Combining chances

[jcreech]

I was on defense on this hand, and was guilty of putting declarer a less than desirable position.  The auction at the other table was 1 - 2 - 4 - All pass, so declarer did not have to think much about how to play the hand. 

At our table, as East, I bid 5 and South carried on to 5 .  Now you have to think about how to avoid a third loser. 

  T6543
  AT87
  6
  965


  AQJ72
  KJ4
  K8752
  -

Lead: A

This is a hand where chances can be combined. 

1.  There is a two-way finesse in hearts, as well as play for Qx.
2.  You can either finesse the K or play for the drop.
3.  You can play for the A to be onside and the suit to break 4-3 or you can play for the Axx on either side (ducking the first trick completely).

In many ways this is a challenge for Ken - to identify the best combination of chances - but it is a challenge for others to select the best combination with essentially only the tools you would have at the table to select a line.

I will go first, and try to remain objective despite having seen the full hand.

My plan would revolve around a line where a specific player has at least one of the three critical cards.

If I were to play West (the overcaller) for at least one of the three cards, then:
1.  after winning the lead, I would lay down the A, hoping for the stiff K;
2. if that did not work, then I  would duck a diamond, planning to ruff out Axx in West - if trump broke 3-0, then I would use trump as transportation;
3. if that did not work, then I would play the K, then finesse the Q.

If I were to play East for at least one of the three cards, then:
1. after winning the lead, I would cross to dummy with the A to finesse in trump;
2. if that did not work, then I would cross to dummy to lead a diamond toward the K, then ruff two diamonds - using the K and the fifth diamonds for heart pitches;
3. if that did not work, then I would cross to dummy once more to finesse the Q.

I think the percentages work better for playing East for at least one critical honor, but I think that the overcall increases the chances that West actually has at least one of the critical honors, but will it really matter unless it includes at least one of the following:
1. stiff K on either side
2. Axx on either side
3. Q

So even though I think that West is more likely to have all three missing honors, I feel like the percentage are with playing East for at least one.

What are your thoughts? 

(And Ken, I would at least know which of my two lines is better, as well as what line is best.  Thank you.)

[Masse24]

Thoughts that run through my mind as I look at the opening lead:
1. Opps are vulnerable.
2. Opps have 22 HCP.
3. Overcaller, presumably since vulnerable, has roughly--at least--an opening hand.
4. For advancer to bid at the 5-level, vulnerable, he must have values and length. So I'm guessing something like a 12-10 split in HCP and 6-4 or 5-5 in the suit.

Just glancing at the hands provided, without calculating percentages, but considering the opening lead, this is how I would proceed:

I ruff the lead. Not low, but with the 7. While it will not be necessary as an entry to dummy, preserving the 2 is just good technique.
Play a to the Ace in dummy.
Then a toward the King. If it works, great. If not, no loss as I've created a suit to ruff in dummy for later finesses.
I'm always taking the finesse, unless something along the way indicates otherwise. The 50% finesse being better than the 26% drop.
I've committed myself to taking the finesse (if necessary) in hand.
But the suit may be key. It's long--5 cards--and I have a stiff in dummy. With seven outstanding, the percentage of a 4-3 break is roughly 62%. It's probable that I can set it up with repeated ruffs so that the finesse is unnecessary. So while the early play of a to dummy permits the play toward my King (I can discard a in dummy on the King if the Ace is onside), it is also the start of my plan to set up the suit. Usually those five-card suits to set up are in dummy. This time it's in my hand.

I would need more coffee to calculate the "best" line combining chances. 

[kenberg]

If you are thinking of me because I am a mathematician, I will say again that while math can be useful I doubt that it is the right place to start here.

I am not seeing where it says who was vul.  If that 5 was by vul against non-vul I imagine he might be thinking 5 is making. And maybe it is if spades are 3-0 and hearts are 4-2.


Anyway we need to think about what conclusions can be drawn from the auction. Probably the A can be on either side but if W has Axx I can establish both the A and the fifth by leading a small . That would just about bring in the contract and it's pretty tempting. Say I duck a at T2, they play another , I ruff a low and lead a , say E follows,  to the Q and maybe it loses.  They return another , I ruff. I am down to one trump in my hand. I can ruff another (with the T and if the A comes down I draw the last trump (with what is now my last trump) and claim since I can toss two hearts on the diamonds and I still have a trump on the board to handle my last heart.

