Percentage play

[frank0]

[hv=pc=n&s=skqjt987h32da2c32&n=s32hakq54d543cakj]133|200[/hv]

South play 6♠, opp lead a ♦, what's the best play?

[ggwhiz]

Win the Ace, club to the jack = 50-50

 

Playing hearts needs a 3-3 split or the doubleton heart to have the stiff ace of spades, less than 40%.

[frank0]

Not exactly, they can ruff club also if club is not 5-3 or 4-4

[barmar]

5-3 and 4-4 together are 80%, so the finesse is 40%. 3-3 ♥ is 36%. 4-2 is 48%, and stiff ♠A is 12.5%, so stiff A with the short suit is 3% (that's treating the suits independently, and I presume it goes down when you calculate conditional probabilities). So the finesse is still a slight favorite.

 

There's a great falsecard opportunity on this hand. If the ♣Q is in the slot and ♥ are not 3-3, play it smoothly when declarer leads towards dummy. Declarer will assume it's a singleton, so he'll switch to the heart line. And even if you don't come up with this play on trick 2, it also works on trick 3 when he cashes the high ♣. If he's not fooled, you haven't lost anything.

 

I wish I could think of things like this at the table.

[EricK]

Win the Ace, club to the jack = 50-50

 

Playing hearts needs a 3-3 split or the doubleton heart to have the stiff ace of spades, less than 40%.

But on half the 4-2 ♥ splits RHO will have to ruff low, you can overruff and still try the ♣.

[Mbodell]

And the short hearts could also be void in trumps, not just stiff A. 4-0 is 10%, so that adds another couple of % to the line. So I think that puts hearts at:

 

3-3 = 36% all work = +36%

4-2 = 48%, short ♥ with no spades or stiff A is (a bit less than) (12.5%+10%)*50% that works out to +5.4%.

Plus when 4-2 is onside with bad spades we fall back on clubs for (48% - 48%*(12.5%+10%*50%))/2 * 40% = +8.52%

5-1 + 6-0 w/ void spades = is quite a bit less than 16%*10%*50% = +0.8%

 

All together, I think this works out to just over 50%. I know we haven't done the conditional probabilities right, but that shouldn't be off by that much.

[Free]

And the short hearts could also be void in trumps, not just stiff A. 4-0 is 10%, so that adds another couple of % to the line. So I think that puts hearts at:

 

3-3 = 36% all work = +36%

4-2 = 48%, short ♥ with no spades or stiff A is (a bit less than) (12.5%+10%)*50% that works out to +5.4%.

Plus when 4-2 is onside with bad spades we fall back on clubs for (48% - 48%*(12.5%+10%*50%))/2 * 40% = +8.52%

5-1 + 6-0 w/ void spades = is quite a bit less than 16%*10%*50% = +0.8%

 

All together, I think this works out to just over 50%. I know we haven't done the conditional probabilities right, but that shouldn't be off by that much.

I don't know where you get the percentage from the 4-2 split with certain ♠ splits from.

 

My calculations (extreme ♦ splits not counted):

♣ finesse:

50% minus some poor splits.

any 8-0 = 0.08 * 2 = 0.16

any 7-1 = 1.43 * 2 = 1.86

any 6-2 = 8.57 * 2 = 17.14

Total = 19.16% where the finesse will fail due to poor splits.

So playing on the ♣ finesse = 80.84% / 2 = 40.42% (remarkable that it goes down that much, did I make a mistake somewhere???)

 

Playing on ♥ (+ ♣ finesse if possible):

3-3 = 35.53

2-4 = 24.22, in combination with 0-4 (5.78%) or 1-3 singleton A (24.87 / 4 = 6.2175) = 2.906%

4-2 = 24.22, in combination with any ♠ split but with the ♣ finesse onside (40.42%) = 9.79%

Total without 5-1 or 6-0 ♥ splits: 48.226%

any 5-1 or 6-0 = very little, I'm getting lazy, I trust you are right ;)

 

The chance of playing on ♥ is already much higher than the ♣ finesse.

[Fluffy]

Free you just forgot that when RHO ruffs third club you can try hearts later. But everything is quite complicated to calculate.

 

anyway it seems clear that attacking hearts first gives the best chance. 3-3, good recovery on 4-2 and a slim chance on 2-4.

[Mbodell]

I don't know where you get the percentage from the 4-2 split with certain ♠ splits from.

 

I was being lazy to get a rough idea and wasn't updating the splits for the vacant spaces. So a 3-1 split is 50%, the A being stiff is 1/4 of those for 12.5. Likewise a 4-0 is 10%. But then we only care about when the good thing happens on the short hand, and for simplicity I was saying that is 50/50. This is obviously incorrect and it gives us that a 4-2 split will be fine, without needing to overruff the 5.4% of the time. Really on a 4-2 split the stiff A with the short hearts is 4.77% of the time (not the 6.25 I was estimating) and the void is 2.6% of the time, so this really works out to +3.57%. The exact value on the 5-1 hearts with void spades and 6-0 hearts with void spades (ignoring any missing auction) is 1.44% of the 5-1 splits have void spades and 0.72% of the 6-0 splits have void spades. That works out to 1.44%*14.53%+0.72%*1.49% = 0.22%. So my naive Q&D estimates were a reasonable amount higher than the exact amount when I use a suit break calculator.