Diagram of the shots described
No. 1 is the perfect position for the spot stroke; the dotted lines in the others show the course that must be followed by the cue ball to recover the initial position.
Man-of-war Gameis a variety of English billiards in which there are three white balls, each belonging to different players.
The followingLAWSare taken, by permission, from the rules published by the Brunswick-Balke-Collender Co.
1.The choice of balls and order of play shall, unless mutually agreed upon by the two players, be determined by stringing; and the striker whose ball stops nearest the lower [or bottom] cushion, after being forced from baulk up the table, may take which ball he likes, and play, or direct his opponent to play first, as he may deem expedient.
2.The red ball shall, at the opening of every game, be placed on the top [or red] spot, and replaced after being pocketed or forced off the table, or whenever the balls are broken.
3.Whoever breaks the balls, i.e., opens the game, must play out of baulk, though it is not necessary that he shall strike the red ball.
4.The game shall be adjudged in favour of whoever first scores the number of points agreed on, when the marker shall call “game”; or it shall be given against whoever, after having once commenced, shall neglect or refuse to continue when called upon by his opponent to play.
5.If the striker scores by his stroke he continues until he ceases to make any points, when his opponent follows on.
6.If when moving the cue backward and forward, and prior to a stroke, it touches and moves the ball, the ball must be replaced to the satisfaction of an adversary, otherwise it is a foul stroke; but if the player strikes, and grazes any part of the ball with any part of the cue, it must be considered a stroke, and the opponent follows on.
7.If a ball rebounds from the table, and is prevented in any way, or by any object except the cushion, from falling to the ground, or if it lodges on a cushion and remains there, it shall be considered off the table, unless it is the red, which must be spotted.
8.A ball on the brink of the pocket need not be “challenged”: if it ceases running and remains stationary, then falls in, it must be replaced, and the score thus made does not count.
9.Any ball or balls behind the baulk-line, or resting exactly upon the line, are not playable if the striker be in hand, and he must play out of baulk before hitting another ball.
10.Misses may be given with the point or butt of the cue, and shall count one for each against the player; or if the player strike his ball with the cue more than once a penalty shall be enforced, and the non-striker may oblige him to play again, or may call on the marker to place the ball at the point it reached or would have reached when struck first. [The butt may also be used for playing a ball in hand up the table in order to strike a ball in baulk.]
11.Foul strokes do not score to the player, who must allow his opponent to follow on. They are made thus: By striking a ball twice with the cue; by touching with the hand, ball, or cue an opponent’s or the red ball; by playing with the wrong ball; by lifting both feet from the floor when playing; by playing at the striker’s own ball and displacing it ever so little (except while taking aim, when it shall be replaced, and he shall play again).
12.The penalty for a foul stroke is losing the lead, and, in case of a score, an opponent must have the red ball spotted, and himself break the balls, when the player who made the foul must follow suit, both playing from the D. If the foul is not claimed the player continues to score, if he can.
13.After being pocketed or forced off the table the red ball must be spotted on the top spot, but if that is occupied by another ball the red must be placed on the centre spot between the middle pockets.
14.If in taking aim the player moves his ball and causes it to strike another, even without intending to make a stroke, a foul stroke may be claimed by an adversary. (See Rule Fifteenth.)
15.If a player fail to hit another ball, it counts one to his opponent; but if by the same stroke the player’s ball is forced over the table or into any pocket it counts three to his opponent.
16.Forcing any ball off the table, either before or after the score, causes the striker to gain nothing by the stroke.
17.In the event of either player using his opponent’s ball and scoring, the red must be spotted and the balls broken again by the non-striker; but if no score is made, the next player may take his choice of balls and continue to use the ball he so chooses to the end of the game. No penalty, however, attaches in either case unless the mistake be discovered before the next stroke.
18.No person except an opponent has a right to tell the player that he is using the wrong ball, or to inform the non-striker that his opponent has used the wrong ball; and if the opponent does not see the striker use the ball, or, seeing him, does not claim the penalty, the marker is bound to score to the striker any points made.
19.Should the striker [whose ball is in hand], in playing up the table on a ball or balls in baulk, either by accident or design, strike one of them [with his own ball] without first going out of baulk, his opponent may have the balls replaced, score a miss, and follow on; or may cause the striker to play again, or may claim a foul, and have the red spotted and the balls broken again.
