Category: MATHEMATICS OF GIN RUMMY

Since gin rummy is a game of percentages and odds, an expert player takes advantage of any odds that favor him. However, there is no rule of thumb in the game that is so rigid that you must adhere to it regardless of any other circumstances. If so, there would be no real skill in playing the game. You would only need to learn the rules and then follow them 100%. The only rule that should be given any degree of rigidity is to always take full advantage of the odds that favor you. Despite all the knowledge of the score and the count, these things are relative and are based on and dependent on the various situations as they exist at any given time. A situation may occur, for instance, where the hand at a given moment is most favorable to you, but this favoritism may be completely destroyed by attempting to play for a count.

For example, you are dealt the following card with the knock being a 7♣:
A♠, A♥, A♣, Q♦, J♦, 10♦, 10♠, J♠, 8♦, 3♦

You are not on score in any game, the hand is single, and you have a count of 28 in the first game column. Your opponent, whom you have dealt to, discards as his first card the J♣. Do you know what you would do?

At this point, a pick of either of two cards, the Q♠ or the 9♠, would allow you to knock. Your chances of being able to knock without such a run are not too good, but if you pick the 9♦, K♦, or the A♦ you would be under the count. Under such conditions, you must consider the odds of getting under count compared to that of getting a meld. The number of cards unknown to you at this point is 41 since you have seen the 10 you hold, plus the one that your opponent has discarded. The practical odds are about 14 to 1 against your getting a meld card that you need on this pick from the stock. However, if you pick the J♣, and discard the 8♦, you would need only one of the following cards to knock: J♥, Q♠, 9♠, 10♣, and the 10♥. The percentages in your favor will have almost doubled.

There are still disadvantages to picking the J♣. First, you would have to throw the 8♦, which even though it is the second play of the hand, is a fairly wild cad. Second, you will be giving up a pick from the deck, which might be one of the three possible cards that will allow you to knock. Third, you are giving your opponent some indication of what you are holding and you cannot expect back from him any other cards in that particular area. The latter could be somewhat of an advantage because your opponent might feel forced to hold clubs above and below the Jack. The expert player would weigh all of these advantages and disadvantages against the particular odds, as well as against the rigid principle of getting under the count.

Nine out of ten times, this hand will be won, the count gotten under, or the hand won on a knock, before your opponent actually gins the hand and go outs in the game. With odds such as these, the expert player would certainly be very foolish not to take advantage of the opportunities offered by this type of hand.

It cannot be stressed enough the importance of keeping under count. That means you need to reduce the point total in your hand so that even if your opponent should go gin, your total point count will be low enough so that he will not win the game. Just being aware of the necessity of keeping under the count will improve your chances of winning by 25% to 33%. Except for expert play, observations show that every third or fourth final hand of the game is lost because of the avoidable failure to keep under count.

Let us suppose that we have gotten to the point where we have seven melded cards are sitting with a three picture-card combination, which is of course, a count of 30 points. Our problem is not to win the hand, but to get under a count. The count that we have to get under may be any number, under 10, 13, 14, 23, or 24. What are the possibilities then of our getting under these various counts without ginning our opponent?

First of all we have to decide if in the combination there is any one card that is 100% safe. If, for example, in a J, J, Q combination one of the Queens has already been played, the Queen is a fairly safe card. If Jacks had been played we certainly would not hold our tow Jacks but would have two relatively safe cards to throw. If our problem is to get under a 30 count there would then be no problem whatsoever. However, if you suppose we have to get under a count in the teens then we will obviously have to get rid of at least two cards while picking tow little cards in order to get under the teen count. Is this feasible based on the play up to this point? This is something that must be looked into.

If the count is one that is under 10 and actually very low, then obviously the only way we will get under this count realistically is by getting nine melded. This is a case where, rather than breaking up a three-card combination and trying to pick little cards or add-ons to the two other runs, we are better off retaining the combination together with one low card that will keep us under the count in the event we pick the third meld. If we pick a fourth card to any of our first two melds and the low card we have been retaining is not safe, we are better off in this circumstance throwing a fourth card from a four-card meld, providing it is safe, rather than the unsafe low card. Generally speaking, you must weigh very carefully the advantages in breaking up such a combination against the risks in staying over the count and holding on to the combination.

As you can see, there are many variances that go into staying under count. The more you know, the more you are able to do so. There is no guarantee that you will win, but it will give you a better chance in the long run.

One major factor that separates the expert from the good player is the ability to “beat the price”. This means that every single play in every hand from the first card to the last card has a specific probability of success or failure. These probabilities are weighed by the expert against the odds for or against him. Then, in each case, he makes the play that is indicated to be in his favor considering the probable outcome of the play.

