THE LAW OF TOTAL TRICKS - a bidding tool for decision

In 1966 Jean-René Vernes (1914-2012), from Paris, published in his book "Bridge Moderne de la Défense" a very interesting statistical analyses about the correlation of total tricks and trumps in bridge. In June, 1969 he wrote a synopsis of this "Law of Total Tricks" in The Bridge World magazine. In the 80's the partnership Marty Bergen and Larry Cohen achieved astonishing results in competitive bridge applying this Law of Total Tricks that was after popularized by Larry Cohen in his 1992 book "To Bid or Not to Bid" that explain how to use this important bridge principle to take better decision in double/pass/bid during competitive auction.

What is the the Low of Total Trick?
It is a statistical constation that the number of tricks of both sides are correlationed with the long fit (possible trumps) of each partnership.

During the bidding both partnership compete for a partial score or game when the high card points (hcp) are divided 20-20 or 21-19 or 22-18 or even 23-17 hcp, but the incridible facto is that the sum of possible tricks of each partnership is statistically determined by the sum of the fit of each partnership, considering the best card play by declarer and defense.

What does that means?
That means, for example, if E-W has a fit of nine cards (5-4 ou 6-3 ...) and N-S has a fit of eigth cards (4-4 ou 5-3 ...) then the number of total tricks possible is 9+8=17. Thus, with normal card play by defense and declarer N-S + E-W has a statistical hope to make 17 tricks. Not always, of course, they may have less 1 trick when not have all top honors trump in their fit. They may get 1 or 2 more tricks when there are double fit or in special cases when the hand has voids or 10+ trumps. 

Thus, playing in MP both Vul if E-W with a fit of 9 cards are going to make 9 tricks for 140 points and N-S with a fit of 8 cards bids 3, then N-S  should be doubled to avoid bad score for E-W, because N-S down 1 Vul not doubled will result for E-W only 100 points.

	KQ106    6 K743 KJ54		all Vul and the bidding goes: West    North  East     South  1      double   2      double*  3**    3 pass     pass   double  all pass -     lead J *shows 4 cards ** shows 6 cards North with singleton takes a risky and bids 3 with only 8 cards fit but was doubled and after a  hopeless play down 1 after giving: 2 Diamonds, 2 Clubs, and 1 Heart. As we can see E-W can make 9 tricks and N-S makes only 8 tricks for a total of 9+8=17 tricks.
92 AQ7532 A83 A10	N   W    E       S	J54    J109 Q54 Q986
	A873    K84 J109 732

But if the honor position in a finesse is changed?
This doesn't matter because if a specific king is in a good position or in bad position for a finesse it is a question in add one trick for a side or for other side. But the number of total tricks stay constant.
In the same way, if a suit has a king dubleton with a ace tripleton the sum of 3 tricks with a ruff by defense or declarer does not change the hand's total tricks.

How can we use this estatistical principle for take advantage?
In the same way we use the Milton Work points (A=4 K=3 Q=2 J=1) to evaluate hand's strength and determinate the safe level to play we can make inference of the trumps fit of both partnership to have the amount of the total tricks and consequently to find the safe level to bid.

Supposing both sides have 20 hcp and E-W has 9 cards fit and N-S has 8 cards fit, then total tricks are 9+8=17 tricks. IF we assume that trumps fit defines the safe level a partnership can play, then E-W with 9 cards fit can play at level 3 but N-S with 8 cards fit can play only at level 2.

In case N-S also has a fit of 9 cards then the total tricks are 9+9=18 and now both sides have great probabity in make 9 tricks playing at level 3. Thus, if no one are Vul and both sides will make score positive playing at level 3, after the more hirarchical suit is bidded at level 3 the other must defend in level 4 for down 1 doubled = 100 points, that is better than let opponents score at level 3. 

