 A T 3 2 | |||
 A J 3 2 | |||
 9 7 5 | |||
 J 3 | |||
 | |||
| K 7 6 5 4 | |||
| K Q 4 | |||
| A 4 3 | |||
| K Q | |||
| South | West | North | East |
| 1 | Pass | 3 | Pass |
| 4 |
| Home | Contents | Chapter 14 | Chapter 16 | 03-20-2006 12:06:30 |
Whenever anyone talks probabilities assume that they are wrong and work it out for yourself. A good example is Marilyn vos Savant's long discussion of a prize behind one of three doors. You pick a door; the game show host opens another door that does not have a prize, and asks if you want to switch choices. Math professors were dropping their Ph. D.'s and school names on the table to justify their incorrect analyses.
This deal is presented first on almost pure restricted choice principles, and then West (or East) show an unusual notrump or a preemptive jump in diamonds. The aim is a set of rules that get the probabilities approximately right.
South West North East 1 Pass 3 Pass 4
West leads the diamond king. To make your contract, you must pick up the trump suit without any losers. You can list West's sixteen possible spade combinations, or simply the three combinations that matter. On those combinations West plays the nine and eight of spades in random order to the first two tricks, and you have to decide between the ten and ace at the second trick.
| West has | Number | |
| Q-9-8 | 1 | |
| J-9-8 | 1 | |
| 9-8 | 1 |
We see that two times out of three the finesse succeeds.
But wait. If the bidding was
South West North East 1 2NT 3 Pass 4
Now we assume West is five-five in the minors, and therefore East is two-four in the minors. We know twelve of West's cards and there is only one space for the missing spade honor in West's hand. We know seven cards in the East hand one spade, two diamonds and four clubs. There are six spaces for the spade honor in East's hand. The restricted choice principle says divide six by two. The spade honor will drop three times out of four.
Can we create a situation where the "restricted choice" finesse is close to 50-50? Try a weak jump overcall in diamonds.
South West North East 1 3 3 Pass 4
East follows to the first diamond and West to the first two spades. West has five spaces and East has eleven spaces, so the odds are 5.5 to 5 in favor of the drop.
We could turn the problem around and give East five-five in the minors. North uses a forcing notrump to give East the space to describe his hand:
South West North East 1 Pass 1NT 2NT Dbl 3 3 Pass 4
I calculate five spaces in the West hand and two spaces in the East hand. We halve two for restricted choice. The finesse five to one to succeed.
West has: Finesse Drop Drop West Ace first King first Void 0 1 0 Q 1 1 1 8 1 0 0 6 4 Q-8 Q-6 Q-4 8-6 8-4 6-4 Q-8-6 Q-8-4 Q-6-4 8-6-4 Q-8-6-4
Consider the trump suit:
A 10 5 3
opposite
K J 9 7 2
We assume no information from the opponents from their bidding or their card play. This is not realistic, but is the usually starting assumptions for probability calculations. For example, it is our Monday night game, the opponents are not allowed to bid, and your opponents system does not allow trump leads, and you can deduce nothing about suit splits from the opening lead.
What is the probability of success in the trump suit if you play for the queen to drop? What is the probability if you finesse West (or East) for the queen?
The way you solve this problem is to list all of the West (or East) holdings in the suit.
Winning cases: 9 9 9
You see that 6 out of 16 times or 37.5% of the time the suit breaks 2-2, 4 out of 16 times the suit breaks 1-3, and 4 out of 16 times the suit breaks 3-1. You look up the answer in the Bridge Encyclopedia, and find a 2-2 break is 40.70%, a 3-1 break is 49.74%, and a 4-0 break is 9.57%. The 2-2 break is more likely than we estimated because the cards in the other suits are more apt to break 11-11 than 12-10, 10-12, 13-9 or 9-13. (The non-sum to 1 is a rounding error.)
You play an honor from one of the hands and you may drop a singleton queen or discover a void in the suit. You count the cases that the drop works and find 9 out of 16; and you also get this result for finessing West (East) for the queen. Because suits break tend to break evenly, the odds of the drop are slightly better than for the finesse.
Exercise: If you have never worked out restricted choice, then do it. You are missing Q-J-x-x in a suit, and the queen or jack (quack) appears when the ace is cashed. The assumption is that a player holding the queen-jack doubleton will play the queen half of the time. (San Diego players always play the queen holding both cards. Then the finesse is 100% when the jack appears, and 48% when the queen appears.) The moral is that you have to play the jack from queen-jack at least 4% of the time.
Probabilities always make simplifying assumptions. There are many inferences in defender's bidding and opening lead.
The best essay in the Granovetter's book "For Experts Only — Selected Essays on Bridge" is Philip Martin's "The Monty Hall Trap". The key hand, slightly modified, is:
The book hand did not include the nine of clubs, and I want to think that the original magazine article did. You play 3NT after a non-informative auction, but you are provided information from the opening lead.
