Bob Mackinnon

The Shifting Sands of Probability

Although it is certainly not the best of all possible worlds, it might be the most probable

– Jules Henri Poincairé, French mathematician and philosopher (1854-1912)

As a schoolboy I was trained to think in terms of numbers, the very bases of scientific knowledge. Do they still teach that way? I wonder. Maybe not as much as before, but the numerical way of thinking is certainly useful when it comes to following presidential elections, and more importantly, to playing bridge. During the recent elections we have heard repeatedly the results of opinion polls and have been constantly warned about their unreliability. The statistics changed with time, but they have proved to be an accurate gauge of how things were going to turn out in the end. Well, the same applies to bridge probabilities – they may change during the play of the cards, but they are usually a good guide to what will happen, baring the occasional nasty surprise.

In the world of bridge we have probabilities fed to us by the writers and commentators. I had always accepted these percentages at face values, but once I had the time to look more deeply into the whole concept of probability, I discovered the assumptions that lie hidden behind the figures and that all too often are lost in discussions which lead to misleading conclusions. The idea that ‘probabilities never change’ was one of the first assertions that fell by the wayside, as, of course, this is contrary to common sense. The correct way of thinking is ‘probabilities change according to what information becomes available’. If they didn’t, we’d be living in a strange world where ignorance would surely be bliss.

Some are reluctant to admit that bridge is a game of guesses, but it is that because it is played in an atmosphere of uncertainty. ‘Guess’ is not a dirty word. Our lives are governed by chance. Probability is a way of organizing our guesses and assigning them proportions. Of course, at the end of the hand when all 4 hands have been revealed one may realize that there were clues along the way that should have pointed us in the right direction. (By the way, did you dump your stocks in a timely fashion?) The skill of an expert is to adopt his play to what has been revealed as the bidding and play proceed. The main theme in the following blogs is that probabilities change with circumstances.

There are very successful players who haven’t mastered Probability Theory, although they apply it in a practical way based on experience. Sabine Auken in her great memoir I Love This Game has described how her curiosity has been piqued at a later stage of her career. The question arises, why should we ordinary players be interested? If one is to use probability successfully then it is necessary to understand how hidden assumptions come into play so as not to be distracted unduly by the numbers lifted from textbooks. Probability should be a way of expressing common sense in numbers. Of course, once one has mastered the basics, there are many applications to be found away from the bridge table.

One foundation of analysis is the table of suit combinations to be found in The Official Encyclopedia of Bridge and elsewhere. Expert players are expected to know these backwards and forward. If they don’t play accordingly, they are open to criticism from the pundits, but maybe the expert had his reasons. Well, I am nothing if not critical, and in many situations there can be arguments both ways. We begin by looking at expert play in an 8-card suit missing the Ace and Jack when the 2008 bridge championships of the world were at stake.

About Five Missing the Ace-Jack

The English Open Team that reached the finals of the 2008 WMSG was formed after

trials involving teams of 4. A third pair was added to the winning foursome on the basis of current form. The added pair were Tom Townsend and David Gold who were judged by some to be the best English performers in Beijing. Let’s see how this pair fared against their future teammates in the trails where under different circumstances 2 experts played differently a heart suit of 8 cards missing the AJ.

Vul: East/West

West East
A96 K42
KQ9653 T8
AQJ K642
T AK54

The bidding without interference took different routes at the 2 tables, one EW pair reaching a slam in hearts played by West, the other game in diamonds played by East. The key at each table was the play in the heart suit.

Townsend-Gold reached 6 played by West. The lead was the 7 won by declarer in hand. He went to dummy with the A. The problem is straightforward –  what is the best play to avoid more than 1 loser in trumps. Declarer immediately suffered defeat by running the T to the J. This is the safety play for 4 tricks as it picks up 4 cards with the Jack in the South hand but the references will tell you that the best play for 5 tricks is to go up with the K on the first round.

Of interest is what to do on the second round of trumps if the K loses to the A. An imprecise short-cut argument based on Reese’s Rule of Thumb leads to the correct decision on this hand as follows. Suppose North takes his Ace on the first round. With AJ he would be obliged to do so, but with Av he might hold off the Ace. This affords the assumption he played the Ace of necessity from AJ.