When opponent opens Vul in a major and partner nVul overcalls in a minor where we have 5 cards support with few points (5-6) but with a good distribution like singleton in other suit and 3 cards in their suit (5431), if the other opponent jumps to their major suit showing 4 cards support we should infere that total tricks are 10+9 = 19 tricks. Now using what the Law lays down we think: if they make 10 tricks at level 4 then we make 9 tricks at level 5 for down 2 nVul (620 x 300), but if they go to level 5 to make 11 tricks based on Total Tricks we will make only 8 tricks so we should not bid 6 because then we will down 4.

In bridge we should play for the best chance and of course there are moments that what is more probable doesn't happen, but if we repeat many times the same situation, playing for the best chance and using the Law, we will have more success than bad luck.

Supposing we are Vul with 10 hcp in 4414 then partner opens 1nt and opponent nVul interfere with long suit in 3, but we find a 8 cards fit in a major to bid game and advancer bids 5.
partner overcaller   you      advancer
  1nt          3       double      pass
   3        pass          4           5
pass     pass           ?

Partner pass for you take a decision. This is the moment we should use the Law of Total Tricks as a guideline to take decision.

They may have a fit of 9 or 10 cards and we have a fit of 8 cards.
In case the total tricks are 18 if we make 11 tricks then they will make only 7 tricks to go down 4 that doubled is 800 for us.
In case the total tricks are 17 if we make 11 tricks then they will make only 6 tricks to go down 5 that is doubled 1100 for us.
The Law of Total Tricks tell us that double is much better so why take a risk whitout any advantage?

But what statistical precision does it have?
Vernes makes analyzes in 340 hands played at world championships and observed that 1/3 of the hands obey the Law of Total Tricks. He also observed that 80% have a variation in 1 trick with a mean standard deviation of 0,93 per trick. Therefore this study of the Law needs some positive corrections of 1 trick when there are double fit 4-4 cards in their hands and 1 trick more when declarer's side has all top honors in their trump suit and also 2 more in case of a super double fit 4-5+ cards. 

That means you should always valorize your hand when it is a case of have double fit or a case in have all top honors.

Suppoging you have 54  AKQ5 109842 98 and the bidding is:
Operner Advancer You Overcaller
  1          pass        1     1
  2           3*         ?                 * 4 cards support
How you do avaliation using the Law? Well, your side have a fit of 8 trumps and they has a fit of 9 trumps so total tricks = 17? WRONG
You are in a case that need adjustment. You have a super double fit and you have trump control of top honors. In this case you must make an adjustment of 2 tricks and your fit goes to 10. Total tricks are now 19 or 20 if they also have a double fit.
Supposing total tricks = 19 then if you make 4 they will make 9 tricks in 3. So you must push then to 4 hoping for down 1 aconsidering that your side has more than 21 hcp, so bid 4 without thinking too much.

Books:
In 1981 Dick Paine and Joe Amsbury published "TNT and Competitive Bidding" to explain Total Number of Tricks (TNT) in bridge;
In 1992 Larry Cohen published "To Bid or Not to Bid" that was a best seller (eighth printing in 1995) and an advance in bridge for all readers;
In 1994 Larry Cohen published "Following the Law" a sequence of his 1992 book were he answered many questions about the Law;
In 2000 Paul Mendelson published "Practise Your Law of Total Tricks" that consider the Law of Total with adjustment of 1 trick mosty perfect.
In 2004 Anders Wirgren with Mike Lawrence published "I Fought the Law of Total Tricks" that consider the accuracy of the Law of Total Tricks good only 35% - 40% of the deals.

Of course, the Law need adjustment sometimes:
- less 1 trick when trump honors is not solid and the others suits are not well divided as observed in the bidding
- positve 1 more trick in cases of double-fit (4-4+) and even adjustment of more 2 tricks in the special cases of voids with 10+ cards fit.

So the Law works very well for situations of 7. 8 and 9 cards fit that are near 82% of the given cards accordingly to simulation by computer.

Can we have safety levels to play depending on our fit?
Vernes also established that besides the traditional safety level to play based on hcp we can use to determine the level of bidding where hcp has oscilation between 20-20/ 21-19 / 22-18 / 23-17 safety levels given by the distribution of the trumps. This means, excluding vulnerability (Vul against nVul), we can compete having less hcp without many risks:
- with   8 trumps at level 2,
- with   9 trumps at level 3,
- with 10 trumps at level 4.

