In our last blog we demonstrated the use of the 2 level weak jump shift by responder and contrasted it with the treatment of bidding the suit at the one level and then rebidding the same suit at the 2 level. In either instance, responder breathes a sigh of relief and assumes he is off the hook. But what if opener has a hand where he simply is not satisfied to quit despite the grim news from partner? Well, first let me tell you that you better get this right, because if you do anything other than lay down the pass card, you have just taken control of the hand and your partner as well.
As noted in the last post, the weak jump shift should show a 6 card suit and 2 to a bad 5 hcps. In order to make game against responder’s WJS you will need a hand that was almost a 2 club opener, something in the range of 19-20 points, probably a 4 LTC hand and at least three card support for partner’s suit. If you hold that nice 19 hcps hand, but only a single in partner’s suit, then trouble is just over the horizon. Either you pass or wish you had.
Contrast this with the responsive hand that bid and rebid the suit showing 6 cards and a good 5 to 9 hcps. The only difference in the requirement for making a forward going bid is that you can make it with hands that are a little weaker. It is always hard to talk about hcps, because so much depends on playability, but I would want 16+ to show further interest.
The results of forward going bidding can also depend on responder’s hand. He will not always have 2 hcps when he makes a WJS and his other 7 cards will not always be 322 (a real goat). If responder has some shortness in one or more suits, it could fit well to produce some additional tricks provided that opener started with the requisite 3 or 4 card support. The death knell for opener to try to find a safer place to play when he lacks support. Ridiculous! Repeat it again, Ridiculous!
With 3 or 4 card support, opener also has an opportunity to further the preempt with minimum values. Even with three card support we have 9 trump don’t we. Being LOTT guys, we are not going to let opponents buy a contract at the 3 level. Old stuff, but equally applicable to this setting. Opponents may not have a game, but down one is often very good, right Bob?
I see you, jumping up and down in your seat sitting on KTx, Axx, A, AKQxxx. You have 20 hcps, 8 1/2 tricks and 4 LTC. I would hope that my partners would have the discipline to open this 1 club -- my aspirations may be too high, but I love them all anyway. I hold QJxxxx, x, xxx, xxx. In response to 1 club, I jump at the chance to bid 2 spades and get out of Dodge! Partner, with justification, has continued interest, but needs to find out more about my hand. What is the best approach?
Well, the start is easy; you bid 2NT which is forcing of course. Now you need agreement with partner what this means. In this context if this is “feature”, you may wish you had some other agreement.There is a better way to proceed.
A preferred way to have partner further define his hand is "Ogust." This is not hard to remember if you use Ogust over weak 2’s. However, there is a twist here.You have to redefine the trems “good hand” and “good suit.” A “good suit” is one where the meager high card points you hold are in your suit. A “good hand” is one where you have a single or a void. Thus, in Ogust speak we respond to 2NT as follows: 2c= bad hand and bad suit, 2 diamonds= bad hand and good suit, 2 hearts, good hand and bad suit, 3 spades= good hand and good suit. With this hand I proudly bid 3 spades over 2NT. I obviously have either the Ace or Queen of spades, perhaps accompanied by the Jack. I also have a single or a void. It makes a comfortable 4 spade bid, but the slam potential of the hand is limited by the fact that I do not know the location of the shortness.
If you are really “uptown” card player (boater, spats and a cane) there is an alternate bid that you can add to your arsenal. This bid does not subsume 2NT, that bid remains Ogust. It is a "do-dad" that lets you pin point any shortness in responder's hand if that is important. Instead of bidding 2NT, you bid 3 clubs. This asks partner do you have any shortness and if so, where is the shortness. Without shortness responder replies 3 spades. With responder’s shortness in hearts as in my example, responder bids 3 hearts. If responder had Qxxxxx and nothing else, you have a hand that is too good to play anywhere other than 6 spades.
No matter how well you play, if you do not get to the best contract, it makes no difference. All your talent is wasted on inferior contracts. The first half of a bridge deal is all about communication. As in real life, those who have the best toys often enjoy it the most. If you don’t think so, send me your best real life toy.
Friday, October 26, 2007
Sunday, October 21, 2007
The Weak Jump Shift and its Counterparts
Strong jump shifts by responder have been left for dead. They are now road-kill. The reason? With enhanced bidding techniques for strong hands (2/1, Bergen, Jacoby, etc.) they have outlived their usefulness. A corollary to owning an ACBL card is that you never leave a bid unused for more than a week. Weak jump shifts are an excellent example where that principle makes sense.
Let me get by some definitional issues. The only weak jump shifts that I am talking about are the 6 jump shifts by responder that do not go beyond the 2 level. I am not an advocate of Criss Cross Inverted Minors, but if you use that convention, the 1c/2d sequence is already taken to show a limit raise for clubs, so that would reduce the number to 5
Partner opens 1 club and as responder you hold K10xxxx, xx, J82, xx. Assuming you don’t bid 2 spades with this hand, what action do you take? Well, first you can pass with 2 card club support, a 6 card major and no defense. Don’t like that? Neither do I. Well fine, then bid 1 spade! Now partner bids 2 hearts (a reverse)! Now do you find any action you like? Bidding 2 spades at this point would clearly misrepresent your hand. Now you have worked yourself into a definite case of the “creeping shorts.” What you would like to do is back this auction up, mark WJS on your convention card and respond 2 spades immediately. As I just demonstrated, the WJS allows you to make a bid on a hand like the one described above and not risk a “runaway” auction that progressively slumps you farther and farther down in your seat.
There are some other benefits to using the WJS. First, for simplicity my example did not show competitive bidding, but in a real life setting if the bidding goes 1c/p/p, or 1c/p/1s, there is no way your are going to get LHO to be silent. Alternatively, if you had responded 2 spades you probably would not have heard any chirping on your left. We hate chirping! Second, you have achieved the ultimate in bridge bidding, a “twofer.” You will have told partner not only about the distribution of your hand, but also about your lack of hcps in a single bid.
The biggest mistake by those using WJS is that they don’t use the bid on very weak hands. With Q9xxxx, xx, 82, xx (bid 2 spades without blinking an eye). The opposite is also an equal problem, they use the WJS on hands that are too strong such as KQ9xxx, xx, Q82, xx (bid 1 spade, not two). Max Hardy in his world renowned book, Advanced Bidding for the 21st Century (2000) states that the proper range for the WJS jump shift is 2 to a bad 5 hcps.
You might ask what do I achieve by limiting the bid to such rotten hands. Why not 4-8 or some other more flexible range? The reason is that we have another bidding treatment for the 6 card suit in the good 5 to 9 hcp range. With hands like KQ9xxx, xx, Q82, xx we are going to respond 1 spade and at the next opportunity we will bid 2 spades. By rebidding our suit we will show the 6 card spade suit, but with a hand that may have as much as 9 high card points. Notice that by adopting this discipline, we have very precisely sliced and diced the range of 2-9 hcps, differentiating the WJS from the “suit rebid” response, and thus giving partner a very precise picture of your hand strength and card distribution.
