Thursday, September 6, 2007

Odds and Ends of Bridge (Part 2)

I. Remembering Suit Splits When Declaring

Are you tired of trying to memorize all those a priori suit distribution tables? Here is a short cut that is easier to remember and just as good.

If the number of cards you and dummy hold are an even number, the probability is that the odd number of cards held by opponents will split as evenly as possible. The probability of this division is always more than 50% and gets greater as your holding progresses from 6-8-10 cards. Remember the Rule, forget the specifics.

If the number of cards you and dummy hold are an odd number, the probability is that the cards in the suit held by opponents will split unevenly. The probability of an even split increases as your holding progresses from 5-7-9 cards, but never reaches 50%.

The 11 card holding is an exception to that rule. Always play the suit to split evenly, 1-1. The odds of a 1-1 split are 52% and the 2-0 split 48%.

There are other applications for these rules. If you are declaring 3NT and LHO makes a 4th best lead of the 4 of spades, you can’t always tell with clever defenders whether the suit is 5-3 or 4-4. We know from our rule that an 8 card suit held by defenders is more likely to be 5-3 than 4-4, so in calculating the number of times to hold up in the suit, we might take that into consideration. Actually, there is another rule that makes that “hold up” decision simple for you. It is called the Rule of 7. You total the number of cards in the suit in your hand and in the dummy and subtract that number from 7, the result is the number of times you hold up in the suit. (e.g. If you count 5 cards, hold up twice if you can!)

One final thought before leaving this subject. Sometimes you have the option of splitting the outstandikng cards in your 7 card suit 3-3 or taking a finesse, and either chance will enable you to make your contract, but if you let the opponents in, they will take their setting tricks. There is a right way to do this so that you can preserve both chances. Always try to break the suit first. If that doesn’t work, you can still try the finesse, but if you do it the other way around and the finesse loses, you lose your second chance. While the finesse alone has only a 50% probability, by combining both chances you increase the probability of success to 67.5%.

II. Rules for Single Finesses

To avoid a lot of card table arithmetic, we have some general rules for simple finesses. These should comfort you, because most of them will be familiar to you.
a. If you hold 11 cards in a suit missing the King, you play for the drop. It is a slight favorite over the finesse.
b. If you hold 9 or 10 cards in a suit, you are correct to finesse against the King, but not the Queen or Jack. For example with AQJxx opposite xxxx finesse against the King. With AKJxx opposite xxxx, play AK from the top. Remember “ 8 ever, nine never”?
c. If you hold 7 or 8 cards in a suit, it is usually correct to finesse against the King or the Queen, but not against the Jack. For example with AQJxx, or AKJxx, as long as you hold 7 or 8 cards in the suit, finesse for either the K or Q. With AKQ10x opposite xxx, play for the 3-2 split.
d. If your opponents hold 5 or 6 cards in a suit, it is correct to finesse against the King or Queen, but not against the Jack. For example, if you hold AKQTx in your hand opposite xx, you would play the suit from the top and refuse the finesse on the Jack. Missing other honors, you finesse for them.
There are some further exceptions to these rules, but they will put you on overload. One prominent (but often misunderstood) exception is the “Restricted Choice” play. See Odds and Ends of Bridge Part 3 in next blog post.

III. Working with the Probability of Split Honors

If 2 honors are missing in a suit, bridge probabilities greatly favor finding the two honors split between the two defenders. If you see in the dummy AJ10, AK10, AQ10, these combinations all benefit from the split honors presumption. To demonstrate look at AJ10x in the dummy and xxx in the hand. There are only 4 relevant possibilities, K/Q both with West, K/Q both with East, and the two possibilities (yes 2!) of the missing honors being split between East and West. You will fail to make 2+ tricks only in the single instance where both the K and Q are held by East, one chance out of four so we say the success rate is of making 2+ tricks is 3:1 or roughly 75%. You surely want to try this when one suit that presents a split honor finesse before taking a 50% simple finesse in another suit.

Even if you only have AQ9 opposite xxx, put in the 9. With the A/Q you always have the finesse as a 50% probability. If you also hold the 9, you increase you chances for 2 tricks. If West holds J/10 doubleton, it will go x,10,Q,K on the first lead. On trick 2, you now drop the J making the A/9 good. The probability of the J/10 doubleton is 12.5%, so total probability of making 2 tricks from this holding is now 62.5%, not 50%. You should always go to this combination before trying a simple finesse in any other suit.
One split honor combination that is often overlooked is QJx in declarer’s hand. You will make a trick and have a stopper in 75% of the cases (where the honors are split or the A and K are both held by the East). Note the difference between this holding and Qxx in the hand and Jxx in the dummy. In this last holding, if the honors are split (again a 75% probability), you get no tricks and have no stopper. This is a suit that you never want to lead since if opponents lead it you gain a trick. If LHO doesn’t lead that suit on the opening lead, declarer can be reasonably certain the honors are split, since if LHO held the A and K in that suit, he would probably have led one of them.

You have probably been wondering why in certain situations we don’t just play for the drop of the Queen when we have the A and K in the suit. If it is a 9 card suit, we do – it is a 52% probability. If it is an 8 card suit, playing for the drop of the Queen doubleton with either opponent (a 27% probability) is a decided underdog under almost any circumstances.. Thus, if you have a choice to break a 7 card suit 3-3 (a 35% probability) or to drop the Queen doubleton from an 8 card holding (an 27% probability), go for splitting the 7 card suit 3-3. It is a significant favorite to be the preferred line of play.

Bridge is not an arithmetic test, but having a general awareness of the laws of probabilities will give you a better feel for the game and significantly better results. I know that any developing player’s mind is already on overload, so just start out by trying to recognize whether any particular play has a probability of greater or less than 50%. You don’t always have a choice, but when you do, try to take the plays where you have the odds working for you.


Cindy Dy said...

Thank you for the information. You have a very good article. I found it informative and useful. Keep up the good work and God bless!


Donnie Isaac said...

Good high level article. Liked the encouragement of aggravating results. The article is correct that players - really of any level - have a lot to think about. This explanation helps a bunch to be able to quickly decide superior lines of play.