[inquiry]

Scoring: XIMP

7NT south

Opening lead diamond Queen. You have 4S, 2H, 2D, 2C with chances three for extra tricks in hearts and in clubs.

If you try to get count in other suits, all follow to diamonds for two rounds, and west has only 1S --if you have entries to try all that stuff

What is your sequence of plays in clubs and hearts, and what influenced your decision? (8C versus 7 hearts, chance of long hearts with WEST on east having six spades, chance of long clubs with WEST due to East having six spades? Others

Best if someone would calculate the precise best line. Is it close enough to ignore?

---

[gnasher]

Both lines work against any singleton queen, or any doubleton queen onside.

A priori, I make it:

Top hearts, then club finesse gains against:
♥Qx offside: 48.5/6 = 8.1
♥Q guarded and ♣Qxx/Q10x onside: (48.5 x 2/3 + 35.5) x 67.8 x 3/10 = 13.8
♥Q guarded and ♣Qxxx onside
(as long as you kept ♦A in dummy): (48.5 x 2/3 + 35.5) x 28.3 x 1/10 = 1.9
Total = 23.8%

Top clubs, then heart finesse gains against:
♣Qx/Q10 offside: 67.8 x 2/10 = 13.6%
♣Q guarded and ♥Qxx onside: (67.8 x 3/5 + 28.3 x 4/5) x 35.5/2 = 11.2%
Total = 24.8%

These figures aren't exact, because I've ignored the effect of one break on the other, but I think they're close enough.

It's easy to find out about a 6-1 or 5-2 spade break: ♦K, ♠KQ, ♣A, ♥K, ♠A throwing a diamond. I'm not sure what effect this will have on the relative odds.

[Jlall]

At the table I'm sure I would have cashed AK of clubs then tried hearts. No idea which is actually better.

[suokko]

Finding singleton or doubleton ♠ in west changes odds hugely in favor of ♥ finesse.

say 6-1 spade break gives you 12/19=63% for any card in west compared to east with only 7/19=37%. So with these changed odds it favors playing AK♣ and then take the ♥ finesse.

Of course if east holds short spades then odds favor toping ♥ and taking ♣ finesse.

But this hand has quite huge extra problem. We need 3 extra tricks so just falling singleton Q is not enough for making. You still have to take to take the finesse for the spot. It would be quite a defense if someone held Q9♥ and droped Q under king.

So we either need Qx in one of key suits or Q offside and finesse to win from playing one of suit from top.

priori odds:
♥ toping providing direct making is 17.4% + 2.8% (♣Q single offside)
while ♣ toping provides direct making. 30.0% + 1.2% (♥Q single offside)

Now we have to combine that with chance to make after finessing another suit later.
♣ finesse wins 36.7% (Qx, QT, Qxx, QTx or Qxxx onside) off remaining 80%. 0.367*0.798=29.3% which totals to 49.7%.
♥ finesse wins 25.8% (Qx or Qxx onside) off remaining 69%. 0.258*0.698=18.0% which totals to 49.2%.

Of course there is cases when one of suit is void and you still would have chance to make with another suit but these lines are equal in both and should add 2-3% to both lines.

So I don't know where this calculation is off because I got different result to what Andy had. But I suspect there is some combination missing or too much in one of our calculations.

[inquiry]

Jlall, on Dec 12 2009, 06:22 AM, said:

At the table I'm sure I would have cashed AK of clubs then tried hearts. No idea which is actually better.

In the real world of the BBO (where this hand is from) cashing either AK would have worked as East had Qx of both key suits.

Without doing math, we would all try the longer suit (here clubs) first and then hook the other suit (here hearts) just as you suggested (chance of dropping Qx or Q from an eight card suit is nearly twice as likely than in a seven card suit... 32.8 percent versus 18.6).

