“Think what a chance thou changest on.”
Cymbeline. Act I., Scene 51h.
LET US consider briefly what are the chances for each different kind of hand at poker.
First, the total number of ways in which a set of five cards can be formed out of a pack containing 52 cards has to be determined. This is easy enough. You multiply together 52, 51, 50, 49, and 48, and divide the product by that obtained from multiplying together 1, 2, 3, 4, and 5. You thus get 2,598,960 as the total number of poker hands.
It is very easy to determine the number of flushes and sequences and flush sequences which are possible.
Thus, begin with the flush sequences. We can have in each suit, ace, 2, 3, 4, 5; 2, 3, 4, 5, 6; 3, 4, 5, 6, 7; and so on up to 10, knave, queen, king, ace; or in all there are ten flush sequences in each suit, forty flush sequences in all.
The number of sequences which are not flush may be thus determined. The arrangement of numbers may be any one of the ten just indicated. But taking any one of these, as 3, 4, 5, 6, 7, the three may be of any suit out of the four; so that each arrangement maybe obtained in four different ways as respects the first card; so with the second, third, etc.; or in all 4 times 4 times 4 times 4 times 4, or 1,024, four of which only will be flushes. Thus there are 1,020 times 10, or 10,200 sequences which are, not flush.
Now as respects flushes their number is very easily determined. The number of combinations of five cards which can be formed out of the 13 cards of a suit are given by multiplying together 13, 12, 11, 10, and 9, and dividing by the product of 1, 2, 3, 4, 5; this will be found to be 1,287. Thus there are 4 times 1,287, or 5,148 possible flushes. Of these, 5,108 are not sequence flushes.
The total number of “four” hands may be considered next. The process for finding it is very simple. There are of course only 13 fours, each of which can be taken with any one of the remaining 48 cards; so that there are 13 times 48, or 624 possible four hands.
Next to determine the number of “full hands.” This is not difficult, but requires a little more attention. A full hand consists of a triplet and a pair. Now manifestly there are four triplets of each kind - four sets of three aces, four of three kings, and so forth (for we may take each ace from the four aces in succession, leaving in each case a different triplet of aces; and so with the other denominations). Thus in all, 4 times 13, or 52 different triplets can be formed out of the pack of 52 cards. When one of these triplets has been formed there remain 49 cards, out of which the total number of sets of two which can be formed is obtained by multiplying 49 by 48 and dividing by 2; whence we get 1,176 such combinations in all. But the total number of pairs which can be formed from among these 49 cards is much smaller. There are four twos, which (as cribbage teaches us) will give six pairs of twos; so there are six pairs of threes, six pairs of fours, and so on; or as there are only twelve possible kinds of pairs (after our triplet removed), there are in all 6 times 12, that is 72, possible pairs which can with the triplet form a full hand. Hence, as there are 52 possible triplets, the total number of full hands is 52 times 72, or 3,744.
The number of triplet hands which are not also fours or fulls (for every four hand contains triplets) follows at once from the above. There are 52 possible triplets, each of which can be combined with 1,176 combinations of two cards out of the remaining 49, giving in all 52 times 1,176, or 61,152 sets of five, three at least of which are alike. But there are 624 four hands, each of which is not only a triplet hand, but will manifestly make four of the triplet hands our gross reckoning includes (for from every four you can make three triplets), and there are 3,744 full hands. These (to wit, 4,496 fours, and 5,744 fulls, or 6,240 hands in all) must be removed from our count, leaving 54,912 triplet hands (proper) in all.
This last result might have been obtained another way, which (as I shall use it for counting pair hands) I may as well indicate here. Taking any triplet of the 52, there remain 49 cards, one of which is of the same denomination as the triplet. Removing.,this, there are left 48 cards, out of which the number of sets of two which can be formed is obtained by multiplying 48 by 47 and dividing by 2; it is, therefore, 1,128, and among these 72 are pairs. There remain then 1,056 sets of two, any one of which can be combined with each of 52 triplets to give a triplet hand pure and simple. Thus, in all, there are 52 times 1,056 triplet hands, or 54,912, as before.