20.The striker, when in hand, may not play at a cushion within the baulk (except by going first up the table) so as to hit balls that are within or without the line.
21.If in hand, and in the act of playing, the striker shall move his ball with insufficient strength to take it out of baulk, it shall be counted as a miss to the opponent, who, however, may oblige him to replace his ball and play again. [Failing to play out of baulk, the player may be compelled to play his stroke over again.]
22.If in playing a pushing stroke the striker pushes more than once it is unfair, and any score he may make does not count. His opponent follows by breaking the balls.
23.If in the act of drawing back his cue the striker knocks the ball into a pocket, it counts three to the opponent, and is reckoned a stroke.
24.If a foul stroke be made while giving a miss, the adversary may enforce the penalty or claim the miss, but he cannot do both.
25.If either player take up a ball, unless by consent, the adversary may have it replaced, or may have the balls broken; but if any other person touches or takes up a ball it must be replaced by the marker as nearly as possible.
26.If, after striking, the player or his opponent should by any means obstruct or hasten the speed of any ball, it is at the opponent or player’s option to have them replaced, or to break the balls.
27.No player is allowed to receive, nor any bystander to offer advice on the game; but should any person be appealed to by the marker or either player he has a right to offer an opinion; or if a spectator sees the game wrongly marked he may call out, but he must do so prior to another stroke.
28.The marker shall act as umpire, but any question may be referred by either player to the company, the opinion of the majority of whom shall be acted upon.
The game of Pin Pool is played with two white balls and one red, together with five small wooden pins, which are set up in the middle of the table, diamond fashion, each pin having a value to accord with the position it occupies.
Diagram of the layout of the pins
The pin nearest the string line is No. 1; that to the right of it is No. 2; to the left, No. 3; the pin farthest from the string line is No. 4; and the central or black pin, No. 5. These numbers may be chalked on the cloth in front of each particular pin.
Neither carroms nor hazards count; for pocketing a ball (when playing on a pocket table), or causing it to jump off the table or lodge on the cushion, or for missing altogether, nothing is forfeited other than the stroke. The only penalty is that the ball so offending shall be spotted upon the white-ball spot at the foot of the table, or if that be occupied then on the nearest spot thereto unoccupied.
When the pins are arranged, the rotation of the players is determined in like manner as in Fifteen-Ball Pool, after which each player receives from the marker a little numbered ball which is placed in the player’s cup on the pool board, and the number of which is not known to any of his opponents.
The object of the player is to knock down as many pins as will count exactly thirty-one when the number on the small ball held by him is added to their aggregate; thus, if the small ball is No. 9, the player will have to gain twenty-two points on the pins before calling game, and whoever first gets exactly thirty-one points in this manner wins the pool.
A white ball is spotted five inches from the lower end of the table, on a line drawn down the centre; and the red ball placed upon its own spot at the foot of the table.
Player No. 1 must play with the remaining white ball from any point within the string-line at the head of the table at either the red or white ball, or place his own on the string spot. Player No. 2 may play with any ball on the table—red or white. After the first stroke has been played, the players, in their order, may play with or at any ball upon the board.
Unless the player has played on some ball upon the board before knocking down a pin, the stroke under all circumstances goes for nothing, and the pin or pins must be replaced and the player’s ball put upon the white-ball spot at the foot of the table or if that be occupied, on the nearest unoccupied spot thereto. But should two balls be in contact the player can play with either of them, direct at the pins, and any count so made is good.
If a player, with one stroke, knocks down the four outside pins and leaves the black one standing on its spot, it is called a Natural, orRanche, and under any and all circumstances it wins the game.
When a player gets more than 31, he isburst, and he may either play again immediately with the same ball he has in the pool rack, starting at nothing of course, or he may take a new ball. If he takes a new ball he may either keep it or keep his old one, but he cannot play again until it comes to his turn.