The expert also differs from the good player in going beyond the ordinary percentages in involved in any given play. That is, the expert measures this percentage against the advantages and disadvantages to him on every ply of the hand. In fact, this is possibly the most significant difference between the ordinary, everyday gin rummy player and the expert. It is called “money management” which is a term that is used in reference to dice, roulette, blackjack, or any other form of gambling. This type of player who takes full advantage of all the odds in his favor and economizes when the odds turn against him will come out a winner over any given period of time.

A major difference between gin and the other types of gambling games is that the percentages and odds are constantly changing. For example, you are playing a hand which is down to the last four cards in the deck. Yet you have not determined whether the hand should be played to the wall, or whether a winning or losing situation can be resolved. It is your pick. There are 14 cards that might be accessible to you, the four cards in the stock and the 10 cards your opponent is holding. The more skillful a player you are, the more you know what you opponent is holding. At this point in the game, an expert will know at least nine of his opponent’s 10 cards. This leaves him a relatively minor choice as to those cards that are left in the stock. He has merely to determine the odds as to one given card and its location. The less knowledgeable player is less aware of his opponent’s holding and thus faced with a greater number of possibilities. Obviously, the more you know of your opponent’s hand, the great the chances of you determining correctly the odds of obtaining a given card from the four cards left in the deck.

It must be remembered that your opportunity to pick a given card in any one play must include the chance of obtaining the card from your opponent’s discard. Although it is true that skill can in no way change the location of the given card in the stack, if the card is at the top and your opponent picks it, your skill can be a very determining factor in making your opponent discard that card to you.

It is your goal to “beat the price” in every single hand. Whether you can do it is left strictly up to your play and knowledge of the game, but with the above information you are in much better shape to be able to do so.

Using the safety factor point count system should be done in order to win the game obviously, but more so it should be utilized to help you figure out which card to discard so that it will benefit only you. Throwing the correct or the “right” discard will not always prove to be the winning play, because there is always the element of luck involved in any given hand. However, the right discard will win a much higher percentage of hands than will any other play. If followed, this system will provide the correct choice of discard for any given defensive or offensive situation. The decision as to whether or not to make a play for its defensive or offensive value is solely at the discretion of the player, based on all the other factors that must go into making this decision.

An example of a hand that emphasizes all of these problems at the same time would be a six-melded hand together with a combination of the J♣, J♦, Q♦, plus the 6♣ with no known factors relating to any of these cards or Kings or Tens. All we do know is that one of our opponent’s melds is four Aces. We are also faced with a safe count of 30 points which precludes the possibility of throwing off a run. Our pick from the deck is the 10♠. Our first decision is whether to remain under the count and gamble the hand offensively or attempt to go to the wall defensively and risk breaking existing runs and going over the count. Obviously with our game in jeopardy, it is not advantageous to go over the count. Therefore, we must eliminate the thought of taking the hand to the wall. We must then decide whether to play the hand for its maximum offensive possibilities or its maximum defensive possibilities, short of going to the wall. If our decision is to play the hand to its maximum potential defensively without going over the count we can establish just at face value with no other information that the defensive value of each of the five possible discards are as follows: the 10♠ which is a 6 value, the 6♣ which is a 4 value, the J♣ which is a 3 value, the Q♦ which is a 3 value, and the J♦ which is a 2 value. The obvious choice would be the 10♠ because although it has a ranking of a 6 defensively, it is has a value of 0 offensively. It is most important to remember that just as a card may be used six ways by your opponent; it can also be used six ways by you.

When in doubt as to what card to throw between two possible discards, you need to calculate the numerical defensive count of each against the offensive count and whichever one gives you the better odds, that is the card to keep. For instance, you have a choice of throwing the Q♣ or J♦. The Q♣ has an offensive value to you of tow and the J♦ an offensive value of four. Their defensive values being equal, you would of course, retain the J♦. If the Q♣ had a defensive value of one point better than the J♦, then you would still generally retain the J♦. If, however, the Q♣ had a defensive value of three points better than the J♦, then you would want to retain the Q♣.

This safety factor point count system that was previously described can be used to your advantage not only for proper defense, but also to enable you to determine your opponent’s holdings.

For example, if you know that your opponent is predominately a defensive player, then you can be sure that he will never throw a card, even in the early stages of the game that has a safety factor of five or six. In the middle stages of the game he will never throw a card that has a safety factor of three or four, and in the late stages of the game will never throw a card that does not have the safety factors of one or zero. The middle-of-the-road or aggressive player will disregard the safety factors of his discards in the early stages of the game. In the middle stages, he will generally throw cards of even a five or six safety ranking in order to play his most offensive opportunities. The aggressive player will even take calculated risks in the late stages of the game, by throwing cards with a safety factor of two or three.