But maybe E started with Axxx. I have lost the spade finesse, I have not yet brought down the A, I am on the board after ruffing that third round of , but I still come to my hand with a , drawing their last spade.  As I draw the last spade maybe I find W had a stiff K. This has not gone well but I am not dead yet. I could have dropped a stiff K, I could have led a toward my K, but I can still ruff a on the board and establish the K. So now I just have to get the hearts right, the K will take care of the long heart.

If W held the stiff K I will play him for the Q, if W held Kx I will play E for the Q. If by any chance it was W who held Axxx then I will play E for the Q. The more spades and diamonds W holds, the fewer hearts he holds.

Am I confident?  Hardly. But a lot of things could go right, for example maybe the spade finesse works. Occasionally finesses actually work.

[jcreech]

vul vs vul

But for probabilities, I was thinking of a priori. For actual play selection, I would assume that you would make the sort of adjustments to your plan that coincides with the considerations you mention.

And at least one finesse will work (unless BBO placed the Q in both hands like someone did to Hal Sims in the early days of contract bridge).

I will post the full hand in a day or so.

[kenberg]

vul vs vul

But for probabilities, I was thinking of a priori. For actual play selection, I would assume that you would make the sort of adjustments to your plan that coincides with the considerations you mention.

And at least one finesse will work (unless BBO placed the Q in both hands like someone did to Hal Sims in the early days of contract bridge).

I will post the full hand in a day or so.

A priori? It could be done but it would be a challenge. But let me describe how it is done.

Let's take the line where you start be laying down the spade A and then continuing with other tries when that does not work.  The overall plan will fail if all three lines fail. The idea then is to calculate the  probability of failure at the probability of failure at the, the multiply be the probability of failure at the  next step, the multiply by the probability of failure at the third step. But things split, there is more than one way to fail At the first step, laying down the spade A, this fails if the K does not fall. But maybe the suit is 3-0, maybe the suit is 3-1 but the K is not falling. Which way failure occurs has a modest effect on the probability of failure at the second step. For convenience, we will treat the second step as independent of the first step. This means that we just ask "What is the probability of bringing down the A in three rounds of diamonds, w/o knowing anything about how spades are arranged?"

So:

Te probability of a 2-1 break in spades -7 0.78, and one third of the time the K is stiff, so the K falls under that A with probability 1/3 of 0.78  0.26. Which means the probability of the first step failing is 0.74.

Now we go on to the second step. What is the probability of bringing down the K in three rounds of diamonds? Ah, but it is not so simple. If we lead a diamond and then ruff a diamond, it might be over-ruffed (with a higher spot, which can only happen if spades are 3-0, Rho holding the trhee)  and then  we are down, we never get to the third step. But let's ignore that. So the probability of a 4-3 split in diamonds is 0.62, the probability that the K is on the three card holding (we don't care whether E or W) is 3/7 of 0.62, a little under 0.27. So the probability of step 2 failing is 0.73.

So now the probability of both steps 2 and steps to failing is obtained by multiplying 0.74 by 0.73. I get about 0.54.

Now things get complicated.
You say at step 4 you will cross to dummy and finesse [against] the Q.  But now we really have to look at whether spades split 2-1 or 3-0. If they split 2-1, no problem and the diamonds have split 3-4, no problem. And we have to look at which spade we ruffed that third  Not if they  diamond with. If we ruffed low, there was a real danger [well, maybe not that big a danger, W should have at least three so maybe we ruffed low, but we are getting away from a priori] of an over-ruff. So we ruffed high. But now if we lead a diamond it can be over-ruffed. So play a spade first? Not if spades split 3-0 We have cashed the spade ace, we have ruffed two rounds of diamonds, so there are two spades left in dummy. If we lead a spade the K wins and another spade is led. Even of hearts split we will be losing a diamond.

So assume spades were 2-0, but that changes how we view the probabilities at step 1. Anyway, assume it, Then we ruff a diamond and take the heart finesse, making if it succeeds. In some a priori sens, that's a 0.50 shot. So the probability of failure is 0.50 times 0,54 which is 0.27. But it really isn't that good because of the 3-0 spade problem.

So more is needed. And I have been a bit casual about some technicalities.

And then a priori is really not the way to go anyway.