The fact that EAST held six spades to west's one, makes this play EVEN a lot more favored as West is more likely to have either or both queens by available space theory. So if necessary to take a hook, you want to take the one through WEST. In fact, cash both ACEs, then king of clubs, and if necessary hook in hearts, is more than twice as likely to work as cashing AK of hearts and hooking in clubs.

What if if WEST held the six spades. Would that alter the percentage play from cashing AK of clubs (in longer suit) to cashing A club, AK of hearts, and if necessary hook club? Absolutely, in this case, the vacant space principle for who to hook can override the slight extra odds of dropping Qx from a seven card suit versus Qx from a eight card suit.

[gnasher]

suokko, on Dec 12 2009, 03:35 PM, said:

say 6-1 spade break gives you 12/19=63% for any card in west compared to east with only 7/19=37%. So with these changed odds it favors playing AK♣ and then take the ♥ finesse.

inquiry said:

The fact that EAST held six spades to west's one, makes this play [cash top clubs then finesse hearts] EVEN a lot more favored as West is more likely to have either or both queens by available space theory.

I don't think this analysis holds water. We're interested not only in where the queens are, but also in how many small cards accompany each queen.

Cashing ♣AK first works against specifically ♣Qx on the left and specifically ♥Qxx on the left. The 1=6 spade break obviously decreases the chance of either.

Cashing ♥AK first works against specifically ♣xx on the left and specifically ♥xxxx on the left. The 1-6 spade break still decreases the chance of a 2=3 club break, but I imagine it increases the chance of a 4=2 heart break (or if it doesn't, it decrease it by less than the decrease in the chnace of s 3-3 break).

I don't think you can answer this without arithmetic.

[suokko]

gnasher, on Dec 12 2009, 11:36 PM, said:

Cashing ♣AK first works against specifically ♣Qx on the left and specifically ♥Qxx on the left.  The 1=6 spade break obviously decreases the chance of either.

♣Qx left (4 combinations) or ♥Qxx left (15 combinations)

gnasher, on Dec 12 2009, 11:36 PM, said:

Cashing ♥AK first works against specifically ♣xx on the left and specifically ♥xxxx on the left.  The 1-6 spade break still decreases the chance of a 2=3 club break, but I imagine it increases the chance of a 4=2 heart break (or if it doesn't, it decrease it by less than the decrease in the chnace of s 3-3 break).

♣xx or T left (11 combinations) or ♥xxxx left. (5 combinations)

So now if per combination odds change favoring length in LHO we get massive relative boost for ♥Q onside because there is more combinations to get inflated odds than in ♥xxxx case. Of course this is linear solution so it might easily cause different solution if side suit are splitting very badly this might very well make double finesse in ♣ sometimes the best line.

Is there any tool for creating some nice graphs for this case? I guess some spreadsheet solution might give nice odds graph for the best lines in different spade breaks.

[gnasher]

I think you've missed my point. When spades break 1=6, the probability of ♥Qxx onside goes down, but the probability of ♥xxxx onside goes up.

[suokko]

I did use Henk's dealer to generate some statistics for different lines and how often they work.

Sum from data is that
♠ 0-7 and 1-6 should take ♣ finesse
♠ 2-5 and 4-3 is not giving statistical clear answer in this small set of deals
♠ 3-4 should drop ♣Q
♠ 5-2, 6-1 and 7-0 should take double finesse in ♣

Here is raw data if anyone wants to see how much the spade break affects odds.

$ ./dealer 7NT.test 
priori drop club: 0.288431
priori finesse club: 0.290641
priori double finesse club: 0.265207
0-7 drop club: 0.000414
0-7 finesse club: 0.000508
0-7 double finesse club: 0.000427
1-6 drop club: 0.007338
1-6 finesse club: 0.00784
1-6 double finesse club: 0.006611
2-5 drop club: 0.039797
2-5 finesse club: 0.039971
2-5 double finesse club: 0.033588
3-4 drop club: 0.090335
3-4 finesse club: 0.088058
3-4 double finesse club: 0.075547
4-3 drop club: 0.094708
4-3 finesse club: 0.094059
4-3 double finesse club: 0.085768
5-2 drop club: 0.045756
5-2 finesse club: 0.048525
5-2 double finesse club: 0.0492
6-1 drop club: 0.009419
6-1 finesse club: 0.010907
6-1 double finesse club: 0.012894
7-0 drop club: 0.000664
7-0 finesse club: 0.000773
7-0 double finesse club: 0.001172
Generated 1000000 hands
Produced 1000000 hands
Initial random seed 1260665359
Time needed [space] 9.13 sec