Weak jump shifts are commonly used in competition and over a t/o double. Should they also be used when there is no competition from your RHO opponent? Of course they should. And for all the same reasons. Just because your RHO has not bid doesn’t mean that all is quiet on the Western Front. If you pass that nice 3 point hand, or bid 1 spade, it is a guaranteed that the Terrorist on your left is about to try to steal this auction from your side. Challenge your LHO to bid at the 3 level; it’s like throwing a grenade. Are you safe? Off course you are, you have described your hand to partner in detail. Unless he makes a forcing call, your obligation is over. Remember to check the box on the convention card that says that WJS are used when not in competition. Note that it is marked in “red” and must be alerted.
To wrap up this section let’s talk about what action responder takes when he has a 6+ card suit and 10-11 hcps. Assume you have KQ9xxx, xx, Kxx, Q8. Your partner opens 1 club, there is a pass on your right and you respond 1 spade. Partner now bids 2 diamonds. You must not bid 2 spades. That bid says I have 6-9 hcps and in the above example you have a nice 10 hcps. Jump your second response to 3 spades to show your 6 card suit and invitational values. Partner with a single spade and minimum hand can pass, but more times than not you will happily be in 4 spades. With the above hand if I had another spade or another point or if partner had rebid 1NT, I would bid 4 spades rather than 3 spades.
In the next post to our blog, we will continue this discussion and look at opener’s rebid options when he holds some significant extras and wants to ignore your warnings.
Let me get by some definitional issues. The only weak jump shifts that I am talking about are the 6 jump shifts by responder that do not go beyond the 2 level. I am not an advocate of Criss Cross Inverted Minors, but if you use that convention, the 1c/2d sequence is already taken to show a limit raise for clubs, so that would reduce the number to 5
Partner opens 1 club and as responder you hold K10xxxx, xx, J82, xx. Assuming you don’t bid 2 spades with this hand, what action do you take? Well, first you can pass with 2 card club support, a 6 card major and no defense. Don’t like that? Neither do I. Well fine, then bid 1 spade! Now partner bids 2 hearts (a reverse)! Now do you find any action you like? Bidding 2 spades at this point would clearly misrepresent your hand. Now you have worked yourself into a definite case of the “creeping shorts.” What you would like to do is back this auction up, mark WJS on your convention card and respond 2 spades immediately. As I just demonstrated, the WJS allows you to make a bid on a hand like the one described above and not risk a “runaway” auction that progressively slumps you farther and farther down in your seat.
There are some other benefits to using the WJS. First, for simplicity my example did not show competitive bidding, but in a real life setting if the bidding goes 1c/p/p, or 1c/p/1s, there is no way your are going to get LHO to be silent. Alternatively, if you had responded 2 spades you probably would not have heard any chirping on your left. We hate chirping! Second, you have achieved the ultimate in bridge bidding, a “twofer.” You will have told partner not only about the distribution of your hand, but also about your lack of hcps in a single bid.
The biggest mistake by those using WJS is that they don’t use the bid on very weak hands. With Q9xxxx, xx, 82, xx (bid 2 spades without blinking an eye). The opposite is also an equal problem, they use the WJS on hands that are too strong such as KQ9xxx, xx, Q82, xx (bid 1 spade, not two). Max Hardy in his world renowned book, Advanced Bidding for the 21st Century (2000) states that the proper range for the WJS jump shift is 2 to a bad 5 hcps.
You might ask what do I achieve by limiting the bid to such rotten hands. Why not 4-8 or some other more flexible range? The reason is that we have another bidding treatment for the 6 card suit in the good 5 to 9 hcp range. With hands like KQ9xxx, xx, Q82, xx we are going to respond 1 spade and at the next opportunity we will bid 2 spades. By rebidding our suit we will show the 6 card spade suit, but with a hand that may have as much as 9 high card points. Notice that by adopting this discipline, we have very precisely sliced and diced the range of 2-9 hcps, differentiating the WJS from the “suit rebid” response, and thus giving partner a very precise picture of your hand strength and card distribution.
Weak jump shifts are commonly used in competition and over a t/o double. Should they also be used when there is no competition from your RHO opponent? Of course they should. And for all the same reasons. Just because your RHO has not bid doesn’t mean that all is quiet on the Western Front. If you pass that nice 3 point hand, or bid 1 spade, it is a guaranteed that the Terrorist on your left is about to try to steal this auction from your side. Challenge your LHO to bid at the 3 level; it’s like throwing a grenade. Are you safe? Off course you are, you have described your hand to partner in detail. Unless he makes a forcing call, your obligation is over. Remember to check the box on the convention card that says that WJS are used when not in competition. Note that it is marked in “red” and must be alerted.
To wrap up this section let’s talk about what action responder takes when he has a 6+ card suit and 10-11 hcps. Assume you have KQ9xxx, xx, Kxx, Q8. Your partner opens 1 club, there is a pass on your right and you respond 1 spade. Partner now bids 2 diamonds. You must not bid 2 spades. That bid says I have 6-9 hcps and in the above example you have a nice 10 hcps. Jump your second response to 3 spades to show your 6 card suit and invitational values. Partner with a single spade and minimum hand can pass, but more times than not you will happily be in 4 spades. With the above hand if I had another spade or another point or if partner had rebid 1NT, I would bid 4 spades rather than 3 spades.
In the next post to our blog, we will continue this discussion and look at opener’s rebid options when he holds some significant extras and wants to ignore your warnings.
Sunday, September 30, 2007
My Vote for Best Bridge Book of 2007
Summer has gone and so has my reading list. I didn’t read every new bridge book published in 2007, and perhaps they are saving the best for last, but if Mel Colchamiro’s book, “How to Play Like an Expert (without having to be one)” is not at the top of the list, somebody should conduct an investigation. This is Mel’s first effort at writing a book, but many have enjoyed his monthly column in the Bridge Bulletin, “Claim with Colchamiro.” The book does touch on some of his previous writing, but not to excess and always for the right reasons, adding more clarity and depth (and occasionally correction) to the practical advice he has previously given to us.
As the title indicates, the book does not address bridge experts. If you can declare at the “double dummy” standard, defend like a demon, recite all the percentage plays and intuitively figure out everything “on the fly” without ever being out of tempo, then this book is clearly not for you. But if you were in that category, your name would appear on the back cover along with Eric Kokish, Paul Soloway and Bobby Wolff.