Your knowledge as to exact holdings or presumed holdings of your opponent’s hand can seriously affect the safety factor of any given card. For instance, a card that you may be holding as a lay off on a known run must be considered also in that manner. A card with a safety factor as low as one, if it also completes an opponent’s meld is a bad throw. In other words, a card with a safety rank of one which may be used by an opponent to complete a combination or change a run from a dead one to a live meld should have a substantially higher numerical value or rank.

It may appear when first looking at it, that this numerical value system would require a computer with every card. Fortunately this is far from the truth. In fact, you would never actually calculate and adjust the safety value of every card in your hand with every play because all those cards which you have melded or are holding for purely offensive purposes automatically exempt themselves from such a calculation. Only when you are required to make a decision as to which of two or three cards to throw would you need to calculate their relative safety value. You will do this by, at the time and that time only, using your actual recollection of all the cards that have already appeared plus your knowledge of your opponent’s holdings.

This numerical system was devised to provide a simple way of determining the correct and safest discard, as well as to show you what kind of advantage you have compared to someone who does not know this system. This same system can be used in offensive play when you decide, at a strategic point in the game, to take full advantage of your hand’s offensive possibilities.

The calculations mentioned in the mathematic and odds article are gin mathematical problems and solutions, but they are not of great strategic value to the player in the game. In gin rummy every play requires a decision that could, if carried to the extreme, be based on a pure mathematical formula. Every card has certain offensive benefits if retained as well as having certain defensive detriments if throw. These benefits and detriments can be reevaluated by considerations given to score and count.

The point count system that we will explain below was developed with an eye towards helping the gin player make the correct decisions during play. It is based primarily on establishing a defensive value for every card based on the number of ways that this card can be used by an opponent. Without any knowledge of either discards or your own, every Ace or King can be used in any one of four ways. Every Queen and Deuce can be used in five ways. Every other card in the deck from the 3 to the Jack can be used in six ways.

In calculating the relative safety factor of a discard, we will start with the fact that a card which can be used in any of six ways by an opponent is the wildest or most unsafe card that can be thrown. Such a card is given a numerical value or rank of six. When a card is 100% dead and cannot be used by an opponent in any way, it is given a numerical rank of zero. Therefore, a card that can be used by an opponent in any one of four ways, will receive a rank of four. A card that can be used in any one or two ways will be ranked two. A one way card receives a rank of one, and so on.

The basic value in using this system is that in practically every gin hand many occasions come up when we have a choice of cards to throw and the determining factor is its overall safety. The point count system method represents a simplified way to truly determine the relative safety value of any one card as compared to another.

The major factor in this system is the fact that after the first discard is made and after each subsequent discard and pick, the numerical safety value of the cards change. For instance, if you had to decide between throwing the 7♥, and the Q♦, your obvious choice would be the Q♦ since in itself it has a safety factor of five as against six for the 7♥. But, assume that the 8♥, has already been discarded and you are holding the 7♣ in a meld in your hand, the safety value of the 7♥ has been reduced to a numerical rank of two. That is, the original possible combination in which the 7♥ could have been used as a meld consists of either a 7♣ and 7♦, a 7♣ and 7♠, a 7♦ and 7♠, a 8♥ and 9♥, 5♥ and 6♥, 6♥ and 8♥.

By elimination of the 7♣ and 8♥ by meld and discard, the 7♥ now has a safety factor of two, and thus would be a much safer throw than the Q♦ and its five value. It is very important to remember that as the unused deck diminishes, the relative safety factor of each card becomes more important since the possibility an opponent’s using a wild discard increases substantially.

When gin rummy became really popular in the 1970’s, there were some players who decided to do a little computer work on the actual mathematics and odds of gin rummy. They are extremely interesting, albeit rather unimportant. However it does give you an idea of the possible hands and odds during the game.

For instance, there are 15,820,024,220 possible ten-card hands in gin rummy. The non-dealer, who receives an eleven-card hand, has the possibilities increased to 60,403,728,840 different hands. After the deal, the combinations and permutations for either hand must be based on 41 cards. Actually each player can start with one hand out of a separate total of 1,471,442,973. After the cards have been dealt, the number of possible hand-against-hand combinations climbs to a whopping 2,165,144,422,791,078,729! Therefore, to give the probabilities and odds of each of the various possibilities in gin rummy would be an exercise in futility and of little value to the player in the game. There would be just too many figures to remember, but there are certain valid general principles that all expert players keep in mind.

We already know the mathematics of gin is based primarily on the law of probabilities. For instance, the probabilities of the dealer being dealt a three-card meld in his first 10 cards are about two out of every five hands, while the chances of the non-dealer being dealt a run in the first 11 cards that he sees, are approximately one out of two. The odds of the non-dealer throwing a card that his opponent can use in a meld in his first play are:

If the discard is a:	Odds:	Probability:
King or Ace	6 to 1 against	0.141
Queen or Deuce	5 to 1 against	0.171
Any other card	4 ½ to 1 against	0.182
.	.	.