$ ./dealer 7NT.test 
priori drop club: 0.288201
priori finesse club: 0.29044
priori double finesse club: 0.265413
0-7 drop club: 0.000414
0-7 finesse club: 0.000501
0-7 double finesse club: 0.000422
1-6 drop club: 0.007293
1-6 finesse club: 0.008035
1-6 double finesse club: 0.006679
2-5 drop club: 0.039928
2-5 finesse club: 0.039928
2-5 double finesse club: 0.033694
3-4 drop club: 0.090413
3-4 finesse club: 0.08759
3-4 double finesse club: 0.075425
4-3 drop club: 0.09439
4-3 finesse club: 0.094491
4-3 double finesse club: 0.086353
5-2 drop club: 0.045685
5-2 finesse club: 0.048345
5-2 double finesse club: 0.0489
6-1 drop club: 0.009454
6-1 finesse club: 0.010788
6-1 double finesse club: 0.012742
7-0 drop club: 0.000624
7-0 finesse club: 0.000762
7-0 double finesse club: 0.001198
Generated 1000000 hands
Produced 1000000 hands
Initial random seed 1260665388
Time needed [space] 9.17 sec

$ ./dealer 7NT.test 
priori drop club: 0.288373
priori finesse club: 0.291129
priori double finesse club: 0.265955
0-7 drop club: 0.0004
0-7 finesse club: 0.000535
0-7 double finesse club: 0.000465
1-6 drop club: 0.007269
1-6 finesse club: 0.008123
1-6 double finesse club: 0.006836
2-5 drop club: 0.040272
2-5 finesse club: 0.039758
2-5 double finesse club: 0.033461
3-4 drop club: 0.09033
3-4 finesse club: 0.088009
3-4 double finesse club: 0.075951
4-3 drop club: 0.094512
4-3 finesse club: 0.09459
4-3 double finesse club: 0.086117
5-2 drop club: 0.045827
5-2 finesse club: 0.048727
5-2 double finesse club: 0.049352
6-1 drop club: 0.009147
6-1 finesse club: 0.010601
6-1 double finesse club: 0.012633
7-0 drop club: 0.000616
7-0 finesse club: 0.000786
7-0 double finesse club: 0.00114
Generated 1000000 hands
Produced 1000000 hands
Initial random seed 1260665401
Time needed [space] 9.20 sec

And code to produce this output if someone else wants to check it for logic errors.

predeal
 [space] [space] [space] [space]north SKQ, HAJT83, DA64, C864
 [space] [space] [space] [space]south SAJ76, HK2, DK2, CAKJ95

spades_left_0 = shape(west,0xxx)
spades_left_1 = shape(west,1xxx)
spades_left_2 = shape(west,2xxx)
spades_left_3 = shape(west,3xxx)
spades_left_4 = shape(west,4xxx)
spades_left_5 = shape(west,5xxx)
spades_left_6 = shape(west,6xxx)
spades_left_7 = shape(west,7xxx)

top_cl = (shape(west,xxx2) && hascard(west,QC)) || (shape(west,x3xx) && hascard(west, QH))

finesse_cl = ((shape(west,xxx2) && hascard(east,QC)) || (shape(west,xxx1) && hascard(west,TC))) || (shape(west,x4xx) && hascard(east,QH))