Martin argues that West will lead his long suit, and therefore information about spade length is biased information. He concludes that when West leads a four-card suit he will average more than two clubs, and when he leads from a five-card suit he will average about two clubs.
Sticking to my advice of never trusting anybody in probability discussions I did the calculation with the following assumptions:
4-4 spades problem Finesse West 73.14% Drop - Honor from South first 59.71% Drop - Honor from North first 54.35% Finesse East 40.29%
5-3 spades problem Finesse West 52.11% Drop - Honor from South first 58.46% Drop - Honor from North first 60.49% Finesse East 60.19%
4-2 hearts problem Finesse West 68.19% Drop - Honor from South first 61.46% Drop - Honor from North first 60.84% Finesse East 44.65%
The conclusion is that if West has led from a four-card suit finesse him for the queen, and if West has a longer suit (with 8 or less missing) finesse East for the queen, (or, a smidgeon better, if West has led from a five-card suit with eight cards missing cater to East having the four missing clubs and then play for the drop.)
One can make more bridge-like assumptions: Given two four-card suits and a 1NT - 3NT auction I would tend to lead a major, and so assume a major is always led. Then, if 4-card diamond suit is led, the probability of a successful club finesse against West is 100%. Given two 4-card minors to lead, diamonds will probably be led because they are almost guaranteed to be stronger than the Q-8-x-x of clubs. Given 4-3-3-3 with clubs (such terrible clubs), some other suit is led. Given two 5-card suits to lead, the stronger is led. I use 1/2 in the estimate although diamonds will average stronger than spades. There is no correction for possibility that West (or East, if South was not dealer), would have bid with a distributional hand.
The numbers change, but the overall advice is the same: Finesse West if he led a four-card suit, and finesse East otherwise. If West leads a four-card heart suit the drop, playing an honor from South first, is almost as good as the finesse against West.
There is another tuning of the solution which I have not done. There is an order in which we would select leads between four-card suits. I tend to lead the better of the two suits, except I exclude suits with the ace. (This is probably why declarer was given all four aces in Martin's deal.) I think that when West leads a suit near the top of the list the chances that he has length in clubs decreases, and if he leads from the bottom of the list his expected length in clubs increases.
I tried one other variation where the opponent's have nine spades (and seven diamonds.) The one different result was that if West has five spades, you should play for the drop rather than finesse East.
The point of another essay is to look at a bridge hand from the point of view of declarer if you are defending, or from the point of view of defender if you are declaring. (It is not proper to deduce partner's problem from the fact that he went into the tank.)
I just got through talking to John Strauch and Bill Shutts about the Philip Martin problem (I wanted John to calculate the probabilities independently). When we played, this was the first deal. Look at it from West's point of view and try to figure out what declarer has, or look at it from declarer's point of view and try to figure out what West has. From the way the hand was played I knew what declarer had. From my defense, declarer should know what I had.
North West South East (Me) 1 Pass 1NT All Pass
I led the 2. So much for always leading your longest suit.
Bill muttered something about if he has only four spades he has
to have the queen. He won the king of spades (over the queen),
took a heart finesse, and played the king of clubs from dummy.
At this point I believe that I have eliminated his hand entries,
and his hand should look like it does except he can have the ten
of spades and the queen of diamonds with and/or instead of the
king.
I ducked the club king, took the next two tricks with the queen and ace of clubs, and led the jack of spades. He ducked and I led the nine of spades which reveals who has four spades. Why?
He won the spade, cashed the ace of hearts. I really had done everything in tempo until here. I decided not to unblock for the reasons that (1) I had defended well enough to obtain a reasonable result (the result for 120 an average), (2) there was an reasonable chance that John would have the king of diamonds rather than the queen (we would set 1NT one trick), and (3) Bill should be able to read the diamond position on the unblock defense. If I unblock he should cash hearts, exit to John in spades, and guess the diamond return. Giving an opponent credit for having worked out the hand is always questionable; they often are on a different page of the book.
This second essay has a discussion on putting a defender on the wrong page of the book:
Apparently you don't play on Monday night, and you end in 3NT
rather than 4 after West has overcalled 1 . It is IMPs or
rubber bridge and all you have to do is make your bid after
West's lead shows the A-J-10 of hearts. The recommended play is to
go to dummy with the ace of clubs for the spade finesse. If West
wins the spade king he will try to reach East in the diamond
suit.
I have never seen anything written on the following principle which I came to many years ago. Given the choice of two lines of play that you judge to be about equal, if one of them is to run a long suit, then run the long suit.
Everybody received a spade lead against 3NT, and the two lines were (1) run clubs and (2) lead the diamond jack at trick two. If the diamond jack is not covered rise with the ace and take your nine tricks. Two of our group ran clubs and made five or six notrump; the third led the diamond jack and won ten tricks. The opponents had to find six pitches on the run of the clubs and could not conceal their hands. Declarer was able to judge that the diamond finesse was on.
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