This game is the regular Three-Ball Carrom Game with a small pin added, like those used in Pin Pool, which is set up in the centre of the table. The carroms and forfeits count as in the regular Three-Ball Game, but the knocking down of the pin scores five points for the striker, who plays until he fails to effect a carrom or knock down the pin. A ball must be hit by the cue-ball before the pin can be scored; playing at the pin direct is not allowed. The pin must be set up where it falls; but in case it goes off the table or lodges on the top of the cushion it must be placed upon the centre spot. The pin leaning against the cushion must be scored as down, and when the pin lodges in the corner of the table, so that it cannot be hit with the ball, it is to be set up on the centre spot. One hundred points generally constitute a game, but any number of points may be agreed upon.
This game is played in the South, California, and in Mexico and Cuba, and is played with two white and one red ball, and five pins placed similar to those in Pin Pool. The red ball is placed on the red-ball spot, and the first player strikes at it from within the baulk semicircle. The game is scored by winning and losing hazards, carroms, and by knocking over the pins. It is usually played thirty points up.
The player who knocks down a pin after striking a ball gainstwopoints, if he knocks down two pins he gainsfourpoints, and so on, scoring two points for each pin knocked down. If he knock down the middle pin alone he gainsfivepoints. The player who pockets the red ball gainsthreepoints and two for each pin knocked down by the same stroke. The player who pockets the white ball gains two points, and two for each pin knocked over with the same stroke. Each carrom counts two. The player who knocks down a pin or pins with his own ball before striking another ball loses two for every pin so knocked down. The player who pockets his own ball without hitting another ball forfeits three points; for missing altogether he forfeits one point. The striker who forces his own ball off the table without hitting another ball forfeitsthreepoints, and if he does so after making a carrom or pocket he loses as many points as he would otherwise have gained. The rules of the American Carrom Game, except where they conflict with the foregoing rules, govern this game also.
The game of Bottle Pool is played on a pool table with one white ball, the 1 and 2 ball, and pool-bottle. The 1 and 2 balls must be spotted, respectively, at the foot of the table, at the left and right diamond nearest each pocket, and the pool-bottle is placed standing on its neck on the spot in the centre of the table, and when it falls it must be set up, if possible, where it rests.
Carrom on the two object-balls counts 1 point; Pocketing the 1 ball counts 1 point; Pocketing the 2 ball counts 2 points; Carrom from ball and upsetting bottle counts 5 points. The game consists of 31 points. The player having the least number of points at the finish of the game shall be adjudged the loser.
Any number of persons can play, and the rotation of the players is decided as in ordinary pool. Player No. 1 must play with thewhite ball from any point within the string at the head of the table, at either the 1 or 2 ball at his option. The player who leads must play at and strike one of the object-balls before he can score a carrom on the pool-bottle. If a player carrom on the bottle from either of the object-balls, in such a way as to seat the bottle on its base, he wins the game, without further play.
Should the 1 or 2 ball in any way, during the stroke, touch the bottle and the bottle is in the same play knocked over or stood on its base by the cue-ball, the stroke does not count. If the player forces the bottle off the table or into a pocket, the bottle must be spotted on its proper spot in the centre of the table, the player loses his shot and forfeits one point, and the next player plays.
A player who makes more than 31 points is burst, and must start his string anew; all that he makes in excess of 31 points count on his new string, and the next player plays.
American Game:—
English Game:—
In calculating the probability of any event, the difficulty is not, as many persons imagine, in the process, but in the statement of the proposition, and the great trouble with many of those who dispute on questions of chance is that they are unable to think clearly.
The chance is either for or against the event; the probability is always for it. The chances are expressed by the fraction of this probability, the denominator being the total number of events possible, and the numerator the number of events favourable. For instance: The probability of throwing an ace with one cast of a single die is expressed by the fraction ⅙; because six different numbers may be thrown, and they are all equally probable, but only one of them would be an ace. Odds are found by deducting the favourable events from the total, or the numerator from the denominator. In the example, the odds against throwing an ace are therefore 5 to 1. The greater the odds against any event the moreimprobableit is said to be, and the morehazardousit is to risk anything upon it.
When an event happens which is very improbable, the person to whom it happens is consideredlucky, and the greater the improbability, the greater his luck. If two men play a game, the winner is not considered particularly lucky; but if one wanted only two points to go out and the other wanted a hundred, the latter would be a very lucky man if he won.
