[kgr]

IMPs

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West leads ♦K and East overtakes with ♦A.
East returns ♥7.
What is the best play?

[ArtK78]

If your goal is to make the contract (which I assume is your goal), then you need 6 heart tricks. There is also a small chance that you can establish and cash the clubs while one of your opps ruffs in with a natural trump trick.

One winning line is to play for Kx of hearts in either hand. In that case, you want to play A and then small. The a priori odds of Kx of hearts in either hand are 27.128%. After trick one, when both opponents have followed to one round of diamonds, the odds increse slighly to 27.328%. And, after East shifts to the ♥7, the odds remain the same - 27.328%.

Another winning line with identical odds is to play for the Jx of hearts in either hand. In that case, you want to play the A followed by the Q.

Both of these lines are inferior to finessing either the Q or the 10.

Finessing the Q works whenever the K is onside doubleton or tripleton, and whenever the J is doubleton in either hand. This is clearly better than A and then Q, as it works every time A and then Q works, and it works in additional cases. The odds of the finesse of the ♥Q leading to winning 6 heart tricks after the lead of the ♥7 at trick 2 are 40.992%.

Finessing the 10 works whenever the J is onside doubleton or tripleton, and whenever the K is doubleton in either hand. The odds of the ♥10 leading to winning 6 heart tricks after the lead of the ♥7 at trick 2 are the same as the finesse of the Q at trick 2 - 40.992%.

Anytime the hearts are 4-1 you must lose 2 heart tricks (and anytime the hearts are 5-0 you must lose 3 heart tricks). However, if RHO has 4 hearts and your initial play of the Q or the 10 holds, there is a chance of establishing the club suit while RHO follows to three rounds of clubs and then cashing a long club and throwing your diamond loser while RHO ruffs in with a natural trump trick. This can be done if you play the Q at trick 2 and then find RHO with KJxx or Kxxx of hearts along with either 2 or 3 clubs, or by finessing the 10 at trick 2 and then finding RHO with KJxx of hearts and either 2 or 3 clubs. Since finessing the Q at trick 2 gives you an added chance of finding the singleton ♥J with LHO, it is better to play the Q at trick 2.

In the case that RHO has 2 clubs, and you finesse the ♥Q at trick 2 followed by the ♥A at trick 3, RHO cannot ruff in effectively on the third round of clubs. If he ruffs in, you can pitch your diamond loser. This will allow you to establish the clubs and cash a long club to pitch your diamond loser before RHO can ruff in effectively.

Therefore, the play of the ♥Q at trick 2 wins and you discover that RHO has 4 hearts, you will still succeed if RHO has either 2 or 3 clubs.

The bottom line is that the best play at trick 2 is the ♥Q.

[cherdano]

♥Q is the one card I am certainly not playing, given that LHO overcalled 2♦...

[Fluffy]

Depends on who is west, if its me ♥K is about 85% on my left and I wouldn't play the queen. But against random loser who overcalls KQJxxx and out the queen is great, when the queen holds you can even cope with 4-1 breaks setting the clubs.

[nige1]

Deleted (Fluffy beat me to it)

[Lorne50]

ArtK78, on 2013-October-10, 10:51, said:

The a priori odds of Kx of hearts in either hand are 27.128%. After trick one, when both opponents have followed to one round of diamonds, the odds increse slighly to 27.328%. And, after East shifts to the ♥7, the odds remain the same - 27.328%.

Do not agree with these numbers. You have a 2D overcall and a raise so diamonds are likely to be 6-4 (unlikely to be 7-3 or 5-5 given no jump raise or rebid by the overcaller).

Ignoring whether the 2D bid shows enough values to affect who has the heart K, if the diamonds are 6-4 the odds of a doubleton K are 16.2% with W and 11.5% with E and are not affected by both following to trick one. However as other say the overcall affects the position of the heart K since W can't have more than 6 points in his suit and must have more of the remaining values than his partner.

[ahydra]

We only have 20 HCP, the remaining points could be split anywhere from 16-4 to 10-10. I think the HK is more likely to be with West, yes, but not as much as Fluffy and cherdano are suggesting. Still, I reckon the 2D bid would be enough to swing me towards playing the ten.

Quality of opponents comes into it as well. Everyone at my club would lead honestly (so the 7 is singleton or from Kxx or Jxx). Some people at my club wouldn't like to lead away from HK here, so the ten is the 100% play.

ahydra

[WellSpyder]

ahydra, on 2013-October-11, 06:31, said:

Quality of opponents comes into it as well. Everyone at my club would lead honestly (so the 7 is singleton or from Kxx or Jxx).

Are you suggesting it would be normal to switch to the higher card from a doubleton trump here?? I think the only thing you can deduce against honest leaders is that ♥7 is RHO's lowest ♥ - and given the spots in my hand and dummy I could have deduced that even without knowing oppo were honest leaders....