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Playing the Percentages
It's not always possible, in the middle of the play of the hand, to calculate the precise percentages of alternative lines of play, and sometimes intuition is required to make the right choice. Check out your own intuition in the following examples.
Related Play Problems Play Problem 6
Related Extracts from Past Wednesday Games
What would you lead against West’s 4♥? - A Spade? Doesn’t look appealing and might blow a trick (picture the Jack in Dummy and South with the Queen but not the Nine) - A Trump? Also not without risk. - A Diamond? That is seriously risky, all the more so as the suit is long and the strength is on North’s right. - A Club? That would be our choice.
OK, which Club do you lead? If you lead a high one then Partner might play you for a doubleton (the good news is that Declarer might also play you for a doubleton). If you lead a low Club, Partner might play you for to have led from three or four to an honor, and might go wrong as a result. We suppose we’d lead the Nine, if only because it is top of a mini-sequence. So, South takes her two Clubs and returns a Club. What should Declarer do? He doesn’t know that North has led from three Clubs, and fears that North started with a doubleton. Declarer has no reason to ruff the third Club low, that’s never a good idea, it just wastes a trump for no reason. So: - If Declarer ruffs with the Queen then North will have two natural trump tricks (and if she had started with ♥KTx and two Clubs then she would, of course, decline to overruff). - If Declarer pitches a Spade then he will get a pleasant surprise when North follows suit. That will be 10 tricks.
We can see which play works, but which is the technically correct play? Surely, pitching a Spade gives Declarer the best chance of success. It’s always preferable when North started with three Clubs, and will also succeed whenever North has a doubleton Club and South has ♥K or ♥Kx or ♥Kxx. Ruffing with an honor works less often, most notably when North has a doubleton Club and ♥Kxx and South has ♥Tx. Did you triumph in 4♥ after the opening lead of the Club Nine?
The auction is simple enough, but the play isn’t. However, Declarer can still prevail, despite the 5-0 trump break. ♥Q won by Declarer’s King (saving a Dummy entry for later) ♠Q won by West’s Ace Spade return won by Declarer’s Jack (again saving on Dummy entries) At this point Declarer has a decision to make. - If the ♦K is onside then the winning line is to cross to the ♥A, finesse the Diamond, cash ♦A, knock out the ♣A and scramble 10 tricks that way (via two high trumps, a ruff in each hand, two Hearts, two Diamonds and two Clubs). - If the ♦K is offside then Declarer cannot afford to use Dummy’s Heart entry for the losing finesse. Instead, she must cash the ♦A, lose to the ♦K, win the Heart return, and run the Diamonds. West is helpless and does best to pitch two Hearts before ruffing in. Declarer overruffs, knocks out the ♣A, and has 10 tricks one way or the other.
We can see which line works, but which one is theoretically better? Actually, the winning line is also the percentage line, for two reasons: (a) East has no Spades so has more room in his hand for the ♦K; (b) The second line also succeeds when West has a Diamond singleton. Well played those who got to 10 tricks on this one, you won’t have much company.
A somewhat tortuous auction by N-S, finally arriving at the sensible contract of 6♠. West finds the best opening lead of a trump and here is one line of play in which Declarer makes 13 tricks: Win the ♠A on the board Over to the ♣A and ruff a Club Ruff a Heart Ruff a Club When the ♣K comes tumbling down on the third round of the suit, and when trumps are obligingly 3-2, Declarer has all 13 tricks. But that was not a good line of play! If the ♣K had not fallen in three rounds (which it usually won’t) or if the trumps had been 4-1, Declarer would not even have been able to scrape up 12 tricks. A better line of play would be to use the power of Declarer’s Club spots in an attempt to secure 12 tricks: Win the ♠A on the board Cross to the ♣A Lead the ♣Q, covered by the King and ruffed on the board Draw trumps Now Declarer knocks out the ♣J and makes 12 tricks. This line brings home the slam when: - The ♣K is with West and Clubs are 4-3 (even if trumps are 4-1) - The ♣K is with East, but the ♣J is with West, and trumps are 3-2 and Clubs 4-3. We haven’t calculated the odds but they are surely better than the earlier line.
North’s 2♣ was an inverted raise, 4♣ was Minorwood, and eventually the Club slam was reached.
How does the play go? - If N-S play in 6♣ there are 12 easy tricks, with just a Spade to be lost. - If N-S play in 6NT they will need to score three Spade tricks. The winning line (and also the percentage play) is to run the ♠Q (covered by the King and Ace), and then to finesse against the Nine. That’s about a 40% chance, and about 10% better than cashing the ♠A and leading towards the Queen. That being the case, this is a top board for the optimistic pairs who reached 6NT, provided that they are good at suit combinations.
North’s 1NT opening is a bit off-shape and somewhat deficient in the majors but it still looks like the most practical opening bid to us. After South’s 2♥ transfer bid is doubled, North has extra options available. Here is one way to use those options: - Pass shows a doubleton Spade (after which Redouble by the Transfer bidder is a re-Transfer) - Redouble shows 3 Spades and a maximum (seems more useful than trying to play 2♥ redoubled once every couple of decades) - 2♠ shows 3 Spades but less than maximum - Other bids can be your regular super-accept methods, whatever they may be.
Using those methods, South Redoubles as a re-Transfer and then investigates 3NT by bidding 3♥. North cannot bid 3NT, of course, so 4♠ is the final contract.