Of course, each case should be considered fully on its own merits. The key decision needn’t be made until declarer plays the second round from the dummy and South follows with a second low card. Let the low cards be denoted by the letters u, v, and w. Here are the 2 conditions remaining that afford a winning position after the sequence of u-A-w:

  I II
  South    North South North
  Juw       Av  uvw AJ
Plausible Sequences            4 6
  u-A-w u-A-w u-A-v
  w-A-u w-A-u  v-A-u
  u-v-w  v-A-w  w-A-v
Probability Weights        1/4 1/6

With Av North might hold up the Ace on the first round, giving him 2 choices, so Condition I has at most 4 plausible sequences available. Under Condition II South has 6 ways to play 2 insignificant cards from a selection of 3, whereas North has no plausible play other than taking his Ace. If under Condition I one judges that North is equally likely to take the Ace as not, the 4 plausible sequences are equally probable. The probability that the defenders chose to play in a particular observed sequence, u-A-w, has the probability of choice of 1 out of 4. As a consequence the probability of Condition I is subject to a reduction of 1/4 once the given sequence has been observed. A similar deduction is that the initial probability of Condition II must be reduced by 1/6 once the sequence u-A-w has been observed.

Reese’s Rule would give the correct theoretical answer if only 2 cards had been played, u-A, say. Condition I has 4 plausible plays at that point, while Condition II has 3. Thus, the probability weights are 4:3 in favor of AJ doubleton. It is the random choice of play of the insignificant cards that determines the weights.

Quantification of North’s Choice

From the bidding the North defender has a good idea of the lie of the heart suit and could very well be inclined to hold up the Ace depending on the circumstances. The question is how his inclination alters the probability weights. Declarer cannot be certain of how North would play from his holding of Av, but he may (should) assume a particular tendency. An assumed frequency of hold up on the part of the North defender gives an indication of whether or not to finesse for the Jack on the second round, as the following table shows.

Hold-up Frequency Weight I  Weight II  Decision
4 out of  5 3 5 Drop the Jack
3 out of  4 3 4 Drop the Jack
2 out of  3 3 3 Toss up
1 out of  2  3 2 Finesse
never 3 1 Finesse

The hold-up frequency of 1 out of 2 assumes that North will play the Ace at random on a 50-50 basis. The fractional weights given previously are expressed in integer format for convenience, the ratio of 3:2 being preserved. If North would hold up the Ace 2 out of 3 times, it would be a toss up as to whether or not to finesse for the Jack. Any greater frequency of hold-up would favor an attempt to drop the Jack. If North would never hold up the Ace, there are 2 possibilities under Condition I and 6 under Condition I, so the odds becomes 3:1 that the finesse will succeed, based solely on the number of permutations in play of insignificant cards in the South hand.

All’s Well That Ends Well

At the other table the lead against 5 by East was the 5 won by declarer in hand. Immediately he began the heart suit by leading the T to the K, losing to the A. The 7 was returned to the A in dummy, creating a loser in that suit. The temptation for declarer is to return to hand with the A, discard the spade loser on the K, and finesse South for the J if he judges North would hold up his Ace fewer than 2 times out of 3. On the other hand, if declarer thinks North would hold up at least as frequently as 2 times out of 3, under the circumstances he can save time by playing the Queen immediately with the additional chance of dropping the bare Jack in the South if the cards had been dealt Ju opposite Avw (Condition III).

The declarer, Nick Sandqvist, didn’t lose his focus because he might have missed reaching a makeable slam. He drew two rounds of trumps with the A and Q, South following with the 9 and T, cards of significance. He could judge that South’s diamonds were more likely to have been played from a doubleton T9 rather than from a tripleton T98, leaving North with 87 remaining. This is another example of restricted choice in action where with T98 South would have had 6 choices in the sequence of his plays, whereas with T9 doubleton he would have had just 2 choices.

We don’t know whether Sandqvist had Reese’s Rule in mind, but at trick 6 he made the right decision when he played the Q from dummy dropping the J in the North, so was able to continue the established heart suit to embarrass North who couldn’t afford to ruff even though the 87 were equals in front of the K6. By not blindly following the standard advice and making the right play in the wrong contract led to a 13 IMP gain early in the match, which the Sandqvist team went on to win by a large margin.