Next for double and single pairs.
From the whole pack of 52 cards we can form 6 times 13 pairs; for 6 aces can be formed, 6 pairs of twos, 6 pairs of threes, and so forth. Thus there are in all 78 different pairs. When we have taken out any pair, there remain 50 cards. From these we must remove the two cards of the same denomination, as neither or both of these must not appear in the hand to be formed. There remain 48 cards, from which we can form 72 other pairs. Each of these can be taken with any one of the 46 remaining cards, except with those two which are of the same denomination, or with 44 in all, without forming a triplet. Each of these combinations can be taken with each of the 78 pairs, giving a two pair hand, only it is obvious that each two-pair hand will be given twice by this arrangement. Thus the total number of two-pair hands is half of 78 times 72 times 44, or there are 123,552 such hands in all.
Next as to simple pairs. We get, as before, 78 different pairs. Each of these can be taken with any set of three formed out of the 48 cards left when the other 2 of the same denomination have been removed, except the 72 times 44 (that is 3,16
pairs indicated in dealing with the last case, and the 48 triplets which can be formed out of these same 48 cards, or 3,216 sets in all. Now the total number of sets of three cards which can be formed out of 48 is given by multiplying 48 by 47 by 46, and dividing by the product of the numbers 1, 2, and 3. It is found to be 17,296. We diminish this by 3,216, getting 14,082, and find that there are in all 78 times 14,082, or 1,098, 240.
The hands which remain are those which are to be estimated by the highest card in them; and their number will of course be obtained by subtracting the sum of. the numbers already obtained from the total number of possible hands. We thus obtain the number 1,302,540.
Thus of the four best classes of hands, there are the following numbers:
Of flush sequences there may be 40
Of four 624
Of full hands 3,744
Of common flushes 5,108
Of common sequences 10,200
Of triplets 54,912
Of two pairs 123,552
Of pairs 1,098,240
Of other hands 1,302,540
Total number of possible hands 2,598,960
It will be seen that those who devised the rules for poker play set the different hands in very proper order. It is fitting, for instance, that as there are only 40 possible flush sequence hands out of a total number of 2,598,960 hands, while there are 624 “four” hands, the flush sequences should come first, and so with the rest. It is noteworthy, however, that when sequences were not counted, as was the rule in former times, there was one hand absolutely unique and unconquerable. The holder of four aces then wagered on a certainty, for no one else could hold that hand. At present there is no absolutely sure winning hand. The hold of ace, king, queen, knave, ten, flush may (though it is of course exceedingly unlikely) be met by the holder of the same cards, flush, in another suit. Or when we remember that at whist it has happened that the deal divided the four suits among the four players, to each a complete suit, we see that four players at poker might each receive a flush sequence headed by the ace. Thus the use of sequences has saved poker players from the possible risk of having either to stand out or wager on a certainty, which last would of course be very painful to the feelings of a professional gambler.
We might subdivide the hands above classified into a much longer array, beginning thus: 4 flush sequences headed by ace; 4 headed by king, and so on down to 4 headed by five; 48 possible four-aces hands; 48 four-kings hands; and so on down to 48 four-twos hands; 24 possible “fulls” of 3 aces and 2 kings; as many of 3 aces and 2 queens; and so on down to 24 “fulls” of 3 twos and 2 threes, and so on. Any one who cares to do this can, by drawing the line at any hand., ascertain at once the number of hands above and not above that hand in value; and thus determine the chance that any hand taken at random is above or below that particular hand in value. The comparatively simple table above only shows how many hands there are above or not above pairs, triplets, and the like. But the more complete series could be very easily formed.
We note from the above table that more than half the possible poker hands are below pairs in value. So that Clay was right enough in wagering on an ace-high hand, seeing that there are more hands which will not beat it (supposing the highest next card a king, at any rate) than there are hands that will; but he was quite wrong in calling on such a hand, even against a single opponent.