It is a remarkable fact that luck is the only subject in the world on which we have no recognised authority, although it is a topic of the most universal interest. Strictly speaking, to be lucky simply means to be successful, the word being a derivative ofgelingen, to succeed. There are a few general principles connected with luck which should be understood by every person who is interested in games of chance. In the first place, luck attaches to persons and not to things. It is useless for an unlucky man to change the seats or the cards, for no matter which he chooses the personal equation of good or bad luck adhering to him for the time being cannot be shaken off. In the second place, all men are lucky in some things, and not in others; and they are lucky or unlucky inthose things at certain times and for certain seasons. This element of luck seems to come and go like the swell of the ocean. In the lives of some men the tide of fortune appears to be a long steady flood, without a ripple on the surface. In others it rises and falls in waves of greater or lesser length; while in others it is irregular in the extreme; splashing choppy seas to-day; a storm to-morrow that smashes everything; and then calm enough to make ducks and drakes with the pebbles on the shore. In the lives of all the tide of fortune is uncertain; for the man has never lived who could be sure of the weather a week ahead. In the nature of things this must be so, for if there were no ups and downs in life, there would be no such things as chance and luck, and the laws of probability would not exist.
The greatest fallacy in connection with luck is the belief that certain menarelucky, whereas the truth is simply that theyhave beenlucky up to that time. They have succeeded so far, but that is no guarantee that they will succeed again in any matter of pure chance. This is demonstrated by the laws governingthe probability of successive events.
Suppose two men sit down to play a game which is one of pure chance; poker dice, for instance. You are backing Mr. Smith, and want to know the probability of his winning the first game. There are only two possible events, to win or lose, and both are equally probable, so 2 is the denominator of our fraction. The number of favourable events is 1, which is our numerator, and the fraction is therefore ½, which always represents equality.
Now for the successive events. Your man wins the first game, and they proceed to play another. What are the odds on Smith’s winning the second game? It is evident that they are exactly the same as if the first game had never been played, because there are still only two possible events, and one of them will be favourable to him. Suppose he wins that game, and the next, and the next, and so on until he has won nine games in succession, what are the odds against his winning the tenth also? Still exactly an even thing.
But, says a spectator, Smith’s luck must change; because it is very improbable that he will win ten games in succession. The odds against such a thing are 1023 to 1, and the more he wins the more probable it is that he will lose the next game. This is what gamblers call thematurity of the chances, and it is one of the greatest fallacies ever entertained by intelligent men. Curiously enough, the men who believe that luck must change in some circumstances, also believe in betting on it to continue in others. When they arein the veinthey will “follow their luck” in perfect confidence that it will continue. The same men will not bet on another man’s luck, even if he is “in the vein,” because “the maturity of the chances” tells them that it cannot last!
GAMES.ODDS.One1 to 1Two3 to 1Three7 to 1Four15 to 1Five31 to 1Six63 to 1Seven127 to 1Eight255 to 1Nine511 to 1Ten1023 to 1
If Smith and his adversary had started with an agreement to play ten games, the odds against either of them winning any number in succession would be found by taking the first game as an even chance, expressed by unity, or 1. The odds against the same player winning the second game also would be twice 1 plus 1, or 3 to 1; and the odds against his winning three games in succession would be twice 3 plus 1, or 7 to 1, and so on, according to the figures shown in the margin.
GAMES.1st2nd11100100
That this is so may easily be demonstrated by putting down on a sheet of paper the total number of events that may happen if any agreed number of games are played, expressing wins by a stroke, and losses by a cipher. Take the case of two games only. There are four different events which may happen to Smith, as shown in the margin. He may win both games or lose both; or he may win one and lose the other, either first. Only one of these four equally probable events being favourable to his winning both games, and three being unfavourable, the odds are 3 to 1 that he does not win both; but these are the oddsbefore he begins to play. Having won the first game, there are only two events possible, those which begin with a win, and he has an equal chance to win again.
GAMES.1st2nd3rd111110101100000001010011
If the agreement had been to play three games, there would have been eight possible events, one of which must happen but all of which were equally probable. These are shown in the margin. If Smith wins the first game, there are only four possible events remaining; those in which the first game was won. Of these, there are two in which he may win the second game, and two in which he may lose it, showing that it is still exactly an even thing that he will win the second game. If he wins the second game, there are only two possible events, the first two on the list in the margin, which begin with two wins for Smith. Of these he has one chance to win the third game, and one to lose it. No matter how far we continue a series of successive events it will always be found that having won a certain number of games, it is still exactly an even thing that he will win the next also. The odds of 1023 to 1 against his winning ten games in succession existed only before he began to play. After he has won the first game, the odds against his winning the remaining nine are only 511 to 1, and so on, until it is an even thing that he wins the tenth, even if he has won the nine preceding it.
