Spoiler.... don't read if you want to work on the problem.. This hand is from BBO
North
S void
H AT974
D QJ543
C QJT
South
S AKQJ542
H Q863
D A
C A
Freee narrowed down the problem when he stated "Seems to be depended on the H suit." Since you have no losers in the other suits, this even seems clear to me. :-)
Trpltrbl followed this up with a statistical analysis that you have 82.78 % chance of winning 4 heart tricks by playing the HA first (of course 3H is enough as long as you lose only 1), and a 6.22% chance for 5 tricks by playing the HQ first (or course to win five hearts, you have to throw your away your ACE of diamonds on the fifth one, or else you will be ruffing it). Let's deal with this analysis first. After West showed long minors and showed up with two spades, do we really believe there is a 6.22% that EAST will have a singleton heart Jack giving WEST five cards in the majors on this auction (2S and KJx of hearts)? Obviously that isn't likely, I would say “impossible” as long as WEST didn’t psych.
The second issue, can we take advantage of the 82.78% by cashing the heart ACE first? The truth is, we are certain that hearts are probably 3-1 or 4-0 with EAST having the heart king, so if we had suitable entries, we would be 100% chance of losing only one heart. But entries to dummy, that is
THE PROBLEM. Free suggest leading the heart ten from dummy and let it ride after winning the heart ACE. Well, if WEST showed out on the first round of hearts, that is a marked hook, and if West followed on the first round of hearts, this can't happen. Either hearts are 2-2 in which EAST will play an honor, or hearts are 3-1 in which case EAST will play an honor. Since WEST really will not hold 3 hearts, what is the danger here? If hearts are 2-2 or 1-3 (with 3 with EAST), you can surely make by playing low to the HEART ACE and then low back to the heart queen.
But, what if EAST has all four hearts? Is that possible? Free thought leading low to the ACE and finessing the heart TEN will deal with all 4-0 split. But if WEST is 2-0-6-5, when you play low to the HEART ACE, and lead a heart, EAST will jump up with the heart King and exit safely with a minor and sit back and wait for his heart jack to win a second trick. This is what mikestar realized when he said “assuming no mistakes on the part of your opponents, you are down one”
So is all lost if hearts are so unfriendly as to be split 4-0? No, you might still get lucky if you can read the position. If EAST has four hearts, all he need do is keep his four hearts and a club. When you lead a heart from dummy he wins the heart KING and exits the club. But, you can overcome exactly one 4-0 heart split. If EAST has the diamond Kx and the four hearts, he can be squeezed out of his exit card in clubs.
Run all but one trump and throw away some hearts from dummy, keeping QJ of diamonds and AT9x of hearts. You will end up with four hearts, diamond ace and a spade in your hand. Now, you have to decide. Did EAST start with 3 hearts, in which case a heart to the ACE and a heart back, or did EAST begin with 4 hearts. Fortunately, you get some clues in here. East has to find two discards on the two long trumps. If he throws a heart, you play heart to the ACE and a heart. If he throws a diamond, you try diamond ACE and if the king hasn’t been played, you play heart to the ace and a heart. But if he throws two clubs, now you have an interesting option. At this point, if you decide that EAST was 4-4-2-3 with the diamond king (else why not a DIAMOND discard). The position will be:
Quote
S void
H AT97
D QJ
C void
S void
H KJ52
D Kx
C void
South
S 2
H Q863
D A
C void
Here you cash the last trump, discarding anything but the heart ACE (I would discard heart ten, but it makes no difference). If East throws a heart, just play on hearts with low to the ace and heart. If he throws a diamond, cash the diamond ACE, then lead a heart and hook heart on way back. Note, setting up the diamond is not even necessary. It is ok to throw the diamond off the dummy under the last trump, in which case, and play D only if EAST discards one.
In view of the double, and two club discards on trumps, and the lack of a heart opening lead, I would play EAST for all four hearts. If east keeps all four hearts, after the last trump, you have to cash the diamond ACE, then play on hearts.
Note: If WEST has diamond king, and EAST has four hearts, you can not make, as EAST can keep a club exit card.