Instructional books should be judged on clarity, good organization and how well they meet the expectations of their target audience. Mel tells you that his targeted audience is “us regular folk”, and at the risk of being intellectually whipped by the words “overly simplistic”, he takes dead aim on that audience. Mel’s teaching experience shows through and enables him to carry his message in a series of clear, concise rules, somewhat in the style of Ron Klinger’s “Better Bridge with a Better Memory.” It takes over where the Rules of 11 and “Eight Ever, Nine Never” leave off and takes us through more than a dozen easy to apply guidelines that for the most part can be calculated on your fingers. No move worries about “do I or don’t I” or “should I or shouldn’t I” or “what would Mel do now.” For many of those common dilemmas, Mel tells you how to make the same decision he would make without paying the tuition.
One thing I like about the book is the application of most of the rules are not dependent on a partnership understanding. Would it be better as a partnership read? Yes, but its value does not generally require mutual partnership understanding. Another plus, the book is replete with graphic examples that greatly aid the understanding.
If you think you are not an appropriate audience target at the conclusion of Chapter 4, then skip ahead to Chapter 14 and start to work your way through 60 pages of “Balance of Power” Doubles. While Mel makes a gallant effort to put “action doubles”, “BOP doubles” and “penalty doubles” into distinct and identifiable cubby holes, it still is a humbling experience. My advice is don’t get lost in the forest. Look for the major summarizing rule at the end of the chapters, and then go back and worry about the subsets. Here is where it would be nice if you and partner got on the same page.
Those who know my bidding style know that I come from the Larry Cohen camp, generally subscribing to the maxim that "you can’t leave opponents undisturbed if 1NT is opened on your right." The only thing worse that defending 1NT is defending 1NT doubled. Those waiting for the perfect hands to utilize Cappelletti, Hamilton and other “beefsteak” 5-5 no trump killers are doing a lot of waiting, no balancing and getting a lot of average minus boards. Mel has developed a system to get you into the auction whether you are in 2nd or 4th seat. The entire system is circumscribed by two rules explained on pages 10-25. Here is a summary of those rules:
(a) If your RHO opens 1NT (15-17), count the number of cards in your two longest suits, subtract your Losing Trick Count (LTC) and bid if the result is greater than 1. For reasons explained in the book, Mel calls this the Rule of 8, but I prefer to think of it as the Rule of 1.
(b) How do I get into the auction? Use the “DONT” defensive bids. I like this, Mel likes this and more importantly so does Larry Cohen, arguably the strongest match point player ever.
(c) If the bidding goes 1NT/P/P, partner obviously did not meet the Rule of 1, but that doesn’t mean that he doesn’t have some stuff. Here you apply the Rule of 2. Bid (DONT of course) regardless of your hcps or vulnerability if you have at least 2 “shortness” points. Example: T643, QT95, T, Q965. Bid 2 clubs! If partner followed the 2nd seat discipline, the probability is that our side has a total of about 19.5 hcps. Remember there is no penalty double in DONT, and partner could hold KQxx, Kxx, Axxx, Kxx and not have a call. In this example his longest 2 suits = 8, less 7 LTC= 1. This is not more than 1, so partner is required to pass. He is very thankful when you take a bid. So now you know that defending against No Trump is in fact simpler than 1-2-3, it is as simple as 1-2! I would recommend reading Chapter 4 before you read Chapter 3, I think it puts it in better perspective.
If you are looking for page after page of tedious “play of the hand” problems, then this book is not for you. If you are completely undisciplined and do not like rules of application and want to be left in your current quandary of applying instinct in an ad hoc fashion, then save your money.
The book is available at www.melbridge.com and at other book resources. The cost is $21.95, it is paper bound and 276 pages. Did I buy the book? Yes, but if I had it to do over again, I would recommend it to you and then read your copy.
As the title indicates, the book does not address bridge experts. If you can declare at the “double dummy” standard, defend like a demon, recite all the percentage plays and intuitively figure out everything “on the fly” without ever being out of tempo, then this book is clearly not for you. But if you were in that category, your name would appear on the back cover along with Eric Kokish, Paul Soloway and Bobby Wolff.
Instructional books should be judged on clarity, good organization and how well they meet the expectations of their target audience. Mel tells you that his targeted audience is “us regular folk”, and at the risk of being intellectually whipped by the words “overly simplistic”, he takes dead aim on that audience. Mel’s teaching experience shows through and enables him to carry his message in a series of clear, concise rules, somewhat in the style of Ron Klinger’s “Better Bridge with a Better Memory.” It takes over where the Rules of 11 and “Eight Ever, Nine Never” leave off and takes us through more than a dozen easy to apply guidelines that for the most part can be calculated on your fingers. No move worries about “do I or don’t I” or “should I or shouldn’t I” or “what would Mel do now.” For many of those common dilemmas, Mel tells you how to make the same decision he would make without paying the tuition.
One thing I like about the book is the application of most of the rules are not dependent on a partnership understanding. Would it be better as a partnership read? Yes, but its value does not generally require mutual partnership understanding. Another plus, the book is replete with graphic examples that greatly aid the understanding.
If you think you are not an appropriate audience target at the conclusion of Chapter 4, then skip ahead to Chapter 14 and start to work your way through 60 pages of “Balance of Power” Doubles. While Mel makes a gallant effort to put “action doubles”, “BOP doubles” and “penalty doubles” into distinct and identifiable cubby holes, it still is a humbling experience. My advice is don’t get lost in the forest. Look for the major summarizing rule at the end of the chapters, and then go back and worry about the subsets. Here is where it would be nice if you and partner got on the same page.
Those who know my bidding style know that I come from the Larry Cohen camp, generally subscribing to the maxim that "you can’t leave opponents undisturbed if 1NT is opened on your right." The only thing worse that defending 1NT is defending 1NT doubled. Those waiting for the perfect hands to utilize Cappelletti, Hamilton and other “beefsteak” 5-5 no trump killers are doing a lot of waiting, no balancing and getting a lot of average minus boards. Mel has developed a system to get you into the auction whether you are in 2nd or 4th seat. The entire system is circumscribed by two rules explained on pages 10-25. Here is a summary of those rules:
(a) If your RHO opens 1NT (15-17), count the number of cards in your two longest suits, subtract your Losing Trick Count (LTC) and bid if the result is greater than 1. For reasons explained in the book, Mel calls this the Rule of 8, but I prefer to think of it as the Rule of 1.
(b) How do I get into the auction? Use the “DONT” defensive bids. I like this, Mel likes this and more importantly so does Larry Cohen, arguably the strongest match point player ever.
(c) If the bidding goes 1NT/P/P, partner obviously did not meet the Rule of 1, but that doesn’t mean that he doesn’t have some stuff. Here you apply the Rule of 2. Bid (DONT of course) regardless of your hcps or vulnerability if you have at least 2 “shortness” points. Example: T643, QT95, T, Q965. Bid 2 clubs! If partner followed the 2nd seat discipline, the probability is that our side has a total of about 19.5 hcps. Remember there is no penalty double in DONT, and partner could hold KQxx, Kxx, Axxx, Kxx and not have a call. In this example his longest 2 suits = 8, less 7 LTC= 1. This is not more than 1, so partner is required to pass. He is very thankful when you take a bid. So now you know that defending against No Trump is in fact simpler than 1-2-3, it is as simple as 1-2! I would recommend reading Chapter 4 before you read Chapter 3, I think it puts it in better perspective.