LET'S SEE SOME HANDS PLAYED IN WORLD CHAMPIONISHPS WHERE THE LAW OF TOTAL TRICKS IS IGNORED BY HIGH LEVEL PLAYERS

There are a famous hand played in 1978 at New Orleans - World Pairs Olympiad where the brazilian partnership Marcelo Castelço Branco and Gabino Cintra won first position with the help of this board where the bidding was:
Branco Oponente Gabino Oponente
  1          2           2         4
?
No one Vul and Marcelo opens 1 with: AKQ86 10763 J8 A10 but opponents bidded 4 and Marcelo saw the singleton heart in partner's hand. Then he bids 4 even knowing that partner has only 3 cards. Thus he takes a risky decision in 4 that was doubled and could be down 1, but opponent makes a blunder in defense and he made 4x to win 590 points and also won the World Pairs Olympiad.

742 5 K1097              K8754 J1053  =====  9       A9842 ! W - E ! KQJ AQ       !    S    ! 65432 93       =====  QJ62 AKQ86    10763 J8 A10

The bidding was: NORTH    EAST   SOUTH   WEST                                1         2   2 4 4       double    pass         pass pass lead: A

After the lead of A the doubler plays A and Q. Marcelo plays Clubs for his Ace and ruff a Heart then he plays K discarding a Heart that was ruffed by the doubler (west). Now if west plays Clubs - that breaks dummy entry - Marcelo without a safety return to his hand will give another ruff for down 1, but the doubler plays a Heart and now Marcelo ruffs Heart in dummy and plays trump to strip trumps from West before enter in dummy to discard his last loser Heart.

The questions is: what Marcelo Branco should think for his decision?
In that time few players use the Law to take decision. Marcelo should use it to avoid a possible overbid and a very bad negociation. 

The answer is: N-S and E-W probably have only 8 trumps, so the Total Tricks are 16. If N-S makes 10 tricks then E-W will makes only 6 tricks! this is 4 dows. If N-S makes only 9 tricks then E-W makes only 7 tricks and so doubled will be 500 that is better than 420. The bad negociation was take a risk to make 10 tricks and let scape sure 500 points.

If Marcelo double and Gabino leads trump E-W will down 4 or 5 in this hand. Conclusion: the decision to bid 4 was a unwarranty risk ignoring the Law.
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2- In 1980 at Walkenburg in the World Team Olympiad, Gabino Cintra have to decide between to bid 5 or double 5, because he was Vul against nVul. The bidding was:

AQ762 A A763 1096       Vul against nVul

G.Britain Gabino G.Britain Branco
  pass         1          2         2
    3          3          4         4
    5           ?

Gabino bids 5, probably, seeing the singleton Clubs in Marcelo hands, he supposes that will make 5, but bad division in Hearts makes Gabino  down 1. The complete hand was:

K105 KQ9874  K54    8        J984     =====   3    3           ! W - E !   J10652 Q109    !     S    !   J82  AK543  =====   QJ72 AQ762           A A763 1096

Bidding NORTH  EAST    SOUTH   WEST pass       1          2 2         3 3 4 4         5 5 pass pass  pass

What shoul be the decision based in the guideline of Total Tricks?
South should infer 8 Spades trumps and 9 Clubs trumps given 17 Total Tricks. So if N-S makes 11 tricks then W-E will make only 6 tricks for down 4 in 5 that is 800. Of course the lead must be trump and when N-S takes the hand must play another trump, and after another trump. W-E will then lose 3 Spades more 2 Diamonds and 1 Heart.

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3-  In 1979 at Rio de Janeiro in Bermuda Bowl, the expert player Mike Passell from USA team has to make a decision:

J93 Q86 A743 A53 nVul contra Vul

Passell Belladonna Brachman Pitall
--            1            1            1
    2 3 pass   pass
     ?

