Tuesday, September 25, 2012

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 5th 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 12th trick coming from? It could be from ruffing 2 spades in hand, but there is big transportation problem with that. A 12th 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.

4 comments:

  1. Your odds are incorrect.

    The odds of a 3-2 split are not 40%.

    The correct odds for the play described (with the 3-2 condition) are:

    (0.4*0.678)+(0.355*0.678*0.6).

    The actual probability of the above play is 41.5%.

    And the finesse is 50%.

    So if you don't take the remaining cases into account, the play you took IS the inferior play.




    http://www.bridgehands.com/P/Probability_of_Card_Distribution.htm


    ReplyDelete
    Replies
    1. I also understand that missing all the spots, you're fated for 1 trump loser if the suit breaks 4-1. But since you're able to handle some 4-1 breaks (with the fall-back on diamonds), the "normalization" to 3-2 breaks only is a big approximation.

      Example you play for 3-2 with Q onside, and find out the Q is onside but the suit is 4-1. You could still fall-back on diamonds.

      That's why I prefer to count my odds in the "real" dimension rather than in the conditional dimension as you did.

      Delete
  2. The 40% number is the probability that, given that hearts are 3-2, the Q is doubleton. That is definitely correct. The trick 1 assumption I made is that hearts are 3-2, otherwise the contract is doomed for failure. It may still be makable if hearts are 4-1 and diamonds 3-3 so the actual percentage of making the contract with the finesse or more like .67*.5 + .355*.283 = 43.9%

    ReplyDelete
  3. bridging finance which is an ultimate method of maintaining liquidity while you are waiting in the mean time to get funds to complete your dreams. This type of finance bridges the gap of sixty days until you receive your expected funds. Many companies use bridging finance for the maintenance of community operations. The funds are released by the investment bank that under writes the new issue. Bridging loans

    ReplyDelete