Combining Chances

The made us play a lot of bridge in the Swiss in Seaside on Sunday. Seven 8 board matches with a long lunch break and starting at 10:30 instead of the 10:00 that we are used to. So, instead of finishing between 4:45 and 5:00, finishing between 6:45 and 7:00 made travel afterward kind of a pain. Jim and Nancy booked an 8:30 flight out of Portland, roughly an hour and a half drive from the playing site, and had to skip the last match to catch their flight. We didn’t even know that we were playing 7 rounds until after lunch, just assuming it was six rounds of 8 like it typically is in the Mid-Atlantic. Anyway, we were able to get a pair to fill in who had played in the fast pairs and I was able to get another ride to the airport because my flight wasn’t until 10:55pm.

I said I would write about another failed slam, but really, this one – the last hand of the tournament – is more interesting, and I actually got it right.
 

Dealer: S Vul: NS	North ♠ xx ♥ KJxx ♦ KQ9xx ♣ xx
	South ♠ Qxxx ♥ Axxx ♦ Ax ♣ ATx

East	South	West	North
	1NT	Pass	2♣
Pass	2♥	Pass	4♥
Pass	Pass	Pass

1NT was 14-17. West led the ♠A, got a discouraging 9 from east, and shifted to a low club to east’s Q. It appears that this contract now depends on picking up hearts for no losers, meaning a 3-2 break with the Q onside is needed with. The contract has almost zero chance with a bad trump break so for now, let’s consider only 3-2 breaks. There is another option of cashing diamonds quickly to discard my two remaining clubs so I could then ruff dummy’s second club. This means taking the HAK first. If the Q falls, I draw the last trump. If the Q does not fall, I’ll start trying to run diamonds. This wins anytime the Q is doubleton (40% of the 3-2 splits), or anytime diamonds are 3-3 (36%), or when the person with only 2 hearts has JT doubleton of diamonds (negligible%). Combining the Qx chance with the 3-3 diamonds chance, it is 62% likely to make the contract (.40 + .36*.60). By taking just a heart finesse, it’s 50% because the defense would have 2 easy black kings to cash for the setting trick. The 62% and 50% numbers are based on an assumption that hearts are splitting 3-2.

In practice, the contract is makable on some 4-1 trump splits if diamonds are 3-3 because you'd still be able to pich all of south's clubs before they can trump in and lose only a heart and 2 spades. My line works when the singleton is the Q in either hand and 3-3 diamonds (.36*.28*.2 = .02). The heart finesse like works when Qxxx is onside or there's a singleton Q and 3-3 diamonds (.02 + .36*.28*.8*.5 = .04).
Overall, my line wins 43.5% of the time: .678*.62 + .02
The heart finesse wins 39.9%: .678*.5 +.02 + .04
Yeah, that covers all the possibilities.

When I made this play (it was Qx offside and diamonds were not 3-3 so this was the only way to make), I got a look of disapproval/dismay from my opponents that I often give when an opponent makes an anti-percentage play that works against me. Little did they know that this is the technically correct play.

Here’s the slam I referred to. Through 4 rounds we were right at average and felt like we were losing the 5^th match so Alli pushed a bit to bid this slam.

Dealer: S Vul: NS	North ♠ Txxx ♥ AQxx ♦ Jxxx ♣ J
	South ♠ A ♥ Txx ♦ KQT9 ♣ AKQ9x

East	South	West	North
	1♣	Pass	1♥
Pass	2♦	Pass	3♦
Pass	3♥	Pass	4♦
Pass	4♥	Pass	4♠
Pass	5♣	Pass	6♦
Pass	Pass	Pass

West led the ♠Q. Assuming 3-2 diamonds and the ♣T falling in 4 rounds, both pretty much necessary to have any chance at this, there are 11 tricks easy enough – 3 diamonds, 1 ruff, 1 heart, 1 spade, 5 clubs. Where is the 12^th trick coming from? It could be from ruffing 2 spades in hand, but there is big transportation problem with that. A 12^th trick could obviously come from a simple heart finesse but that’s not exciting and only a 50% shot, possibly even less given that east has the ♦A and ♠K?

Is there a line that is better than 50%? I don’t think so, but I pondered this for several minutes before eventually just drawing trumps and relying on a heart finesse, which lost. Mainly I was trying to calculate the odds of drawing 2 rounds of trumps, cashing the ♥A, then 4 rounds of clubs, discarding the rest of dummy’s hearts, then ruffing 2 hearts in dummy. That at least would require the person with 4 clubs to also have 3 diamonds and I that already puts it below 50%. Actually, that doesn’t work either because there still aren’t enough entries to do everything. I guess that one’s just not makeable… lost 12 imps and lost the match by 16.

updated at 9:15am 9/26 with more accurate percentages for the first deal.