More About Five Missing the Ace-Jack

The final of the women’s 2008 WMSG Championships was won by England over China by the slim margin of 1 IMP over 96 boards.    The purist might say that the difference was a vital overtrick, implying that missing an overtrick can be a critical play even in such a long match. Well, yes, but the difference could be more than made up by avoiding many of the simple errors one observes in the bidding and play. In that regard England was the more steady team throughout and deserved their win, although they almost blew it in the last 16-board segment. To me the lesson is this: bid aggressively, avoid major errors in the play of the hand, cooperate with you partner, and your team will be hard to beat. There is too much variability built into the results to be worried about overtricks.

One of the sources of percentages in play are those related to how a declarer should play combinations of cards in a given suit. An abundant source of this type of information is available in The Dictionary of Suit Play Combinations a great reference book written by J.-M. Roudinesco. If that is not enough, there is the compute program, Suit Play, created and made available on the Internet by Jeroen Warmerdam. In the following segments I continue to look at the recommended play in suit combinations where 8 cards are missing that include the Ace and the Jack. If the Chinese Women had got one of these right they would have gained enough IMPs to win their final match handily. By examining the hands in detail, we should get a deeper understanding of how probability analysis should be applied under changing circumstances.

Going with the Odds

The first combination we shall ponder is QT opposite K97652. The textbook play is low to the Queen, and regardless of whether that wins or loses, to run the Ten on the next round. The chances of this providing 4 tricks is close to 96%. That figure is derived from considering the suit combinations in isolation from the full deal. As this is based on the prior expectations of how the cards were dealt, the approach yields a valid approximation when very little is known about the defenders’ hands. If they have passed throughout, one may assume that the suits are likely to split more evenly than expected a priori, but that may not greatly affect the calculation of odds. Here is the hand from the final where the Chinese declarer went against the odds and lost 6 IMPs as a result.

Dealer: East

Vul: Both

North (Wang)  
West (Brock) East (Smith)
T832 Q765
83 AJ4
A975 43
K32 AT86
  South (Sun)  
West North East South
    Pass 1
Pass 1NT* Pass 2
Pass 2 Pass 3
All pass      

Sun held a good hand in the context of a  Precision opening bid, only 5 losers, so she raised herself to the 3-level where others holding the hand were content to stay put in 2 hearts. The lack of aces is a defect not overcome by the distribution, and perhaps the diamond suit is overly rich with honor cards while the heart suit is rather sparse in that regard. In the Open Final where Italy faced England neither South was allowed to play in 2 as West balanced and EW played in a contract of 2 , going down 1. At the other table in the women’s final, Nevena Senior played undisturbed in 2 making 3. So Sun did the right thing in theory as her 3 contract appears to be solid and prevents the opposition from balancing into their optimum contract of 2 .

The opening lead was an innocuous 2 won by the K when Nicola Smith played her Q. The play in the trump suit was now front-and-center, and if Sun had gone with the percentages, China would have won the championship. A point well made by Linda Lee in her blog of October 19 was that there was some urgency in the trump play as the lack of controls for the declaring side made it a race to the finish line with declarer hoping to prevent the opponents establishing the setting trick before declarer makes sure of her 9 tricks.

Sun didn’t feel the urgency even though the defenders held minor suit aces that provided them with transportation. This wouldn’t have mattered if Sun played the trump suit optimally according to the a priori odds that is, low to the Q, planning to run the T next. The A was still there as a safe entry to dummy to allow this sequence. Unfortunately for China, Sun chose to play a heart to the T and the J. Nicola Smith had defended well throughout the Final and here she was quick to take the opportunity of obtaining a ruff in diamonds. She led the 4 and Sally Brock took her ace to returned a diamond immediately just in case that 4 was a singleton. Not so, but it didn’t matter as Smith on winning the A underled her  A and got the ruff that set the contract and brought 6 IMPs England’s way, a critical swing late in the match.

When 2 high honors are missing, it is a great temptation to finesse against the lower honor first. The motivation is that this guards against one defender holding A-J-x in front of the queen. Another reason for adopting this approach would be that declarer places the Ace behind the QT, in which case the odds usually favor the honors being split. We shall examine this argument in the next segment.

