there are six ways of losing by the old seven route.
Where’s the catch?
Six is made with ones, 5s, 2s, 4s, and 3s. The combinations can be made with say a 1 and a 5, and again with a 5 and a 1. There are two of these combinations, and then the third which is made of 3. But 3s can be used only once; since they repeat themselves.That is not so in the case of the 7s. There are three combinations of numbers, but all three of them are used twice, whereas in 6 and 8, two of the combinations are used twice and the other only once.
Therefore, if you threw for a 6, 11,000 times, you would win 5 000 times and the banker 6,000. He would be 1,000 ahead of you for every 11,000 throws, or one win ahead of you for every eleven throws.
So, with figures still not lying, don’t go around betting even money that you can throw a 6 or 8. You’ll end on a park bench telling the boys about the big games you used to be in when you had it.
Now, we have proved several things:
The odds are slightly against the shooter winning and in favor of the man betting against the shooter.The odds against your making a 4 or 10 should properly be two to one.
The odds against your making a 5 or 9 should properly be (and usually are) three to two.
The odds against your making a 6 or 8 are properly six to five (and are never offered.)
YOU have been patient and have been waiting until now to remind me about something that upsets all the mathematics ever figured.
You want to mention luck with a lot of capital letters.
It is going to be sad, but you’re wrong again, and that can be proved with mathematical certainty.
For the sake of simplicity let’s jump from dice to matching coins; an even chance bet, and exactly fair to each.We will take twenty million people who are on WPA jobs and therefore have little to do but figure things out while they wait from week to week for their checks. Divide them up, half and half, and give each pair a nickel to match with.
Now as sure as you start them matching, you are going to have some that you would want to call lucky, and others that you would call un-lucky. You don’t know who they are, and it doesn’t matter whether they have a rabbit’s foot in their pockets, or have only one arm to match with or not.
It is a mathematical fact that if you have that gang match their coins twenty times you will have twenty men who are sure to win every time. But whether they were men or fence posts, or machines, or what not, the coins would come out that way. The men are not lucky. Mathematics is just obeying its laws as it always does.
Give them a whistle and start them to matching.
They all match once. Ten million of them have won, and ten million have lost. On an equal chance of pulling a white or black ball out of a bag where there is one of each, you can figure that if you pulled 1,000,000 times you would get the white one half the time. So, ten million have won and ten million have lost, ten million were lucky and ten million were not.
Try again. We are not now facing a new fact, but the same possibility that we faced before. Ten million will win and ten lose. But out of that ten million that lost, each has an equal chance to win, and out of the ten million who won, each had an equal chance now to win or lose.
So out of that ten million who lost, having a new equal chance, five million will lose again, but five million will win. And it is equally true that out of the ten million who were "lucky” half of them will win and half will lose. Of that group, the losers broke even and were neither lucky nor unlucky. The same holds true of the first losers who now broke even by winning.Let’s see how they stand.
Five million lost twice.
Five million won twice.
Ten million lost once and won once.
You would say that one fourth were lucky, one fourth unlucky and the half of them neither lucky nor unlucky. Right.Sound the bugle and match again.The twenty million match again, half to win, half to lose. Don’t forget, each man has an equal chance to win or lose.
Out of your five million unlucky ones, all having an even chance, about half will win and half will lose on this third throw. And of the five million lucky ones about half will lose and half win. Those in the middle are playing in varying luck, neither good nor bad, but shifting.
Now, half our five million lucky boys won, half lost. We have now:Two and a half million who lost three times.Two and a half million who won three times.
Seven and a half million who won once and lost twice.
And we have only two and a half million who won three times.The next or fourth time we throw, half these winners will lose, cutting down the persistent winners to one and a quarter million. Remember, that the two and a half million had an equal chance to win or lose, and, therefore, half of them would lose.The rest of the mob gets into the intermediate class where luck varies back and forth, always getting nearer even. But notice out of our winners and losers, at every flip of the coin, half of them also join those whose luck has not stayed the same.
So we go on matching, or throwing dice, or betting on any even chance.
At the fifth event, there will be 625,000 out of the ten million first winners who have won every time, and an equal amount of losers on the other side.
REMEMBERING always, that facing the next event, the chances are equal, and that half of the people will win and half lose, we keep on matching, and at each event the number of each class is cut in half.
By the time we have matched those twenty million people twenty times there will be only twenty left out of them, or one out of a million, who has won every time, and only twenty who have lost every time.
But according to the laws of probability it had to happen. There was no way to make the chances different. There were two, but only two things that could happen, and each had an equal chance of happening.All right, then you say the twenty were lucky. But hold a minute. The very next time you match, half of those twenty will lose, and the next time half of the ten, and then half of the five and so on.And the same change, with reverse English, will be talking place with the losers, and with the middle boys whom we’ve forgotten.
So, it must have been set from the beginning. In fact, it could have been worked out on paper at the start, and proved that what happened would happen.Perhaps you now say, the personality of the player comes in. He is just a lucky man. Very well, let us examine that statement.
You and I are two of those lucky men, and we now face each other with a coin, drawn and loaded and ready to match. One of us must lose. The fact that I have won twenty times might prove that up to that point I was lucky, but there is nothing in the laws of nature that will make us both win, so I am now facing exactly a fifty-fifty chance, and no more, of being lucky again. You have the same chance.
PEOPLE who take chances never seem to reason this out, and they never stop to look at the greatest contradiction in their reasoning about the matter. It was bound to happen like it did before the men began matching.
A man will watch another play and see that the man is lucky. The observer will figure he is a good man to follow, and will bet that he wins. He doesn’t stop to think that the man has only a fifty-fifty chance of winning next time. He follows him.
But the same observer will do this: he will see a long run of numbers come up lucky for a player and he will figure the man has won twenty times and his luck can’t last. He’ll play against him, because the more he wins the surer he is to lose on each succeeding throw.
He has contradicted himself right there. Even though the player has a fifty-fifty chance to win next time, the observer will bet with him because he has won before, or bet against him because he has won so many times before. And the simple truth is that neither of these past performances has anything to do with what the next throw will be. It will be exactly a fifty-fifty throw. It always has been and always will be.