An interesting problem arose recently in an ACBL robot tournament.

North

♠

753

♥

8

♦

K764

♣

AQJ94

South

♠

AKJ96

♥

K

♦

AQJ5

♣

K87

W

N

E

S

P

P

P

1♠

P

2♣

P

3♥

P

4♣

P

4♦

P

5♦

P

6♠

P

P

P

The auction isn't important. I only include it here for the sake of full disclosure and to demonstrate that nothing was known about the opponents' hands.

I received the lead of ♣2, dummy played low and I took the T with my K.

What now? Cashing a high trump then crossing to dummy with a diamond (seems safer than a club) and taking the spade finesse appears to be a 50% proposition. Not perhaps exactly 50% but pretty close. The problem with this line is that I don't trust the robots to fail to cash their ♥A if the trump finesse is off.

How about combining some different chances?

I played off two top trumps hoping for the queen to drop doubleton (dropping singleton on left would be OK, too, providing I can get to dummy twice without being ruffed). That's a 30% chance. It didn't happen. (Probability of doubleton Q is 27.1%, probability of stiff Q on my left is 2.83%)

So, now I was in the (approximately) 63% zone. I still had a decent chance of pitching my losing heart on the fourth round of clubs while an opponent ruffs in. For that to happen, I needed clubs to be 3-2 or 4-1 (we'd already had one round where all followed) *and* I needed the one with the trump to have either three or four clubs.The probability of 3-2 clubs is 68% (vacant places doesn't apply because I don't know the division of any of the other suits, although I'm pretty sure each opponent has at least five hearts since nobody preempted). But only half of those 3-2 splits are good for me (depending on who has the outstanding trump). So, that's 34%. There's also the chance of 4-1 clubs (14%). Total probability of surviving the club ruff is 48% x 63% = 30.2%. I haven't calculated the probability of the zone because it's complicated. Both 5-0 splits are fatal (I think), as are some of the 4-1 splits, so I've just approximated it as 63%.

Since these two events are (almost) independent, we can combine the chances as follows: 30% (Q drop) + 30.2% (survive club ruff)

This should provide a total chance of 60.2%, rather better than the probability of the finesse.

Unfortunately, the player with only two clubs also had the ♠Q. Even worse news, it was my RHO so a simple finesse would have worked. The hand cost me 14.7 IMPs. Ouch!

Yet, I was the only declarer to try this line. It's true that a couple of declarers received the ♥A lead, so now, their best hope lay in a simple finesse (which worked, of course). It's also true that most of the declarers were not in slam, so they (should) hope that the best chance will fail. Several declarers including most of the others in slam, took a *first*-round spade finesse!

Comments/corrections would be welcome especially regarding strategy at MPs, bearing in mind that other declarers may have already lost a heart on the opening lead.

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