Against 4♠, the defense will cash two Hearts and shift to a Diamond. Now it will be 9 or 10 tricks depending upon how Declarer plays the trump suit. All things being equal, the percentage play here is to lead twice towards the long hand, playing a high honor both times. This has a 60% chance of success, whereas finessing the Nine has only a 49% chance. However, the odds change when West is presumed to be long in Hearts, and some Declarers may be persuaded by the auction to finesse the Nine. That may be a well-thought out choice, but it is also an unlucky one!
In the play of 2♠, South makes 8 or 9 tricks, depending on her ability to guess the Q♦. The Q♣ is the most likely lead, covered by the King and the Ace. Say that East shifts to the K♠, in an attempt to stop the Club ruff on the board (he doesn't know that the ruff is no longer needed, thanks to Declarer's Club spots) ... then a Spade to Ace, and a third Spade ... then the Heart finesse, lose a Club, win the Heart return, ruff a Heart (West showing out), cash a Club. At this point, Clubs could be 4-4 or 5-3 either way, which means that West started with 3 or 4 or 5 of the missing 6 Diamonds. That should be enough to play West for the Q♦, which turns out to be the winner.
Suppose that South leads a fourth-best Spade. Now, East will show us whether he’s a Mouse or a Pig: - If East is a Mouse he’ll settle for a safe 11 tricks … he’ll play a low Spade from Dummy, winning the Ace … then he runs the Q♥, even if it loses it will be into the hand which cannot attack Spades profitably … but the Queen is covered by the King and Ace, and Declarer has his 11 tricks (he cannot risk the finesse against the T♥ because if it loses and a Club is returned his communications will be ruined and he’ll end up with only 10 tricks). - If East is a Pig, he’ll play the Q♠ at Trick One … if this loses to the King, it’ll be 9 or 11 depending on whether the Heart finesse works. However, the Q♠ holds the trick, Declarer crosses to the A♦, then the Q♥ is covered by the King and Ace. Declarer is up to 12 tricks already, and runs the Diamonds and cashes the A♠, in the process squeezing North in Hearts and Clubs. 13 tricks, no less! Oink, oink!
Actually, the odds clearly favor being greedy on this one. The safe line (low Spade from the board at Trick One) renders exactly 11 tricks pretty much all the time, whereas the greedy line gives the following (ignoring the surprising additional squeeze trick): 25% of the time (both major Kings with North) it’s 9 tricks 25% of the time (K♠ with North, K♥ with South) it’s 11 tricks 50% of the time (K♠ with South, K♥ anywhere) it’s 12 tricks.
Against 4♠, South can bring the proceedings to a swift close by cashing the A♣ and continuing the suit, at which point a disgruntled Declarer will claim 11 tricks. But that’s an unlikely defense, perhaps a Heart opening lead would be more likely (not that this lead looks like such a great bargain either, by the way). Anyway, a Heart is led, trumps are drawn, and the Hearts are cashed (Declarer is looking for distributional clues before he tackles the crucial Diamond suit). Now the KQ♦ are cashed, both defenders following, and it’s decision time! Here are Declarer’s choices: - Lead a Diamond to the Ace, scoring 11 tricks if the Diamonds are not 3-3, and 12 when they are. - Finesse the T♦, scoring 10 or 12 tricks.
It might seem that “11 or 12” is better than “10 or 12”, and if the relative odds were even money then of course it would be. But there’s no even money here, North showed up with all four trumps, and also followed to all 3 Hearts. As the trumps were being drawn, South had some pitches to make, probably 3 Hearts and a Club. If that was the case, North will be known to have started with 8 major cards, and South with just 5. We’d say that makes South a clear favorite to hold both remaining Diamonds. Making 6 if Declarer trusts his judgment, for a positively gruntled +480.
Against 3NT, East does not have an attractive lead, and may well start out with a top-of-nothing Club. Declarer has three ways to play this hand: - Line 1: Declarer wins the A♣, takes the Spade finesse, cashes the A♠ dropping West’s King, and now (thanks to the onside A♦) crosses to Dummy to enjoy the Spades, racking up an astonishing 11 tricks! Of course, if East has the K♠ and at least two others he will duck that first Spade, in order to lock out Dummy’s Spades. Any Declarers who found that line of play will get a top board, but we’d suggest that they don’t brag about it too much, as that line of play is inferior in the extreme. We calculate that it brings in 11 tricks no more than 4% of the time, 10 tricks another 4% of the time (when East has doubleton King so cannot duck the first Spade) and the other 92% of the time it will result in defeat. Not very good odds! - Line 2: Win the Club lead in hand, cash A♠, then overtake the J♠ with the Queen. This line makes 10 tricks whenever Spades are 3-3 (36% of the time) or when East has doubleton King (another 8%). Even if Spades are 4-2, then the contract still has chances given some luck in the red suits. - Line 3: Win the Club lead in hand, cash A♠, then play the J♠ and low from Dummy (the defense should duck this if they can). Now, if the A♦ is with East, the Spades can be established and enjoyed. That will be 10 tricks 50% of the time, and there is the additional chance of a doubleton K♠ in either hand which takes the chances for 10 tricks up to 58%. And there are additional red suit chances if the Spades don’t come in. Yes, Line 3 looks like the winner, but Line 2 is certainly a vast improvement on that all-or-nothing Line 1.
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