The effect of increase in the number of hands can also readily be determined. Many, even among gamblers, know so little of the doctrine of chances as not to be aware of, still less to be able to measure the effect of, the presence of a great number of other contestants. Yet it is easy to illustrate the matter.
Thus, suppose a player casts a die single against one other. If the first has cast four, the odds are in favor of his not being beaten; for there are only two casts which will beat him and four which will not. The chance that he will not be beaten by a single opponent is thus 4/6 or 2/3. If there is another opponent, the chance that he individually will not cast better than 4 is also 2/3. But the chance that neither will throw better than 4 is obtained by multiplying 2/3 by 2/3. It is therefore 4/9; or the odds are 5 to 4 in favor of one or other beating the cast of the first thrower. If there are three others, in like manner the chance that not one of the three will throw better than 4 is obtained by multiplying 2/3 by 2/3 by 2/3. It is, therefore, 8/27; or the odds are 19 to 8 in favor of the first thrower’s cast of four being beaten. And so with every increase in the number, of other throwers, the chance of the first thrower’s cast being beaten is increased. So that if the first thrower casts 4, and is offered his share of the stakes before the next throw is made, the offer is a bad one if there is but one opponent, a good one if there are two, and a very good one if there are more than two.
In like manner, the same hand which it would be safe to stand on (as a rule) at poker against two or three opponents may be a very unsafe hand to stand on against five or six.
Then the player has to consider the pretty chance-problems involved in drawing.
Suppose, for instance, your original hand contains a pair - the other three cards being all unlike; should you stand out? or should you draw? (to purchase right to which you must stand in); or should you stand in without drawing? Again, if you draw, how many of the three loose cards should you throw out? and what are your chances of improving your hand?
Here you have to consider first whether you will stand in, which depends not on the value of your pair only, but also on the chance that your hand will be improved by drawing. Having decided to stand in, remember that discarding three tells the rest of the company that in all probability you are drawing to improve a pair hand; and at poker, telling anything helps the enemy. If one of your loose cards is an ace, you do well to discard only the other two; for this looks like drawing to a triplet, and you may chance to draw a pair to your ace. But usually you have so much better chance of improving your hand by drawing three, that it is, as a rule, better to do this.
Drawing to a triplet is usually good policy. “Your mathematical expectation of improvement is slight,” says one work on the subject, “being 1 to 23 of a fourth card” (it should be the fourth card) “of the same denomination, and 2 to 23 of another pair of denomination different from the triplet,” a remark suggesting the comment that to obtain a pair of the same denomination as the triplet would require play something like what we hear of in old Mississippi stories, where a “straight flush” would be met by a very full pair of hands, to wit, five in one hand and a revolver in the other! The total expectation of improvement is 1 to 8; but then see what an impression you make by a draw which means a good hand. Then, too, you may suggest a yet better hand, without much impairing your chance of improvement, by drawing one card only. This gives you one chance in 47 of making fours, and one in 16 of picking up one of the three cards of the same denomination as the odd cards you retain. This is a chance of one in 12.
“Draws to straights and flushes are usually dearly purchased,” says our oracle; “always so at a small table. Their value increases directly as the number of players.” (The word “directly” is here incorrectly used; the value increases as the number of players, but not directly as the number.) Of course in drawing to a two-ended straight – that is, one which does not begin or end with an ace – the chance of success is represented by 8 in 47, for there are 47 cards outside your original hand, of which only eight are good to complete the straight. For a one-end straight the chance is but 4 in 47. With a small chance, too, of improving your hand, you are trying for a hand better than you want in any but a large company. “If you play in a large party,” says one authority, “say seven or eight, and. find occasion to draw for a straight against six players, do so by all means, even if you split aces.” The advice is sound. Under the circumstances you need a better hand than ace-pair to give you your fair sixth share of the chances.