double_finesse_cl = (shape(west,x4xx) && hascard(east,QH)) || (hascard(east,QC) && [space]hascard(east,TC) && shape(west,xxx0 + xxx1 + xxx2))


produce 1000000
action
 [space] [space] [space] [space]average "priori drop club" top_cl,
 [space] [space] [space] [space]average "priori finesse club" finesse_cl,
 [space] [space] [space] [space]average "priori double finesse club" double_finesse_cl,
 [space] [space] [space] [space]average "0-7 drop club" top_cl && spades_left_0,
 [space] [space] [space] [space]average "0-7 finesse club" finesse_cl && spades_left_0,
 [space] [space] [space] [space]average "0-7 double finesse club" double_finesse_cl && spades_left_0,
 [space] [space] [space] [space]average "1-6 drop club" top_cl && spades_left_1,
 [space] [space] [space] [space]average "1-6 finesse club" finesse_cl && spades_left_1,
 [space] [space] [space] [space]average "1-6 double finesse club" double_finesse_cl && spades_left_1,
 [space] [space] [space] [space]average "2-5 drop club" top_cl && spades_left_2,
 [space] [space] [space] [space]average "2-5 finesse club" finesse_cl && spades_left_2,
 [space] [space] [space] [space]average "2-5 double finesse club" double_finesse_cl && spades_left_2,
 [space] [space] [space] [space]average "3-4 drop club" top_cl && spades_left_3,
 [space] [space] [space] [space]average "3-4 finesse club" finesse_cl && spades_left_3,
 [space] [space] [space] [space]average "3-4 double finesse club" double_finesse_cl && spades_left_3,
 [space] [space] [space] [space]average "4-3 drop club" top_cl && spades_left_4,
 [space] [space] [space] [space]average "4-3 finesse club" finesse_cl && spades_left_4,
 [space] [space] [space] [space]average "4-3 double finesse club" double_finesse_cl && spades_left_4,
 [space] [space] [space] [space]average "5-2 drop club" top_cl && spades_left_5,
 [space] [space] [space] [space]average "5-2 finesse club" finesse_cl && spades_left_5,
 [space] [space] [space] [space]average "5-2 double finesse club" double_finesse_cl && spades_left_5,
 [space] [space] [space] [space]average "6-1 drop club" top_cl && spades_left_6,
 [space] [space] [space] [space]average "6-1 finesse club" finesse_cl && spades_left_6,
 [space] [space] [space] [space]average "6-1 double finesse club" double_finesse_cl && spades_left_6,
 [space] [space] [space] [space]average "7-0 drop club" top_cl && spades_left_7,
 [space] [space] [space] [space]average "7-0 finesse club" finesse_cl && spades_left_7,

[suokko]

gnasher, on Dec 13 2009, 02:40 AM, said:

I think you've missed my point. When spades break 1=6, the probability of ♥Qxx onside goes down, but the probability of ♥xxxx onside goes up.

While ♥xxxx goes up then ♣ shortness goes down same time. So question is which of these odds change faster.

Looks like ♥xxxx case wins in extreme ♠ positions.

[ceeb]

For what it's worth, I've calculated almost exactly. Suprisingly, a noticeable edge to playing ♥ first -- about 13% where ♣ first is superior, nearly 16% for ♥. (In addition, 20% of the time both lines work because some Qx is onside or both Qx are offside, and a further 19% the line is moot because testing one round of both suits turns up a void or a Q.)

To understand the idea, compare two similar example cases. Suppose the first suit you play finds xx offside. If you play hearts first, consider that it is 3.6% (given 1-6 spades) that RHO is dealt xxxxxx,xx,xx,Qxx -- heart drop fails, club finesse comes to the rescue. That compares with the 0% chance for the club-drop-first to succeed via LHO holding x,Qxx,???,xx since by stipulation both opponents have followed to two diamonds. Even without testing diamonds the same principle applies since a 7-1 split would be unlikely.

Without the spade distribution information, clubs-first would be about 1/2% better.