If you are looking for page after page of tedious “play of the hand” problems, then this book is not for you. If you are completely undisciplined and do not like rules of application and want to be left in your current quandary of applying instinct in an ad hoc fashion, then save your money.
The book is available at www.melbridge.com and at other book resources. The cost is $21.95, it is paper bound and 276 pages. Did I buy the book? Yes, but if I had it to do over again, I would recommend it to you and then read your copy.
Sunday, September 23, 2007
The Odds and Ends of Bridge (Post Graduate Course)
Until poker became a TV star, everybody thought Poker was mostly bluff and luck and that duplicate bridge was a game of skill. Today we know that even the lowly internet poker players calculate the probability of finding an “out card” and live and die off pot odds. Conversely, club game bridge players largely consider bridge probabilities as an annoyance that only ruins an otherwise beautiful afternoon game. It can only be explained by the difference between money and masterpoints. If you don’t want me to ruin your beautiful afternoon, just exit this blog and go back to the novel. This is not easy stuff, but I hope that sometime it will form a useful reference. This is my last blog post on “Odds and Ends” and soon we will return to entertainment.
While bridge probability tables are helpful and avoid some difficult computations, the odds set forth in these tables are known as “a priori” odds, since they measure the probability of chances on any given hand prior to the play of any of the cards in that hand. As soon as one or more cards are played on the deal, these a priori odds are history. Sometimes the resulting change in the odds will be miniscule, but other times will be substantial. Probabilities calculated after the commencement of play are called “a posteriori” odds. These possibilities are too numerous to fully capture on any printed table.
The Deletion Principle
Let’s look at a simple example where we hold a seven card suit AKQxx opposite xx. If the suit splits 3-3 we will be able to take 5 tricks in this suit. When opponents hold 6 cards in a suit the “a priori” odds for the 3-3 split are 35%. The odds for a 4-2 split are 48%. With 6 cards outstanding there are 64 possible card divisions, 20 of them being 3-3 and 30 of them being 4-2 or 2-4 and the rest being nightmares.
Now, starting our analysis and play, assume that both defenders follow suit on the play of the Ace and King. We now know that the distribution of the suit was not 6-0 or 5-1 (our so-called “nightmares”). We have eliminated the 2 possibilities of the 6-0 break and the 12 possibilities of the 5-1 break, leaving only the 50 possible combinations of 3-3 and 4-2. By reducing the denominator, we have increased the odds of a 3-3 split from 35% to 42%, and the odds of a 4-2 split from 48% to 57%, but also note that the relative probability of the two suit divisions to each other has stayed the same. The probability of the 3-3 split is still less than 50%, and will remain less than 50% even if West follows suit on the third lead.
This example demonstrates an important principle of bridge probabilities known as the “deletion principle.” It can be stated as follows:
“When the opponents follow to the play of a suit with insignificant cards, the impossible distributions are deleted but the probabilities of the remainder of the distributions retain their relative proportionality to each other.”
Note that I have highlighted the words “insignificant cards.” You might at first think the holding of the J or Q by a defender would be significant. But in this example, the only division that will obtain 5 tricks is the 3-3 split, so the appearance of the Jack or Queen at anytime during the play of the suit would have no significance to the final result. If West held Jxx, and false carded the Jack on the second play in the suit, it would be an insignificant false card since we still have only one play that works for 5 tricks, the 3-3 split. We have no choice but to lay down the honors and hope that the jack was a false card. That is why all the opponents cards in this particular 7 card arrangement are considered insignificant.
Now let us change the cards a bit and demonstrate a holding in which the Jack does become a significant card and note how that changes probability of the 3-3 split. Assume we have AKQ10x opposite xx. Again we lead our two top honors with both opponents following suit with insignificant cards (not the jack!). As in the first example we now eliminate the fourteen 6-0 and 5-1 suit distributions, but there is another suit distribution that we can also eliminate in this case. We know that neither defender held Jx or we would have seen the Jack on the second round of play. Thus, we can also delete those 4-2 combinations that would have included a Jx doubleton by either opponent. Of the 30 possible 4-2 combinations, 10 of them involve Jx doubleton. While the probability of the 3-3 split remains unaffected at 35%, the probability of the 4-2 split has been reduced from 48% to 32%. Now in relative terms, the 3-3 split has a 52% probability and the 4-2 split a 48% probability.
Unlike the first example, we could delete the 4-2 combinations involving JX because a defender holding those cards will not make things easy for declarer by playing the Jack on the first or second lead if he held Jxx. Since now we know that the 3-3 suit split has become a slight favorite when our 7 card suit includes the AKQ10, we simply plunk down the queen and hope the odds work in our favor. The unwashed will sniffle that you got lucky, but we should let our non-readers suffer the consequences of not checking into Tommy’s Bridge Blog.
A10xxx opposite Kx. This example demonstrates that the favorable odds will also occur where there are two significant cards outstanding, the Q and J. The deletion process is a little different. Here we can delete the 4-2 combinations that include a Qx or Jx (18 cases out of 30) and the 3-3 combinations where either defender has both of the honors (8 cases out of 20). Again the combination of these deletions works out to a 52% probability in favor of the 3-3 split. The play is to ruff out a small card after taking the A and K, hoping to see the Q and J fall on the third lead. It’s a favorite! You might say, “Duh, how else would you play it?” Well the important thing is that you realize that you have now vaulted over that 50% threshold, and spiting the 7 card suit is now a preferable line of play to taking a finesse in another suit.
What you want to be on the lookout for in these situations are the combinations where opponents hold one or more significant cards and you don’t see any significant cards after the first two plays in the suit. What really makes an honor held by the opponents a significant card is that your 7 card suit holds a “threat card” (such as the 10 in my first example). This eliminates the possibility of a false card.
Here are examples of other layouts where the deletion principle enhances the probability of the 3-3 break to a 52% probability. The point being that you should try to split the 7 card suit, rather than take a finesse in another suit as an alternative. (i) 109xx opposite AKx. The play for 3 tricks in this suit is A,K and small to the 10. The 3-3 split is a 52% probability. (ii) A opposite KJxxxx. Play the A, K and ruff a small card if neither defender shows out. The 3-3 split is a 52% probability. (iii) AQxxxx opposite x. Here the 52% play is to play the Ace and ruff 2 small clubs trying to split the suit. It is also a 52% probability. It is not so obvious that the 6th card in one hand is the threat card, so memorize it!
The deletion principle is also the reason for the popular saying “8 ever, 9 never.” I won’t waltz you through the numbers, just keep repeating the old saying.