Placing an Ace

The following hand played early in the Final matches of the Open, Women’s and Bronze Medal series. It involved declarer play in a suit with the following construction:

K9765 opposite QT3.

The textbook play for 4 tricks with the suit taken in isolation is to lead to the Q and if that holds to pass the T. That assumes declarer has no information on how the cards clubs may be distributed. The odds of success are 57%, but the bidding may change the a priori odds. Let’s see the whole hand and how it was played in the Open Series where Italy faced England.

Dealer: East

Vul: North/South

West East
94 KQJ75
QJ8753 T64
87 Q96
A84 J2
West North East South
Townsend Lauria Gold Versace
    Pass 1
1 2 Dbl 3
Pass 3 Pass 3
Dbl 5 All Pass  

Gold led the K and Lauria held up. Gold switched to the 4 after which Lauria won to run the 9 around to Townsend’s A. The heart return did no damage as a losing spade could be discarded on the A. Obviously Lauria’s play in the club suit was predicated by the bidding and the opening lead from a high honor sequence which placed the A in the West hand. He assumed the J was in the East. He could have taken the necessary finesse in diamonds at trick 3 and led the 3 from dummy, guarding against AJ doubleton in the West.

In his book Playing with the Bridge Legends Barnet Shenkin makes the point that when amateur analysts (like myself, I admit) criticize experts, they are usually wrong – even with the help of Deep Finesse, I am willing to concede. Who are we to question the great Lauria? Nonetheless no player is perfect and we amateurs mustn’t give up on trying to understand the mental processes at play. We shall explore this later.

In the Open Bronze Medal match, Germany against Norway, Kirmse played in 3NT after West showed values and a heart suit. Here was the bidding at his table.

West North East South
Aa Gromöeller Molberg Kirmse
    Pass 1NT
2* Dbl Rdbl** 3
Pass 3 Pass 3NT
All Pass      

The German declarer won the heart lead in dummy and followed Lauria’s line of running the 9. That doesn’t seem right even though it succeeded in gathering in 10 IMPs. The danger is that the opposition may find the spade switch if given 2 tries when West wins the  J without holding the A. So now we have evidence of 2 experts going against the conventional wisdom of ‘low to the Queen.’

Child’s Play?

The term ‘Chinese finesse’ is a derogatory one referring to a play that depends on an error on the part of defenders in not covering an unsupported honor played from the hidden hand. The declarer runs the unsupported honor and makes an undeserved winner. It is the defender who has erred. I now propose that the term ‘Chinese Drop’ be applied to a play that also appears to be ridiculous, except when it works. The Chinese women could have won if they had employed a play that on the surface appears to be one that only a novice would make: low to the Queen, then low to the King.

When it comes to the replay of the hand in the Women’s final, I cannot get rid of the feeling that something was definitely wrong in declarer’s approach of making the standard percentage play in the club suit. Let’s see if you agree.

West North East South
Dhondy Wang Senior Liu
    Pass Pass
2* Dbl 2 3NT
All pass      

Heather Dhondy’s lead was the 7, won by Liu Yi Qian in dummy. She led the 5 to her Q and Dhondy ducked that as the A was her only entry. Liu led a second club from her hand, Dhondy following with the last outstanding low club, the 8, and here Liu missed the seemingly ridiculous but winning play of going up with the K to drop Senior’s now bare J. Let’s see the most probable possibilities when Dhondy followed to the second club play.

  Dhondy Senior
Situation #1 8-4 A-J-2
Situation #2  J-8-4 A-2
Situation #3  A-8-4 J-2
Situation #4 A-J-8-4 2


In Situation #1 there is no winning play, so we can rule that out, as we must play for success. Situations #2 and #3 are equally likely on the deal, but isn’t it more likely on the bidding and play that Situation #3 holds? Without an entry in clubs, West might have led a spade hoping to hit her partner’s suit, whereas with the A she would rely on setting up her own suit.

In my experience when an opponent enters the bidding with a flimsy suit, and leads that suit against 3NT, it is more likely that the bidder holds a top honor in my long suit. So, here when NS holds AK, it is more likely that West holds the A. With regard to the dealing of the cards alone that would have no mathematical foundation.

