King-Queen-Nine-Eight-Sixth
Opposite Singleton Ten

by Jon Matthew Farber

This is a surprisingly complex suit combination to handle, with the optimal strategy not being at all readily apparent. Consider the following:

	NORTH ♠ A Q 4 ♥ 10 ♦ A Q 2 ♣ A K 7 6 5 4
	SOUTH ♠ K 3 2 ♥ K Q 9 8 5 4 ♦ K 4 3 ♣ 3

After a relay auction in which South has shown 3-6-3-1 distribution with 11-12 high card points, the king and queen of hearts, and the two pointed kings, North, behind in the match and gambling that South has the heart jack, places South in six notrump.

Spade four, queen, five, two.
Heart ten, three, ?

Plan the play.

Assuming he is only playing the hand once, South should play a high heart, hoping to drop the jack doubleton offside. For those who wish further information, read on, but be forewarned, it is going to get complicated.

South has two options. He can play for East to have started with jack-tripleton and let the ten ride, or he can play for West to have jack-doubleton or singleton. The former is more likely a priori and suggests finessing, but that is a superficial analysis. Instead, we must first consider the situation from East's perspective.

East knows that South needs five tricks from hearts, and South knows that East knows this. The only heart holding East can have where his play makes a difference (assuming he does not make an outright error) is jack-tripleton without the nine or eight ('ace-jack-small' or 'jack-small-small'). If East has the nine with the jack, he must cover. If he has the eight with the jack, he also must cover; if South has the nine East's play makes no difference, but if South lacks the nine not covering allows South to let the ten ride and make the contract.

If East does not have the eight or nine, there are three hand types for South he must consider. If South has the nine without the eight, it makes no difference whether he covers the ten or not. If South's hearts are king-queen-eight or worse (which we will call weak hearts), he must cover, or South will let the ten ride and make his contract. However, if South has king-queen-nine-eight of hearts ('strong hearts'), he must play low in the hopes that South will try to catch the jack-doubleton offside. As such, East needs to know the likelihood of various possibilities. (To give the calculations a semblance of manageability, we will ignore the times when East does not even have the seven, and would need to consider the possibility that South has king-queen-nine-eight-seven-small, which slightly increases the attractiveness for South of not finessing, in case West started with a singleton jack).

With the distribution known, along with the position of the king, queen, and ten, there are 120 ways in which East can hold three cards; the likelihood that East started with ace-jack-small (no nine or eight) is 6 out of these 120 possible combinations. After these three cards are set, there are seven unknown cards left, of which South has four, giving a total of 35 possible combinations, Of these, South will have strong hearts in 10/35 cases, weak hearts in 15/35, and the nine without the eight in 10/35. Lastly, the a priori probability that East started with three hearts, given that South has six and dummy one, is .356. Putting these together, the likelihood of ace-jack-small/ strong hearts is:

(6/120)(10/35)(.356) = .00509. If East covers the ten C fraction of the time (and ducks 1-C), and South finesses with strong hearts F fraction of the time (and plays for the drop 1-F), then he will make whenever East covers (.00509C), and also when East ducks and South finesses, an expectation of:
.00509F - .00509CF.

South will also make if East has ace-jack-small and his hearts are weak (6/120 x 15/35)(.356), and East does not cover (.00763 - .00763C), and he will make when he has the nine alone regardless of East's play (.00509)

Lastly, we need to consider when South has strong hearts, and East four without the jack. Again, with the location of the king, queen, and ten known, there are 210 combinations of cards for South's remaining four hearts. Of these, they will be strong in 15 cases. For East's four cards after South is slotted his, there are 15 possibilities, in which he will not have the jack in 5. The a priori likelihood of East having started with four hearts is .242. Finally, these must be prorated to compare with East having ace-jack-small (6 cases) versus jack-small-small (15 cases). Putting this all together gives a value of:
(15/210)(5/15)(.242)(6/21) = .00165. South will make if he does not finesse, or:
.00165 - .00165F.

Putting these all together gives our first equation, which represents South's relative expectation of success when East holds ace-jack-small. Equation #1 is then:

South's expectation of success (East ace-jack-low) =
.01437 - .00254C + .00344F - .00509CF

From the above, it can be seen that East's best strategy is to always cover. It looks as though South's best choice is to never finesse, but the position needs to be analyzed further.

The calculations when East has jack-low-low are similar, but the resulting equation is more complex. Here, since East has one more small heart than before, leaving one less small heart for South to have, the relatively likelihood that South has strong hearts increases. The likelihood of jack-low-low/strong hearts is:

(15/120)(6/35)(.356) = .00763, jack-low-low/weak hearts:
(15/120)(5/35)(.356) = .00636, and jack-low-low/king-queen-nine:
(15/120(4/35)(.356) = .00509. For strong hearts/four without the jack we have:
(15/210)(5/15)(.242)(15/21) = .00412.

Equation #2 is then:

South's expectation of success (East jack-low-low) =
.01557 + .00127C + .00351F - .00763CF

The difficulty is that South does not know East's hand, so whatever choice he makes for F in the second equation also applies to the first, and vice-versa. Thus, if he follows the strategy of never finessing, as in the first equation, East can counter by never covering. Then, if South guesses this, he can adjust his approach by always finessing, to which East can counter-counter, etc. If South does not wish to get into a guessing game with East, he does have a way to deal with this. Adding the two equations together (setting C=1 in the first equation) gives:

Overall South expectation =
.0274 + .00127C + .00186F - .00763CF (where here C refers to the probability of East covering when holding jack-low-low).

The above equation has some interesting properties. If South does not want to outguess East, he can finesse 16.7 percent of the time, neutralizing any approach East may take. East for his part can elect to cover 24.4 percent of the time, taking South out of the equation. Either player may deviate from this, but they run the risk of being outsmarted.

Thus, the optimal strategy is for East to always cover when he started with ace-jack-small and to cover 24.4 percent of the time when he started with jack-small-small. South in turn should play for the drop five-sixths of the time.

Of course, it is improbable that all of this will be calculated at the table. There is a simple, psychological reason, for South to play for the drop against an expert. That is, it would be very difficult for East to play low with jack-tripleton and allow South to score an unmakeable slam if he started with weak trumps. Of course, if East is a known subscriber to The Bridge World . . .

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