While bridge probability tables are helpful and avoid some difficult computations, the odds set forth in these tables are known as “a priori” odds, since they measure the probability of chances on any given hand prior to the play of any of the cards in that hand. As soon as one or more cards are played on the deal, these a priori odds are history. Sometimes the resulting change in the odds will be miniscule, but other times will be substantial. Probabilities calculated after the commencement of play are called “a posteriori” odds. These possibilities are too numerous to fully capture on any printed table.
The Deletion Principle
Let’s look at a simple example where we hold a seven card suit AKQxx opposite xx. If the suit splits 3-3 we will be able to take 5 tricks in this suit. When opponents hold 6 cards in a suit the “a priori” odds for the 3-3 split are 35%. The odds for a 4-2 split are 48%. With 6 cards outstanding there are 64 possible card divisions, 20 of them being 3-3 and 30 of them being 4-2 or 2-4 and the rest being nightmares.
Now, starting our analysis and play, assume that both defenders follow suit on the play of the Ace and King. We now know that the distribution of the suit was not 6-0 or 5-1 (our so-called “nightmares”). We have eliminated the 2 possibilities of the 6-0 break and the 12 possibilities of the 5-1 break, leaving only the 50 possible combinations of 3-3 and 4-2. By reducing the denominator, we have increased the odds of a 3-3 split from 35% to 42%, and the odds of a 4-2 split from 48% to 57%, but also note that the relative probability of the two suit divisions to each other has stayed the same. The probability of the 3-3 split is still less than 50%, and will remain less than 50% even if West follows suit on the third lead.
This example demonstrates an important principle of bridge probabilities known as the “deletion principle.” It can be stated as follows:
“When the opponents follow to the play of a suit with insignificant cards, the impossible distributions are deleted but the probabilities of the remainder of the distributions retain their relative proportionality to each other.”
Note that I have highlighted the words “insignificant cards.” You might at first think the holding of the J or Q by a defender would be significant. But in this example, the only division that will obtain 5 tricks is the 3-3 split, so the appearance of the Jack or Queen at anytime during the play of the suit would have no significance to the final result. If West held Jxx, and false carded the Jack on the second play in the suit, it would be an insignificant false card since we still have only one play that works for 5 tricks, the 3-3 split. We have no choice but to lay down the honors and hope that the jack was a false card. That is why all the opponents cards in this particular 7 card arrangement are considered insignificant.
Now let us change the cards a bit and demonstrate a holding in which the Jack does become a significant card and note how that changes probability of the 3-3 split. Assume we have AKQ10x opposite xx. Again we lead our two top honors with both opponents following suit with insignificant cards (not the jack!). As in the first example we now eliminate the fourteen 6-0 and 5-1 suit distributions, but there is another suit distribution that we can also eliminate in this case. We know that neither defender held Jx or we would have seen the Jack on the second round of play. Thus, we can also delete those 4-2 combinations that would have included a Jx doubleton by either opponent. Of the 30 possible 4-2 combinations, 10 of them involve Jx doubleton. While the probability of the 3-3 split remains unaffected at 35%, the probability of the 4-2 split has been reduced from 48% to 32%. Now in relative terms, the 3-3 split has a 52% probability and the 4-2 split a 48% probability.
Unlike the first example, we could delete the 4-2 combinations involving JX because a defender holding those cards will not make things easy for declarer by playing the Jack on the first or second lead if he held Jxx. Since now we know that the 3-3 suit split has become a slight favorite when our 7 card suit includes the AKQ10, we simply plunk down the queen and hope the odds work in our favor. The unwashed will sniffle that you got lucky, but we should let our non-readers suffer the consequences of not checking into Tommy’s Bridge Blog.
A10xxx opposite Kx. This example demonstrates that the favorable odds will also occur where there are two significant cards outstanding, the Q and J. The deletion process is a little different. Here we can delete the 4-2 combinations that include a Qx or Jx (18 cases out of 30) and the 3-3 combinations where either defender has both of the honors (8 cases out of 20). Again the combination of these deletions works out to a 52% probability in favor of the 3-3 split. The play is to ruff out a small card after taking the A and K, hoping to see the Q and J fall on the third lead. It’s a favorite! You might say, “Duh, how else would you play it?” Well the important thing is that you realize that you have now vaulted over that 50% threshold, and spiting the 7 card suit is now a preferable line of play to taking a finesse in another suit.
What you want to be on the lookout for in these situations are the combinations where opponents hold one or more significant cards and you don’t see any significant cards after the first two plays in the suit. What really makes an honor held by the opponents a significant card is that your 7 card suit holds a “threat card” (such as the 10 in my first example). This eliminates the possibility of a false card.
Here are examples of other layouts where the deletion principle enhances the probability of the 3-3 break to a 52% probability. The point being that you should try to split the 7 card suit, rather than take a finesse in another suit as an alternative. (i) 109xx opposite AKx. The play for 3 tricks in this suit is A,K and small to the 10. The 3-3 split is a 52% probability. (ii) A opposite KJxxxx. Play the A, K and ruff a small card if neither defender shows out. The 3-3 split is a 52% probability. (iii) AQxxxx opposite x. Here the 52% play is to play the Ace and ruff 2 small clubs trying to split the suit. It is also a 52% probability. It is not so obvious that the 6th card in one hand is the threat card, so memorize it!
The deletion principle is also the reason for the popular saying “8 ever, 9 never.” I won’t waltz you through the numbers, just keep repeating the old saying.
Tuesday, September 18, 2007
Looking for Additive Opportunites
Howard Christ is the Executive Director of the Ocala Duplicate Bridge Club in Ocala Florida. When I am in Florida, he is further distinguished (my choice of words) by having me as a partner 6 days a month. I think Howard really keeps me around to demonstrate that he does, indeed, have limited tolerance for fools. It is nerve wracking to play with Howard since he is one of those players that always seems to know what is going on at the table and never fails to identify an opportunity if there is even a whiff of it lingering over the table. One of his great pearls of wisdom is his thought about opportunity at the table: “You won’t see it if you are not looking for it!”
Frank Stewart’s bridge column for Saturday, September 15 reminded me of the need to look for those opportunities, no matter how slim they may be, as long as it doesn’t cost you anything to explore. Here is the set up in Frank’s column:
^^^^^^^^^^^^K
^^^^^^^^^^^^54
^^^^^^^^^^^^9432
^^^^^^^^^^^^AK6532
JT962^^^^^^^^^^^^^^^^^Q873
K87^^^^^^^^^^^^^^^^^^^JT9
Q876^^^^^^^^^^^^^^^^^^JT
8^^^^^^^^^^^^^^^^^^^^^JT97
^^^^^^^^^^^^A54
^^^^^^^^^^^^AQ632
^^^^^^^^^^^^AK5
^^^^^^^^^^^^Q4
The contract is 3NT. The opening lead is the 3 of spades, up steps dummy’s K. As all good match point players do, you look at the array of potential contracts that could have been reached by the field on this deal and see if the club suit breaks, you are going to take gas on this board, since either 6 clubs or 6 NT are high probability contracts. You now have to hope that the club suit doesn’t break 3-2 and that those slam bidders get heartburn. You note that your 3NT contract also has problems, as the only remaining entry on the board is in the club suit. If you play small to the club Queen and back to the high club honors, you will get exactly 3 club tricks on a 4-1 club break, This will leave you one trick short unless the heart finesse works.