Passel's strength was good for a simple support in 2, in first round he could cuebid in 2, but the facto was that in second round he bids 3 and USA down 2 losing 100 points while earn 100 points because 3 down 1. These 5 imps decide the Bermuda Bowl giving the vitory to Italy.

Passell should have thought that if his partner has singleton Clubs or has 6th cards suit he would probably bid 3. So their side has only 8 trumps and opponents also has 8 trumps for a 16 total tricks. Then who plays in level 3 with only 8 trumps have great probability ingoing down 1. He should pass.

A2 105 QJ86 KQ972  J93      =====   Q106 Q86     ! W - E !  KJ932  A743   !    S    !   K9  A53     =====   J86  K8754  A74 1052 104

Bidding NORTH  EAST     SOUTH   WEST Italy      USA        Italy      USA    1          1           1 2 3         pass        pass      3  all  pass

Larry Cohen in his book "To Bid or Not to Bid" emphasizes that in a competitive bidding NEVER OUTBID OPPONENT THAT GOES TO LEVEL 3 WITH 8 TRUMPS and ALWAYS OUTBID OPPONENT WHEN THERE ARE 18 TOTAL TRICKS.

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4) In his book "To Bid or Not to Bid" Larry Cohen shows his bad decision in not obey the Law or Total Tricks, in a interesting competitive bidding:

Q83 K1042 764 A94      nVul X nVul
Opponent Bergen Oponent   Cohen
1          2        double
4        double     pass          ?

Cohen first double is a indication that he has a major suit with less than 10 hcp having 4-3 or 4-4 in majors. The second double by Bergen shows a good hand with both majors and singleton Diamonds. So opponent have a fit of 9 cards in Diamonds and Bergen-Cohen have a fit of 8 cards in Hearts. Total Tricks are 17. If Bergen-Cohen makes 10 tricks in 4 then opponents will make only 7 tricks in 4, thus Cohen decision should be  pass, but he bids 4.
The complete deal was:

AJ52 A875 3   KQJ5  10976  =====   K4   QJ93   ! W - E !  6          AK8     !    S    !  QJ10952 87       =====  10632 Q83     K1042 764 A94

LEILÃO - no one Vul Bergen   EAST   Cohen  WEST 1         2      double    4 double    pass      4     all pass Cohen ignore the Law and down 1

Notify: The invitation of Bergen to bid a major shows a good hand that only lose 4 because trumps are 4-1 and Spades finesse does not work, but Cohen ignore the Law and was punished with down 1. If he pass and lead trump and then another trump E-W will lose at least 3 Clubs 1 Heart and 1 Spades for down 2 that will be a excellent score, but his suit was not good for play against possible trumps 4-1.

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 5) In 1979 at Bermuda BOWL in Rio de Janeiro, USA versus Italy, and Vito Pittala must decide: pass / double / 4

Vul against nVul what to bid with: K9732 94 AQ10 J82
do you pass, double or bid 4?.
                             Goldman Belladonna Soloway Pittala
pass       pass
                                  1 1 2          2
                                  3         pass          pass         ?

Vito Pittala bidded 4 that was against the Law because his side has only  8 trumps and if the other side has 9 trumps there are 8+9 =17 tricks so if they made 10 tricks in 4 opponent will make only 7 tricks for down 2 that is 200. If 4 down 1 then 3 makes 8 tricks for also down 1.
Lets see the complete deal:

108 KJ7 K9753 AQ9  AJ4        =====   Q65  A83       ! W - E !  Q10652  J2          !    S    !  864  K10643 =====  75  K9732 94     AQ10 J82

Bidding: West       North          East        South     Goldman Belladonna Soloway Pittala pass      pass   1             1 2          2   3            pass      pass 4 pass        pass pass lead 7

After the lead of 7 Belladona play for Spades 3-3 and didn't touch in trump suit, but then receives a ruff in Clubs for down 1. The problem for Pittala was the bid of 4 with a fit of 8 cards when 3 down 2 = 200 and 4 if makes is only 130. Pittala didn't obey the Law of Total Tricks and was punish for that.

Conclusion: Champions also overbidden for not obey the Law

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