These figures are based on the opponent’s using the card immediately in a meld, and not on his considering the card as an improvement to his hand, or as reducing his total of unmatched cards.

As play in a hand progresses, the chances of picking a wild discard increases greatly. For instance, in the late stages of a hand, it is almost certain that a wild card will be picked up. However, at the beginning of a hand, you do not know the cards that are in the stock and probabilities depend on the number of ways in which the discard can be used. These figures are as follows:

Usable Ways:	Odds:	Probability:
3 sequences and 3 sets	3 ½ to 1 against	0.212
2 sequences and 3 sets	4 to 1 against	0.194
1 sequences and 3 sets	5 to 1 against	0.171
3 sequences and 0 sets	7 to 1 against	0.124
2 sequences and 1 set	7 to 1 against	0.129
2 sequences and 0 sets	11 to 1 against	0.084
1 sequence and 1 set	10 to 1 against	0.091
1 sequence and 0 sets	20 to 1 against	0.047
0 sequences and 1 set	20 to 1 against	0.047

Before you give full consideration to the offensive or defensive play of your hand, you have one other thing to consider. It is called the condition of the score. At any given time you need to be sure that you are playing correctly considering what the score is.

For instance, if a player has already scored on two of the three games, while his opponent has not and on the third hand he wins 20 points, he will receive those 20 points on each of the three games. In the final scoring of the game, since the third game is doubled, the 20 points will be equivalent to 40. Thus, the player has gained a grand total of 80 points in the three game columns plus a box, worth 25 points, in each column. The 20 points in the third game then yields the player a grand total of 180 points. If his opponent were to win the same 20 points, it would be scored only in one game, and with the 25 point value for the box, he would only gain 45 points.

Also, since the opponent is still on a triple schneid, the 180 point total could conceivably amount to 360 points. In this particular example, the odds favoring the first player would be at least 4 to 1 and possibly as much as 8 to 1. This would obviously call for the utmost consideration to the offensive play of the hand. If the situation were reversed, and you were on the schneid you would of course play the hand ultra conservatively. You might then find it necessary during the course of the play to give up the wild shot and break up offensive possibilities because the odds were against you.

Another example of playing based on the condition of the score is when a game is approaching its end and you are well behind on the score and are therefore forced to play a catch up type of a game. For instance, if the winning gin score is 250 and your opponent already has 230 while you only have 15, you have no chance whatsoever to win this game except by playing all out any offensive possibilities you have.

Obviously you want to be ahead in any given hand, and win each hand consistently, but if you can’t, then you need to keep a constant eye on the score. The score will tell you implicitly if you need to play offensively or defensively. If the score is close between you and your opponent then you also need to consider whether you want to play to win or play to stay even. It will also give you an idea of whether or not to play offensively or defensively.

As you know by now, at the deal, a hand may be considered a winning or losing one. If it is a winning type of hand, you should play it in an offensive manner, but if it is a losing hand it should be played with consideration to the defensive value of each discard until the hand changes. The expert player is one who can recognize immediately the changing value of his hand, and can adjust his strategy accordingly.

One of the most important strategic plays in gin is establishing certain cards for your opponent to throw, and holding combinations that can utilize the card that your opponent has been “encouraged” to throw. For example, you have a choice of breaking a 3, 4, combination or a 7, 8 combination and you know only that your opponent is holding a 7, 8, and 9 of a different suit. A good defensive layer would be more likely to throw the 9 that you need, since it would be thrown from a duplicated card in his own hand rather than a 2 or a 5 which he does not have matched up.

Another strategy in which expert players readily take advantage of is the unique situation called “on the turn”. Here is an example: Halfway through the deck you are holding a single King because you have believed from the play up to this point that your opponent is probably holding a pair of Kings, which he is trying to meld. At this point your opponent has decided to break his pair and discards his first King which you do not need. You are aware that because of the type of player he is, he would not at this stage throw one loose King from his hand. Therefore, you know he is breaking a pair, and you that so long as you have not picked this King, his next play will automatically be the other King.

Passing up his discard, you go to the deck and fortunately pick up the fourth King. Such a buy is called “picking on the turn”. You will hold this second king in your hand for one pick, and assuming that your conclusion was correct, your opponent’s next discard will be his second King which automatically makes a meld for you. The single exception to your holding a card for one turn is when your opponent picks from the stock a completely dead card and automatically discards it without putting it into his own hand.

The score at any given time should always be used as a forcing situation. A prime example of this is where your opponent is on a schneid or in danger of losing the game if he exceeds a specific count. If, for instance, his safe count is 8, you can afford to retain picture combinations knowing that if he were to pick a picture card he has no choice but throw it because of his count. This is a forcing situation that every expert player will take full advantage of. With a little luck and some expert play it is very easy to change the status of hand from a losing one to a winning one using these force methods.