Many players would simply resign themselves to the heart finesse, a 50-50 proposition, as the only back up to a failed club play. The best declarers who are always looking for a small edge will see that the heart finesse can always be tried, but that there is a slim chance for another line of play that can be explored first without losing the later chance for the heart finesse.
Noting the 4 diamonds to the 9 on the board, they will see that the QJT of diamonds are outstanding and that if RHO has any two of those honors doubleton, the 9 of diamonds will set up as a trick if an entry is preserved to get at it. Yes, finding the JT doubleton East is a long shot, about an 8% probability, but nothing is lost by trying the diamond play before you take the heart finesse. Play a small club to the Queen and lay down the A and K of diamonds, and lo and behold, the bank vault opens up, the J and T fall under the A and K. Now you can play the third diamond toward the 9 and the 9 becomes a good trick when West takes his Q of diamonds. Since you still have the club 6 to get to t he board ,you can reach that diamond trick. Now you no longer have to count on the heart finessse for your 9th trick.
This comes under the category of “if you are not looking for it, you won’t see it.” Perhaps “timing is everything” would also be appropriate. Note that you have to try the diamond play while you still have a club entry, so if you make the mistake of laying down 2 clubs to try the club suit first, you are dead. You can still promote the diamond trick, but you can’t get at it. When the clubs don’t break, you can still try your heart finesse, but you have then given up on the additive diamond play.
This is not an easy problem, and is in this post simply as a teaching tool. First, look for additive opportunities if they do not jeopardize your existing chances. If you are not looking for them, you will not see them. When you have multiple play options, and one of them involves a 7 card suit, even though the odds are against a 3-3 break, always try that solution first before taking your finesses. If the 7 card suit doesn’t work you can usually still take the finesse, but often if the finesse doesn’t work, it is too late to try the 7 card suit. Alternatively, you can do as I do, get Howard Christ to play these hands for you!
Frank Stewart’s bridge column for Saturday, September 15 reminded me of the need to look for those opportunities, no matter how slim they may be, as long as it doesn’t cost you anything to explore. Here is the set up in Frank’s column:
^^^^^^^^^^^^K
^^^^^^^^^^^^54
^^^^^^^^^^^^9432
^^^^^^^^^^^^AK6532
JT962^^^^^^^^^^^^^^^^^Q873
K87^^^^^^^^^^^^^^^^^^^JT9
Q876^^^^^^^^^^^^^^^^^^JT
8^^^^^^^^^^^^^^^^^^^^^JT97
^^^^^^^^^^^^A54
^^^^^^^^^^^^AQ632
^^^^^^^^^^^^AK5
^^^^^^^^^^^^Q4
The contract is 3NT. The opening lead is the 3 of spades, up steps dummy’s K. As all good match point players do, you look at the array of potential contracts that could have been reached by the field on this deal and see if the club suit breaks, you are going to take gas on this board, since either 6 clubs or 6 NT are high probability contracts. You now have to hope that the club suit doesn’t break 3-2 and that those slam bidders get heartburn. You note that your 3NT contract also has problems, as the only remaining entry on the board is in the club suit. If you play small to the club Queen and back to the high club honors, you will get exactly 3 club tricks on a 4-1 club break, This will leave you one trick short unless the heart finesse works.
Many players would simply resign themselves to the heart finesse, a 50-50 proposition, as the only back up to a failed club play. The best declarers who are always looking for a small edge will see that the heart finesse can always be tried, but that there is a slim chance for another line of play that can be explored first without losing the later chance for the heart finesse.
Noting the 4 diamonds to the 9 on the board, they will see that the QJT of diamonds are outstanding and that if RHO has any two of those honors doubleton, the 9 of diamonds will set up as a trick if an entry is preserved to get at it. Yes, finding the JT doubleton East is a long shot, about an 8% probability, but nothing is lost by trying the diamond play before you take the heart finesse. Play a small club to the Queen and lay down the A and K of diamonds, and lo and behold, the bank vault opens up, the J and T fall under the A and K. Now you can play the third diamond toward the 9 and the 9 becomes a good trick when West takes his Q of diamonds. Since you still have the club 6 to get to t he board ,you can reach that diamond trick. Now you no longer have to count on the heart finessse for your 9th trick.
This comes under the category of “if you are not looking for it, you won’t see it.” Perhaps “timing is everything” would also be appropriate. Note that you have to try the diamond play while you still have a club entry, so if you make the mistake of laying down 2 clubs to try the club suit first, you are dead. You can still promote the diamond trick, but you can’t get at it. When the clubs don’t break, you can still try your heart finesse, but you have then given up on the additive diamond play.
This is not an easy problem, and is in this post simply as a teaching tool. First, look for additive opportunities if they do not jeopardize your existing chances. If you are not looking for them, you will not see them. When you have multiple play options, and one of them involves a 7 card suit, even though the odds are against a 3-3 break, always try that solution first before taking your finesses. If the 7 card suit doesn’t work you can usually still take the finesse, but often if the finesse doesn’t work, it is too late to try the 7 card suit. Alternatively, you can do as I do, get Howard Christ to play these hands for you!
Wednesday, September 12, 2007
Odds and Ends of Bridge (Part 4)
^^^^^^^^^^A10
^^^^^^^^^^943
^^^^^^^^^^AK65432
^^^^^^^^^^K
Q964^^^^^^^^^^^^^87532
Q3^^^^^^^^^^^^^^^K762
9^^^^^^^^^^^^^^^^^J1087
QJ10985^^^^^^^^^^None
^^^^^^^^^^KJ
^^^^^^^^^^AJ108
^^^^^^^^^^Q
^^^^^^^^^^A76432
The above hand is from Frank Stewart’s Bridge Column on Sunday, September 9. Let me digress a moment to discuss probabilities before I get to the play of the hand. The probability of the division of the 8 card diamond suit prior to the play of any card in the deal is:
5-0 or 0-5 = 3.91%
4-1 or 1-4 = 26.26%
3-2 or 2-3 = 67.83%
The Queen of Clubs is led and East discards the spade 2. We now have completely discovered the division of the opponent’s holding in the club suit as West 6- East 0. We learned in Odds and Ends of Bridge (Part 1) about the Law of Attraction, that a long suit in one hand tends to attract another short suit and that a long suit tends to reject another long suit. The probable distribution of the 8 card diamond can now be recalculated to:
5-0 or 0-5 = 8.44%
4-1 or 1-4 = 35.22%
3-2 or 2-3 = 46.34%
This simply drives home the point that the initial table odds that you so frequently see and hear about can change rapidly once the complete division of any second suit is known. In this case the odds of a 3-2 spit in the diamond suit have been reduced from 68% to 46% by reason of the club division. While ordinarily the probability of an 8 card suit dividing 3-2 is far superior to taking a simple finesse in another suit, the aberrant division of the clubs has now made the 3-2 split in diamonds a decided underdog to a finesse. The point is not the arithmetic or memorizing probability tables, but when the cards around the table in one suit start to do funny things, don’t expect everything else to be normal – make a disaster plan.