My thinking for my route is that the second will not be over-ruffed on my right. This is because it would give Lho six diamonds, and so even if he also had six clubs he then would have bid 2NT rather than 2 . And I don't careif Lho is short in , if he ruffs it has to be with the K if it is to stand up. So it is safe to fist duck a and then ruff one small in dummy. After that, I lead a spade from the board. I did not really get into what I would do if my Rho shows out, but if he plays low then I play the Q. Firstly the finesse might work, secondly if the finesse fails the I know the spades are not 3-0. So I think that this is a better start. But I did not calculate the probabilities.

And I am definitely not sure it is best.


I did now look up the hand, couldn't bear the suspense, so I know whether or not it works.

[jcreech]

Here is the full hand:

Dlr: S
Vul: B
            T6543
            AT87
            6
            965

  K                  98
  95                Q632
  JT43              AQ9
  AQT732        KJ84

            AQJ72
            KJ4
            K8752
            -

Lead: A

Declarer decided to guess the hearts at trick 2, and, when I misclicked and could not find the undo on version 3 quickly, may have affected how the rest of the hand was played.

Having won the heart, declarer finessed the spade.  My partner returned a second club, ruffed by declarer.  Declarer, now led a small diamond, which I won with the 9.  I led my last trump, won by declarer.  For some reason, declarer led another trump before switching to the K then J of hearts.  When my partner showed out in hearts, declarer rose with the ace.  Now, with no trump left in their hand, declarer was stuck with two more losers in dummy, a heart and a club.

Once declarer chose to play my partner for the Q, he was doomed until my misclick unless he guessed to play for the stiff trump K offside (a 25% chance compared to a 50% chance for the finesse). 

It is interesting that if you play for West to hold at least one of the critical cards, starting with spades and ending with hearts, you will make six.  The trump king falls under the ace, the Axx comes down after two ruffs to provide 2 heart pitches, and the heart finesse is never needed.

Turn it around and play East for one of those cards, and the contract makes as well - losing the K and the A, but you still get the heart pitches, so those are the only losers.

Several lines will work, but I think the critical piece is that you do not take the heart finesse until after you have tried the rest, largely because you may not need it.  I suspect that Ken's line is best because it postpones any heart decision until after the spades and diamonds have been tried.  Clearly, if the diamonds work, you never need to make a heart decision, while if the spade works, the hearts become a possibility for an overtrick.  It is only if everything else fails, you need the heart, and by playing everything else, you have a partial count on the hand, and can improve your chances of making the correct guess in hearts.

[kenberg]

It's a very complicated hand.  I think one way to approach it is with some thoughts of "If I do this, then I will also do that". For example, if I lead a heart to the board at T2, then I will lead a from the board at T3.  The thinking is that I can always get back later to take the spade finesse if I want to. Also, if I start with a small diamond at T2, as I did, then I am going to take the spade finesse. The thinking is that if I am going to decline the finesse, I can just do that. I do not have to go to the board to lead a spade to my A.

Also, some successes are going to be complete, others less so. For example, E might hold Axxxx. Now if I go to the board and lead a , E rises. I can later pitch a heart from the board on the K, but I could still lose a heart and a spade. The same f course applies to my approach. Maybe W started with Ax. Great, the A will fall as I ruff a on the board. But again I won't be pitching two hearts on diamonds. For that to work I need to establish my A and a long . So I have to guess how to score my K, and diamonds must be 4-3.

And what should I think about clubs, are they 6-4 or 5-5? People get frisky but I woul think if Lho has only five clubs then he probably has the A. Or at least it increases the chances.

Also you said something about a misclick, I guess with the hearts. But surely you should play your spots in random order. Your partner already knows the distribution of clubs and anyway if he has to discard he will be discarding clubs. There is no reason to give any accurate signals, no reason for declarer to trust any signal that you do give.

Bottom line: I was not engaging in false modesty when I said that I was unsure of the best line. It's  a complicated hand. I decided to go with trying to bring down the A on my left by leading small and then ruffing s. Well, it brought down the A, but on my right. Unexpected, but fine.

[jcreech]

The misclick occurred at trick 2.  Declarer led a small heart from hand toward the board.  Partner contributed the 9, declarer played the 10 and as I was moving my cursor to play the Q, a small heart was played.  How declarer would have played had I won the Q as intended can only be speculation.

[kenberg]

The infamous brilliant unintended falsecard!