Dummy holds: ♠ - ♥ AQJ ♦ - ♣ A
You hold: ♠ 2 ♥ 2 ♦ 2 ♣ 2
How do you play to maximize your chance of getting all of the last four tricks? Assume that the only point card left is the king of hearts and there is a diamond higher than the 2 in one of the opponents' hands.
Obviously it depends on your situation, so say that the following happened: your partnership started with 21 high card points between you and during play LHO has played 16 points and RHO has played none. Does this change your answer? What are the probabilities now?
Does your answers change depending on which of the following situations happened?
- Neither opponent bid during the auction
- LHO opened an artificial 1♣ showing 16+ points
- LHO opened a standard bid showing 13-21 points. Does it matter what bid it was?
Uhh, I guess you have the only trump left otherwise you'd just not get all 4 remaining tricks.
ReplyDeleteI'd probably lead 2 of hearts and play the ace, then play the queen and hope LHO doesn't have king and drop the 2 of diamonds.
Would be more likely to finesse LHO if LHO opened the 1club, but more likely to finesse RHO if it was standard. Also depends on number of hearts already played.
Of course, I'm a noob at bridge, so =[
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ReplyDeleteWhen I say both opponents are silent I assume that in that case one of your partnership opened the bidding.
ReplyDeleteSay LHO opened the strong 1 club. At the beginning of the hand, with no other information, he would be a big favorite to hold the king of hearts (The exact odds depend on which other high cards are missing). However, once we find out that he has the other missing 16 HCP, I think the information is no longer useful. For every hand LHO holds with the king, a possible hand without the king can be constructed simply by switching the remaining cards, and this will still be consistent with the bidding.
ReplyDeleteThere are two lines I see: Finesse the Q or take the ruffing finesse. The first line works when LHO was dealt the king, and the second works when RHO was dealt the king or if LHO has the singleton king. Thus, I would say the second line is a slight favorite.
Some people have complained to me that there is not enough information in the problem. They are missing the point of the question. It's not so much a question of the correct line of play, but whether one of the lines of play is actually statistically better (if there are more than 5 hearts out the singleton king is impossible), and if that difference is significant. Furthermore, there is the question of whether the bidding makes a difference at this point. If you think it depends on some other variable that I didn't state, then tell me what variable it is and how it depends on that variable.
ReplyDeleteBefore the hand, it's much more unlikely for LHO to have 19 points and RHO to have 0 points than LHO to have 16 and RHO 3. But after 16 points have been played, is it still less likely?
I would like to point out that in this situation, no heart tricks have been played, so knowing the number of hearts each opponent dropped would also be helpful. I agree with Jack's logic for the most part.
ReplyDeleteTo be honest, the fact that the auction and line of play are missing actually does have a large impact on the problem. We're missing the negative inferences from what the opponents failed to bid (overcalls and takeout doubles and such) as well as which cards they chose to play in what order (What did he choose to lead? Was there any useful signaling going on?) How have the other suits been breaking? (I want my vacant space information.) Also if one or both of them turned out to be holding solid honor sequences and/or long suits their failure to bid them tells us something significant about their hand.
ReplyDeleteAlso, on an unrelated note, I suspect that this hand wasn't played optimally. Was it impossible to set up a red suit squeeze?
The only situation I can really think of where the king is approximately equally likely to be in either hand is if LHO opened with a strong 1C and righty managed to deny holding as many as 6 HCP before you guys got to 2S. eg: 1C-1S-P-2S; AllPass. Then all the suits would have to split evenly, so that the vacant space information is equal between the hands. At that point I would make my decision based on what signaling was done where, and whether LHO had an easy time playing the hand up to this point, and whether he avoided playing a heart like the plague.
I would be interested to find out what the exact (or approximate) scenario actually was though, if you still remember it.
Since I really want this to be a math question more than a bridge question, let's look at it this way. Clearly if any signals have been sent that suggest that one side or the other prefers hearts, that signal won't be sent with Txxx, so he'll almost certainly have the king. So let's pretend that we have no useful signaling information (for example, if they play a random spot card from a suit when it doesn't matter).
ReplyDeleteThe actual situation is here: http://www.bridgebase.com/tools/handviewer.html?myhand=M-7242845-1267423436
In it, the opponents' hands were reversed (so the ``strong'' hand was on the ruffing finesse side). I immediately took the ruffing finesse against the strong hand and was disappointed when the plain finesse was the one that would work. So I thought -- if I had played some number of the other tricks beforehand, would I have gotten information that would tell me that the king is actually located on the weak side, or would I not?
Of course playing a large number of tricks in the actual scenario is impossible, as I would probably not play for the singleton king of spades, and also the partnership is so weak that I really can't play that many tricks beforehand -- South could always lead hearts and force me to guess right then.
So supposing that a red suit squeeze is impossible, but that you can run suits down to this four card ending if you wanted to, is it beneficial? This is essentially the most extreme case I could think of where you would have the most information suggesting that the weak hand has the king of hearts, but does the amount of information you have reduce it to simply 50-50?
The issue with looking at signaling is that if this is indeed the situation, and the defense realizes that all they need is for you to fail a finesse somewhere and they won't have any chance for any other tricks anywhere, then they can easily start sending false signals all over the place. So is there a statistical, rather than psychological, reason for playing either side more than the other?
Here's something to think about, which might say that there is a case for playing the weak hand for the king. When making the plays, the weak hand is going to often make unimportant plays of spot cards, while the strong hand is going to be playing mostly honors, and he won't have very much say in which honor he chooses.
Therefore, though there are the same number of starting hands consistent with the play so far with either opponent having the king (a bijection can be created by, for example, switching the top k hearts, where k is the minimum number of hearts remaining in each hand), in the cases where RHO has the king of hearts, there are fewer ways for him to play it down to this four card configuration, as he necessarily has the opportunity to drop hearts at some point, even if he doesn't. That means that once this line of play has occurred, you were more likely to have started in a situation where the RHO had the king of hearts, as then the line of play that actually happened was more likely.
Does that argument hold any water?
As for the actual hand, I think you just got completely fixed. If you were to believe North's bids, and he were to lead a club honor like a normal person, my thinking would go like this: North is almost certain to hold the diamond ace because he clearly cannot have enough defense for a double on this hand otherwise. This means that we have only 2 fast minor suit losers and can take our time playing the hand. So I would just start with trumps and delay the heart guess.
ReplyDeleteAs for the more abstracted question, your restricted choice argument is absolutely right, but almost always negligible in comparison to considerations from the auction and the play of the hand. Obviously against this pair that wasn't really the case...
That's pretty cool, how'd you flip blogs?
ReplyDelete