The bid contract as published was 3NT. Assume the scoring is matchpoints. You are on the board with the King of Clubs. If the diamonds split 3-2 you can make 12 tricks by playing a small diamond to the queen and then back to the board with the spade, running the rest of the diamonds from the top. If you decide to “Pig-out” and go for 12 tricks, you actually can make only 8 tricks since the diamonds split 1-4 and you now do not now have enough board entries to try the “split honors” finesse in the heart suit.
At matchpoints we take the line of play that will both fulfill our contract and, at the same time, present us with the greatest probability to net us the maximum number of overtricks without jeopardizing the contract.
In Frank Stewart’s analysis he did not put all his eggs in the “3-2 diamond split basket,” since he knew that the 6-0 split in clubs was bad news. He played the diamond Ace from the board (Queen from his hand), then the King of diamonds hoping for a 3-2 diamond split and 5 diamond tricks. When East showed out on the second round he put Plan B in motion leading a small heart losing to West’s Queen. He won the club return in his hand, used his spade entry to get back to the dummy, led is 9 of hearts and let it ride, ultimately winning 2 spades, 3 hearts, 2 diamonds and 2 clubs.
Frank’s line of play was a heavy favorite to make 9 tricks, and at the same time he preserved his long shot of finding the diamonds 3-2 and making 5 diamond tricks for a total of 11 tricks. There is a superior line of play to the one chosen by Frank that will preserve the same opportunities, introduce no further risk and at the same time make an overtrick for top board. Did you find it? Hint: You don’t need to smother your own Queen of diamonds!
^^^^^^^^^^943
^^^^^^^^^^AK65432
^^^^^^^^^^K
Q964^^^^^^^^^^^^^87532
Q3^^^^^^^^^^^^^^^K762
9^^^^^^^^^^^^^^^^^J1087
QJ10985^^^^^^^^^^None
^^^^^^^^^^KJ
^^^^^^^^^^AJ108
^^^^^^^^^^Q
^^^^^^^^^^A76432
The above hand is from Frank Stewart’s Bridge Column on Sunday, September 9. Let me digress a moment to discuss probabilities before I get to the play of the hand. The probability of the division of the 8 card diamond suit prior to the play of any card in the deal is:
5-0 or 0-5 = 3.91%
4-1 or 1-4 = 26.26%
3-2 or 2-3 = 67.83%
The Queen of Clubs is led and East discards the spade 2. We now have completely discovered the division of the opponent’s holding in the club suit as West 6- East 0. We learned in Odds and Ends of Bridge (Part 1) about the Law of Attraction, that a long suit in one hand tends to attract another short suit and that a long suit tends to reject another long suit. The probable distribution of the 8 card diamond can now be recalculated to:
5-0 or 0-5 = 8.44%
4-1 or 1-4 = 35.22%
3-2 or 2-3 = 46.34%
This simply drives home the point that the initial table odds that you so frequently see and hear about can change rapidly once the complete division of any second suit is known. In this case the odds of a 3-2 spit in the diamond suit have been reduced from 68% to 46% by reason of the club division. While ordinarily the probability of an 8 card suit dividing 3-2 is far superior to taking a simple finesse in another suit, the aberrant division of the clubs has now made the 3-2 split in diamonds a decided underdog to a finesse. The point is not the arithmetic or memorizing probability tables, but when the cards around the table in one suit start to do funny things, don’t expect everything else to be normal – make a disaster plan.
The bid contract as published was 3NT. Assume the scoring is matchpoints. You are on the board with the King of Clubs. If the diamonds split 3-2 you can make 12 tricks by playing a small diamond to the queen and then back to the board with the spade, running the rest of the diamonds from the top. If you decide to “Pig-out” and go for 12 tricks, you actually can make only 8 tricks since the diamonds split 1-4 and you now do not now have enough board entries to try the “split honors” finesse in the heart suit.
At matchpoints we take the line of play that will both fulfill our contract and, at the same time, present us with the greatest probability to net us the maximum number of overtricks without jeopardizing the contract.
In Frank Stewart’s analysis he did not put all his eggs in the “3-2 diamond split basket,” since he knew that the 6-0 split in clubs was bad news. He played the diamond Ace from the board (Queen from his hand), then the King of diamonds hoping for a 3-2 diamond split and 5 diamond tricks. When East showed out on the second round he put Plan B in motion leading a small heart losing to West’s Queen. He won the club return in his hand, used his spade entry to get back to the dummy, led is 9 of hearts and let it ride, ultimately winning 2 spades, 3 hearts, 2 diamonds and 2 clubs.
Frank’s line of play was a heavy favorite to make 9 tricks, and at the same time he preserved his long shot of finding the diamonds 3-2 and making 5 diamond tricks for a total of 11 tricks. There is a superior line of play to the one chosen by Frank that will preserve the same opportunities, introduce no further risk and at the same time make an overtrick for top board. Did you find it? Hint: You don’t need to smother your own Queen of diamonds!
Friday, September 7, 2007
Odds and Ends of Bridge (Part 3)
Well ,you had to know that Odds and Ends (Parts 1 and 2) were laying the groundwork for more serious discussion to follow. I would be content to let well enough alone and have my readership bask in the comfort of reviewing stuff they already know, but since part of my mission is to drive those bridge intermediates up a notch or two, I can’t let everybody get too comfortable. Actually, the principle of Restricted Choice is not all that complicated, since the hard part of any sophisticated bridge play is recognizing when it confronts you. In Restricted Choice, it really hits you over the head since you unexpectedly see an opponent drop an honor in the first play in the suit.
Playing Rule for Restricted (Free) Choice Situations
The application of this playing rule often presents itself in a 9 card suit where declarer holds A9xxx in one hand opposite KTxx in the other. With QJxx outstanding, the initial plan is to split the suit 2-2 and drop both honors with the second round played. You start with the Ace from your hand, and, lo and behold, East drops the Q or J under the Ace (it does not matter which, since the math solution ignores the possibility of a false card by East). Now you have to decide whether East holds QJ doubleton or a single honor.