In the statistics of 4000 coups at roulette at Monte Carlo it was found that if one colour had come five times in succession, it was an exactly even bet that it would come again; for in twenty runs of five times there were ten which went on to six. In the author’s examination of 500 consecutive deals of faro, there were 815 cards that either won or lost three times in succession, and of these 412 won or lost out. In a gambling house in Little Rock a roulette wheel with three zeros on it did not come up green for 115 rolls, and several gamblers lost all they had betting on the eagle and O’s. When the game closed the banker informed them that the green had come up more than twenty times earlier in the evening. They thought the maturity of the chances would compel the green to come; whereas the chances really were that it would not come, as it had over-run its average so much earlier in the evening. The pendulum swings as far one way as the other, but no method of catching it on the turn has ever yet been discovered.
Compound Events.In order to ascertain the probability of compound or concurrent events, we must find the product of their separate probability. For instance: The odds against your cutting an ace from a pack of 52 cards are 48 to 4, or 12 to 1; because there are 52 cards and only 4 of them are aces. The probability fraction is therefore 1/13. But the probabilities of drawing an ace from two separate packs are 1/13 × 1/13 = 1/169, or 168 to 1 against it.
Suppose a person bets that you will not cut a court card, K Q or J, from a pack of 52 cards, what are the odds against you? In this case there are three favourable events, but only one can happen, and as any of them will preclude the others, they are calledconflicting events, and the probability of one of them is the sum of the probability of all of them. In this case the probability of any one event separately is 1/13, and the sum of the three is therefore 1/13 + 1/13 + 1/13 = 3/13; or ten to 3 against it.
In order to prove any calculation of this kind all that is necessary is to ascertain the number of remaining events, and if their sum, added to that already found, equals unity, the calculation must be correct. For instance: The probability of turning a black trump at whist is 13/52 + 13/52 = 26/52; because there are two black suits of 13 cards each. The only other event which can happen is a red trump, the probability of which is also 26/52, and the sum of these two probabilities is therefore 26/52 + 26/52 = 52/52, or unity.
Another fallacy in connection with the maturity of the chances is shown in betting against two successive events, both improbable, one of which has happened. The odds against drawing two aces in succession from a pack of 52 cards are 220 to 1; but after an ace has been drawn the odds against the second card being an ace also are only 16 to 1, although some persons would be madenough to bet 1000 to 1 against it, on the principle that the first draw was a great piece of luck and the second ace was practically impossible. While the four aces were in the pack the probability of drawing one was 4/52. One ace having been drawn, 3 remain in 51 cards, so the probability of getting the second is 3/51, or 1/17. Before a card was drawn, the probability of getting two aces in succession was the product of these fractions; 1/13 × 1/17 = 1/221. On the same principle the odds against two players cutting cards that are a tie, such as two Fours, are not 220 to 1, unless it is specified that the first card shall be a Four. The first player having cut, the odds against the second cutting a card of equal value are only 16 to 1.
Dice.In calculating the probabilities of throws with two or more dice, we must multiply together the total number of throws possible with each die separately, and then find the number of throws that will give the result required. Suppose two dice are used. Six different throws may be made with each, therefore 6 × 6 = 36 different throws are possible with the two dice together. What are the odds against one of these dice being an ace? A person unfamiliar with the science of probabilities would say that as two numbers must come up, and there are only six numbers altogether, the probability is 2/6, or exactly 2 to 1 against an ace being thrown. But this is not correct, as will be immediately apparent if we write out all the 36 possible throws with two dice; for we shall find that only 11 of the 36 contain an ace, and 25 do not. The proper way to calculate this is to take the chances against the ace on each die separately, and then to multiply them together. There are five other numbers that might come up, and the fraction of their probability is ⅚ × ⅚ = 25/36, or 25 to 11 in their favour.