The Division of Sides

One of the most useful pieces of information available to declarers is the division of sides. It is also the most neglected. When the dummy appears a declarer knows how many cards are missing in each suit. One can link this to the probability that a particular distribution exists in one hand or the other. Here the question is whether Situation #4 is a live possibility, and, if so, how to take that possibility into account. That would give West 6 hearts and 4 clubs. Let’s look at the possible divisions of sides when the opponents hold 7 spades, 9 hearts, 5 diamonds, and 5 clubs and the hearts are split 6-3 and West holds at least 2 clubs.

  3 – 4 2 – 5 2 – 5 3 – 4 2 – 5
  6 – 3 6 -3 6 – 3 6 – 3 6 – 3
  2 – 3 2 – 3 3 – 2 1 – 4 1 – 4
  2 – 3 3 – 2 2 – 3 3 – 2 4 – 1
Weights: 100 60 60 50 15

The Probability Weights given along the bottom are easily calculated from the assumed splits. I’ll show you how in a later blog if you’re interested. By far the single most likely is Condition I, but we see that this is a losing configuration corresponding to Situation I. The same is true of Condition III. That leaves Condition II as the most likely winning position, corresponding to Situations #2 and #3. Condition V, representing Situation #4, is only 1/4 as likely to have been dealt.

How the Unplayed Suits Contribute to Probabilities

Probabilities are derived from ratios of the number of suit combinations. In the case at hand there remain 2 honors missing, 2 possible splits in clubs, and 3 distribution of sides. We can add together the number of suit combinations in the unplayed suits, diamonds and spades, to obtain an estimate of the probabilities of success of the 2 possible winning plays, namely, running the T and going up with the K.

Play II IV V Total Weights* Percentage
Run T 210 175 175 490 14 56%
Run K 210 175 0 385 11 44%
        *Weights are the number of combinations divided by 35

A calculation of the ratio of suit combinations yields the result that the probability of success for running the T is 56%, very close to the a priori probability. An assumption that underlies both calculations is that we are not at liberty to assume that the A is more likely to have been dealt to the West hand. Another way of thinking of this assumption is that we are maximally uncertain as to the location of the A. However, as noted above, we should be surprised if the A were not with the West player for the reasons stated. If one could say that the probability of A being with West was 56% the 2 plays would have the same probability of success. Anything greater than 56% and rising with the K becomes the preferred play. I would estimate that West would hold the A and play the way she did as greater than 3 times out of 5 (60%), and that is my basis for stating that the Chinese Drop represents the best chance at this stage where all the low clubs have been played. Yes, the Chinese Drop is not so dumb as one might expect. It is even the percentage play, given the information that is available at the time of decision.

Luck lies not with the player but in the placement of the cards. Probability Theory, properly applied, is the best tool for extracting it.


The Maximally Likely Distribution of Sides

Returning our attention to the Open Bronze Medal Series we ask why did Kirmse play for the J to be in the East? If you see him, ask him, and let me know. Ask Lauria the same question. I can see a connection between the choice of plays and the most likely division of sides represented by Condition I. In that configuration the clubs are split 2-3 with 3 in the East. Generally it is the better play to start a suit by playing through the defender holding the greater number of cards in the suit. Let’s see how the first card played affects the odds. Here are the possible combinations.

Constituents Number Possibilities
East West    
A-u-v  J-w 3 u,v, or w in the East
 J-u-v  A-w 3 u,v, or w in the East
A-J-u v-w  3 u,v, or w in the West
u-v-w A-J 1 no possibility of a low card in the East

When a club is led from the North hand,  East follows with a low card, specifically card w. That eliminates the least likely combination. The probabilities of A-u-v and J-u-v are the same but have been halved as the low card v could have been as easily played as card w. (A consequence of Bayes’ Theorem). There is no winning option for A-J-u in the West, so, based on the dealing of the cards alone, it is a 50-50 choice as whether to play the Q or run the 9. However, the bidding has indicated the A lies in the West, so the odds clearly favor running the 9 with an edge given by the bidding.

One might say that this analysis is rather naive. It is certainly incomplete as we has considered only one possible division of sides. That being said, I am somewhat amazed by how often the maximally likely division of sides turns out to be a reflection of reality. The longer the defenders follow with low cards to declarer’s plays, the more likely that condition becomes. Those plays have eliminated the possibility of extreme splits.