If he was dealt a doubleton honor, then you have to continue your plan to drop the other honor. If East was dealt a singleton honor, that means that West was dealt Jxx or Qxx, as the case may be, and the correct play is to play toward the K10 in the dummy planning to finesse against West for the other honor (Q or J). While the rule with a 9 card suit generally favors a “drop strategy,” this combination of events is an exception to this rule. It is called the Rule of Restricted Choice, although more modern authors generally refer to it as “Free Choice.”
In my example, on the second lead toward the K10x, you should insert the 10, finessing West for the missing honor. This seeming anomaly is mathematically correct and supported by the laws of probability. The correctness of the rule, often disputed in the past, is now conceded by all experts.
Here are the Rules and Odds for Restricted Choice Situations:
(a) In a nine card suit when there are two missing equivalent honor cards in a suit (e.g. Q, J) along with 2 small cards, if on the first lead one of the opponents plays either one of the honors (Q or J), it is almost 2:1 odds that the honor played is a singleton and you should finesse the other opponent for the remaining honor if you can.
(b) If you have an eight card suit (e.g. K765 opposite A1098) and there are two equivalent honor cards outstanding (Q and J, along with 3 small ones), if on the first lead one of the opponents plays either one of the missing honors (Q or J), the odds are approximately 5:3 that the honor played is a singleton and you should finesse the other opponent for the remaining honor if you can.
(c) If you have a seven card suit (e.g. K76 opposite A1098) and there are two equivalent honor cards outstanding (Q and J, along with 4 small ones), if on the first lead one of the opponents plays either one of the honors (Q or J), the odds are approximately 3:2 that the honor played is a singleton and you should finesse for the other opponent for the remaining honor if you can.
Other restricted choice positions are AKQ9 opposite xxx. If on the lead of the Ace you see the 10 played, lead a small heart from your hand and insert the 9. The J and Ten are equivalent honors.
Here is a little different twist. Dummy has J9x and you hold Qxx. Unfortunately, neither opponent led this suit for you, so you have some work to do. Lead small from the dummy toward the Queen, assume this is taken with the Ace by West. Since the Ace and King are equivalent honors, it is unlikely that West holds both of them. The correct play is to come to your hand and play toward the J9. If West plays a small card, insert the 9, since East is a favorite to have the King. If he also has the 10 there is nothing you can do about it, but if West started out with A10x, you will make a trick in this suit.
One final example. K109 opposite xxx. You play small and insert the 9 which loses to either the J or Q. For the purpose of Restricted Choice applications, the Q and J are equivalent honors. We know that it is more likely that any honor played is not likely to be accompanied by the adjacent honor. We need to lead again toward the K10, and if West plays a small card we put in the 10, not the King. East may have the Ace, but the odds favor West holding the Jack and we hope that the 10 will force the Ace in the East hand. If so, we get a trick in the suit.
What is the common denominator? It is that you are missing two equivalent honors in the suit and on the first lead you see one of them, bet that it is a single and take the finesse (if available) against the other opponent.
In Part 4, I will wrap up all this high math with a discussion of the “Deletion Principle”, demonstrating that middle cards can make a difference and that some 7 cards suits can actually be favorites to break 3-3.
Playing Rule for Restricted (Free) Choice Situations
The application of this playing rule often presents itself in a 9 card suit where declarer holds A9xxx in one hand opposite KTxx in the other. With QJxx outstanding, the initial plan is to split the suit 2-2 and drop both honors with the second round played. You start with the Ace from your hand, and, lo and behold, East drops the Q or J under the Ace (it does not matter which, since the math solution ignores the possibility of a false card by East). Now you have to decide whether East holds QJ doubleton or a single honor.
If he was dealt a doubleton honor, then you have to continue your plan to drop the other honor. If East was dealt a singleton honor, that means that West was dealt Jxx or Qxx, as the case may be, and the correct play is to play toward the K10 in the dummy planning to finesse against West for the other honor (Q or J). While the rule with a 9 card suit generally favors a “drop strategy,” this combination of events is an exception to this rule. It is called the Rule of Restricted Choice, although more modern authors generally refer to it as “Free Choice.”
In my example, on the second lead toward the K10x, you should insert the 10, finessing West for the missing honor. This seeming anomaly is mathematically correct and supported by the laws of probability. The correctness of the rule, often disputed in the past, is now conceded by all experts.
Here are the Rules and Odds for Restricted Choice Situations:
(a) In a nine card suit when there are two missing equivalent honor cards in a suit (e.g. Q, J) along with 2 small cards, if on the first lead one of the opponents plays either one of the honors (Q or J), it is almost 2:1 odds that the honor played is a singleton and you should finesse the other opponent for the remaining honor if you can.
(b) If you have an eight card suit (e.g. K765 opposite A1098) and there are two equivalent honor cards outstanding (Q and J, along with 3 small ones), if on the first lead one of the opponents plays either one of the missing honors (Q or J), the odds are approximately 5:3 that the honor played is a singleton and you should finesse the other opponent for the remaining honor if you can.
(c) If you have a seven card suit (e.g. K76 opposite A1098) and there are two equivalent honor cards outstanding (Q and J, along with 4 small ones), if on the first lead one of the opponents plays either one of the honors (Q or J), the odds are approximately 3:2 that the honor played is a singleton and you should finesse for the other opponent for the remaining honor if you can.
Other restricted choice positions are AKQ9 opposite xxx. If on the lead of the Ace you see the 10 played, lead a small heart from your hand and insert the 9. The J and Ten are equivalent honors.
Here is a little different twist. Dummy has J9x and you hold Qxx. Unfortunately, neither opponent led this suit for you, so you have some work to do. Lead small from the dummy toward the Queen, assume this is taken with the Ace by West. Since the Ace and King are equivalent honors, it is unlikely that West holds both of them. The correct play is to come to your hand and play toward the J9. If West plays a small card, insert the 9, since East is a favorite to have the King. If he also has the 10 there is nothing you can do about it, but if West started out with A10x, you will make a trick in this suit.
One final example. K109 opposite xxx. You play small and insert the 9 which loses to either the J or Q. For the purpose of Restricted Choice applications, the Q and J are equivalent honors. We know that it is more likely that any honor played is not likely to be accompanied by the adjacent honor. We need to lead again toward the K10, and if West plays a small card we put in the 10, not the King. East may have the Ace, but the odds favor West holding the Jack and we hope that the 10 will force the Ace in the East hand. If so, we get a trick in the suit.
What is the common denominator? It is that you are missing two equivalent honors in the suit and on the first lead you see one of them, bet that it is a single and take the finesse (if available) against the other opponent.
In Part 4, I will wrap up all this high math with a discussion of the “Deletion Principle”, demonstrating that middle cards can make a difference and that some 7 cards suits can actually be favorites to break 3-3.
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