As to flushes your chances are better, when you have already four of a suit. You discard one, and out of the remaining 47 cards any one of nine will make your flush for you. Your chance is 1 in 5 2/9. In dealing with this point our oracle goes altogether wrong, and adopts a principle so inconsistent with the doctrine of probabilities as to show that, though he knows much more than Steinmetz, he still labors under somewhat similar illusions. “Theoretically,” says he, “the result just obtained is absolutely true; but I have experimented with six hands through a succession of 500 deals, and filled only 83 flushes in the 500, equal to one in six and one-twentieth draws. Of course I am not prepared to say that this would be the average in many thousand deals; theoretically it is an untrue result; but I here suggest a possible explanation of what I confess is to me a mystery.” Then he expounds the very matter on which we touched above. “In casting dice,” he says, “theoretically, any given throw has no influence upon the next throw, and is not influenced by the previous throw. Yet if you throw a die and it turns up six, while the chances are theoretically one to six” (one in six it should be) “that the next throw will produce a six because the previous throw of six lies absolutely in the past, yet you may safely bet something more than the usual odds against it. Then suppose the second throw turns up a six, that throw also now lies in the past, and cannot be proved to have an influence upon throw number three, which you are preparing to make. If any material influence is suspected, you may change the box and die; and you may now bet twice the usual odds against the six. Why? Because you know by experience that it is extremely difficult to throw six three times in succession, even if you do not know the precise odds against it. Granted, certain odds against throwing six twice in succession, etc., yet at any given moment when the player shakes the box in which is a six-faced die, he has one chance in six of throwing a six; and yet if he has just thrown sixes twice, you may bet twelve to one that he will not throw a six in that particular cast.” If I did not hold gambling to be near akin to swindling, and could find but a few hundred who held this doctrine, how much money might I not gain by accepting any number of wagers of this wise sort!
The fact is, the mistake here is just the ridiculous mistake which Steinmetz called “the maturity of the chances” over again. It is a mistake which has misled to their ruin many thousands of gamblers, who might have escaped the evil influence of that other equally foolish mistake about being lucky or unlucky, in the vein or out of it. Steinmetz puts the matter thus: “In a game of chance, the oftener the same combination has occurred in succession, the nearer are we to the certainty that it will not recur at the next cast or turn-up: this is the most elementary of the theories on probabilities; it is termed the maturity of the chances.” The real fact being that this is not a theory of probabilities at all, but disproved by the theory of probabilities, and disproved, whenever it has been put to the test, by facts.
Take the case considered in “The Complete Poker Player,” and note the evidence on the strength of which the author of that work rejects the theory in favor of a practical commonsense notion (as he thinks), which is, in reality, nonsense. You may expect 9 successful draws to a flush in 47 hands; therefore in the 500 deals lie experimented upon, he might have expected 95 or 96; and he only obtained 83. Now 500 trials are far too few to test such a matter as this. You can hardly test even the tossing of a coin properly by fewer than a thousand trials; and in that case there are but 2 possible events. Here there are 47, of which 9 are favorable. It is the failure to recognize this which led the Astronomer Royal for Scotland to ‘recognize something mystical and significant in the preponderance of threes and the deficiency of sevens among the digits representing the proportion of the circumference to the diameter of a circle. In casting a coin a great number of times we do not find that the occurrence of a great number of successive heads or tails in anyway affects the average proportion of heads or tails coming next after the series. Thus I have before me the record of a series of 16,317 tossings, in which the number of sequences of tails (only) were rendered; and I find that after 271 cases, in which tails had been tossed 5 times in succession, the next tossing gave in 132 cases heads, and in 139 cases tails. Among the 16,317 tossings, two cases occurred in which tail was tossed 15 times in succession, which, as it happens, is more than theory would regard as probable.
Here, however, I must draw these notes to a close. I have been already led on farther than I had intended to go. I shall note only one other of the doctrines (mostly sound enough theoretically) laid down in “The Complete Poker Player.” “Players sometimes,” he says, “act on the strange principle that if they are in bad luck it is well to try the bold experiments usually regarded as bad play – as two negatives in algebra make a positive, so they think that bad play and bad luck united will win.” On this our author makes the significant comment, a slight degree of intoxication aids to perfect this intellectual deduction.” Poker playing generally, as a process for making money more quickly, is much improved and enlivened by a slight degree of intoxication.
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