Take the case of three dice: As three numbers out of six must come up, it might be supposed that it was an even thing that one would be an ace. But the possible throws with three dice are 6 × 6 × 6 = 216; and those that do not contain an ace are 5 × 5 × 5 = 125; so that the odds against getting an ace in one throw with three dice, or three throws with one die, are 125/216, or 125 to 91 against it.
To find the probability of getting a given total on the faces of two or three dice we must find the number of ways that the desired number can come. In the 36 possible throws with two dice there are 6 which will show a total of seven pips. The probability of throwing seven is therefore 6/36, or 5 to 1 against it. A complete list of the combinations with two dice were given in connection with Craps.
Poker.In calculating the probability of certain conflicting events, both of which cannot occur, but either of which would be favourable, we must make the denominator of our fraction equalin both cases, which will, of course, necessitate a proportionate change in our numerator. Suppose a poker player has three of a kind, and intends to draw one card only, the odds against his getting a full hand are 1/16; against getting four of a kind, 1/48. To find the total probability of improvement, we must make the first fraction proportionate to the last, which we can do by multiplying it by 3. The result will be 3/48 + 1/48 = 4/48; showing that the total chance of improvement is 1 in 12, or 11 to 1 against it.
Whist.To calculate the probable positions of certain named cards is rather a difficult matter, but the process may be understood from a simple example. Suppose a suit so distributed that you have four to the King, and each of the other players has three cards; what are the probabilities that your partner has both Ace and Queen? The common solution is to put down all the possible positions of the two named cards, and finding only one out of nine to answer, to assume that the odds are 8 to 1 against partner having both cards. This is not correct, because the nine positions are not equally probable. We must first find the number of possible positions for the Ace and Queen separately, afterward multiplying them together, which will give us the denominator; and then the number of positions that are favourable, which will give us the numerator.
As there are nine unknown cards, and the Ace may be any one of them, it is obvious that the Queen may be any one of the remaining eight, which gives us 9 × 8 = 72 different ways for the two cards to lie. To find how many of these 72 will give us both cards in partner’s hand we must begin with the ace, which may be any one of his three cards. The Queen may be either of the other two, which gives us the numerator, 3 × 2 = 6; and the fraction of probability, 6/72, = 1/12; or 11 to 1 against both Ace and Queen.
If we wished to find the probability of his having the Ace, but not the Queen, our denominator would remain the same; but the numerator would be the three possible positions of the Ace, multiplied by the six possible positions of the Queen among the six other unknown cards, in the other hands, giving us the fraction 18/72. The same would be true of the Queen but not the Ace. To prove both these, we must find the probability that he has neither Ace nor Queen. There being six cards apart from his three, the Ace may be any one of them, and the Queen may be any one of the remaining five. This gives us 6 × 5 = 30, and the fraction 30/72. If we now add these four numerators together, we have:—for both cards in partner’s hand, 6; for Ace alone, 18; for Queen alone, 18; and for neither, 30; a total of 72, or unity, proving all the calculations correct.
In some of the problems connected with Whist, it is important to know the probability of the suits being distributed in various ways among the four players at the table; or, what is the samething, the probable distribution of the four suits in any one hand. The author is indebted to Dr. Pole’s “Philosophy of Whist” for these calculations. As an example of the use of this table, suppose it was required to find the probability of any other player at the table holding four or more trumps if you had six. Take all the combinations in which the figure 6 appears, and add together the number of times they will probably occur. That will be your denominator, 166. The numerator will be the number of times that the combinations occur which contain a figure larger than 3, in addition to the 6. This will be found to be 74, and the probability will therefore be 74/166.
DISTRIBUTIONS.TIMES IN1000.8221283111½832018410½850007222573211973303741147420375101760006322576331356421476430136511765206661015332155542210654311305440125521325530943331054432215444130
MARTINGALES.Many gamblers believe that as the science of probabilities teaches us that events will equalise themselves in time, all that is necessary is to devise some system that will keep a person from guessing, so that he may catch the pendulum as it swings; and to add to it some system of betting, so that he will have the best of it in the long run. Some content themselves with playing a “system” against banking games, which is merely a guide to the placing of the bets, the simplest example of which would be to bet always on heads if a coin was tossed a thousand times, or to bet on nothing but red at Roulette. Others depend more on martingales, which are guides to the amount of the bets themselves, irrespective of what they are placed on.