When a declarer starts playing his suits, he should have a plan in mind. Taking into account the most likely division of sides makes sense. A simplification based solely on the maximum likelihood division is not rigorous, but it may focus the mind in a beneficial way. If the problem faced is a complex one, declarer may gain clarity through simplification. It is a start. By solving a simple problem, one hopes that the more complex problem has the same solution. Probability is on your side. If you can think more deeply, and consider more possible distributions, great.

What one can do in many situations is gather more information before making a critical decision. Every bit of information helps, and the probabilities change accordingly. When Liu played the second round of clubs from her hand and West followed with a second low card, she gained information. The most likely distribution of sides could be eliminated as encompassing a winning choice. Attention should then have been concentrated on the second most likely condition, specifically, the clubs splitting 3-2.

Probability and Information from a Surprise Action

The greater the surprise, the more information the action transmits. Let’s suppose that on Board 2 Dhondy and Senior had not entered the auction. In this day and age, that would have been a surprise as when the dummy came down Liu could see EW held 17 major suit cards and 16 HCP between them. There are 6 equally most likely distributions of sides for a 7-9-5-5 division, namely,

3 – 4 3 – 4 4 – 3 4 – 3 4 – 3 3 – 4
5 – 4 5 – 4 4 – 5 4 – 5 5 – 4 4 – 5
3 – 2 2 – 3 3 – 2 2 – 3 2 – 3 3 – 2
2 – 3 3 – 2 2 – 3 3 – 2 2 – 3 3 – 2

Based on the inaction of this normally active pair, one might downgrade Conditions V and VI for which one player holds 5-4 in the majors. Next consider the opening lead. If it is a heart, this is most consistent with Conditions I and II, as a player is most likely to lead from her longer suit. Note that a 3-2 split in clubs is as likely as a 2-3 split.

But suppose the opening lead was the 5 from the player on the left. That would be a surprise, and surprises greatly affect the probabilities. None of the above 6 conditions makes sense with this lead. Declarer might then consider other options, in particular, that the lead was from a singleton. She would then consider a distribution of sides based on that assumption and work from there. Here are some main candidates together with their probability weights relative to the above set.

  4 – 3 4 – 3 5 – 2 5 – 2 4 – 3
  5 – 4 6 – 3 4 – 5 5 – 4 4 – 5
  1 – 4 1 – 4 1 – 4 1 – 4  1 – 4
  3 – 2 2 – 3 3 – 2 2 – 3 4 – 1
Weights 50 33 30 30 25

The relative weights are based on the number of card combinations available on a random deal. The bidding must also be taken in account. As the opening leader has passed throughout, more weight must be given to Condition XI. It makes sense that with a 4-4-4-1 shape a player would tend to remain silent throughout the auction whereas with 5-4+ in the majors, she would be inclined to make some noise during the auction, so Condition X belongs at the end of the line and Condition XI moves to the top.

Nothing is certain, and that why we must revert to probabilities. It is possible that the opening leader decided to make a ‘safe’ lead from xxx in diamonds because of observed gaps in the majors. Some may even hope to make a deceptive lead. The general rule is that the best deceptions come from actions that appear to be normal. Such players will lead from worthless doubletons for no other reason that they hope declarer gets it wrong. That’s unusual, which is the reason why the technique is sometimes effective, but such occasions are rare. Based on likelihood considerations, what appears to be normal should be assumed to be normal. To be constantly suspicious of a normal action is to present yourself with one more way to lose. Be resigned to being deceived occasionally, and move on. That is the way to keep on the right side of probability, besides which, it’s easier on the nerves.

Sound Advice: If the opponents were sucessful taking a ‘wrong’ view, think that they gave your side a chance for a good score that didn’t quite pay off this time. Be patient. In the long run you will benefit from their poor decisions.


RyanAugust 4th, 2012 at 2:50 pm

Hi Bob.

I’m enjoying your book Bridge, Probability and Information and am fascinated by the idea of “sides”. If you know the opponents cards are divided 6-7-5-8, how do you go about calculating the most likely distributions ? On page 76 of your book you have spades 3-3, hearts 4-3, diamonds 2-2, and clubs 4-4 as the most probable split, but how do you get that ?

RyanAugust 4th, 2012 at 3:46 pm

That should be diamonds 2-3, by the way… .

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