The most common form of martingale is calleddoubling up, which proceeds upon the theory that if you lose the first time and bet double the amount the next time, and continue to double until you win, you must eventually win the original amount staked. If there was no end to your capital, and no betting limit to the game, this would be an easy way to make money; but all banking games have studied these systems, and have so arranged matters that they can extend their heartiest welcome to those who play them.
In the first place, by simply doubling up you are giving the bank the best of it, because you are not getting the proper odds. If you double up five times you are betting 16 to 1; but the odds against five successive events are 31 to 1, as we have already seen, and thebank should pay you 31 instead of 16. You should not only double, but add the original amount of the stake each time, betting 1, 3, 7, 15, 31, 63, and so on. If you do this, you will win the amount of your original stake for every bet you make, instead of only for every time you win. This looks well, but as a matter of fact doubling up is only another way of borrowing small sums which will have to be paid back in one large sum when you can probably least afford it.
Suppose the game is Faro, the chips five dollars a stack, and the limit on cases twenty-five dollars. The limit on cases will then be 400 chips. If eight successive events go against your “system,” which they will do about once in 255 times, your next bet will be beyond the limit, and the banker will not accept it. At Monte Carlo the smallest bet is a dollar, and the limit is $2,400. They roll about 4,000 coups a week, and if you were to bet on every one of them, doubling up, you would win about $1,865, one dollar at a time, and would lose $4,092 simply through being unable to follow your system beyond the limit of the game during the two or three occasions, in the 4,000 coups, that your system would go against you for eleven or more coups in succession. It is useless to say it would not go against you so often, for probabilities teach us that it would be more wonderful if it did not than if it did.
It must never be forgotten that the most wonderful things that happen are not more wonderful than those that don’t happen. If you tossed a coin a thousand times, and did not once toss heads eight times in succession, it would be four times more surprising than if you tossed heads ten times in succession.
BetsWon.Lost.10-9-8--7-89--8-910--94641
Progression.This is a favourite martingale with those who have not the courage or the money to double up. It consists in starting with a certain amount for the first bet, say ten dollars, and adding a dollar every time the bet is lost, or taking off a dollar every time a bet is won. If the player wins as many bets as he loses, and there is no percentage against him, he gets a dollar for every bet he wins, no matter how many bets he makes, or in what order the bets are won and lost, so that the number won equals the number lost. That this is so may be easily demonstrated by setting down on a sheet of paper any imaginary order of bets, such as the ten shown in the margin, five of which are won, and five lost; the net profit on the five bets won being five dollars. No matter how correctly the player may be guessing, and how much the luck runs his way, he wins smaller and smaller amounts, until at last he is “pinched off.” But if a long series of events goes against him his bets become larger andlarger, but he must keep up the progression until he gets even. If ten bets go his way he wins $55; if ten go against him he loses $145.
It is said that Pettibone made a fortune playing progression at Faro, which is very likely, for among the thousands of men who play it the probabilities are that one will win all the time, just as the probabilities are that if a thousand men play ten games of Seven Up, some man will win all ten games. At the same time it is equally probable that some man will lose all ten.
Some players progress, but never pinch, keeping account on a piece of paper how many bets they are behind, and playing the maximum until they have won as many bets as they have lost. Against a perfectly fair game, with no percentage and no limit, and with capital enough to follow the system to the end, playing progression would pay a man about as much as he could make in any good business with the same capital and with half the worry; but as things really are in gambling houses and casinos, all martingales are a delusion and a snare. It is much better, if one must gamble, to trust to luck alone, and it is an old saying that the player without a system is seldom without a dollar. It is the men with systems who have to borrow a stake before they can begin to play.
Such matters as calculating the probability of a certain horse getting a place, the odds against all the horses at the post being given, would be out of place in a work of this kind; but those interested in such chances may find rules for ascertaining their probability in some of the following text books.
The standard American game of Ten Pins is played upon analley41 or 42 inches wide, and 60 feet long from the head pin to the foul or scratch line, from behind which the player must deliver his ball. There should be at least 15 feet run back of the foul line, and the gutters on each side of the alley must be deep enough to allow a ball to pass without touching any of the pins standing on the alley.