The Law of Total Tricks was first described by Jean Rene Vernes in his 1969 Bridge World article and popularized by Marty Bergen and Larry Cohen in his To Bid or Not to Bid. To summarize it very briefly, on any given deal, the sum of the tricks each side can take in their best suit as trumps equals the sum of the cards each side holds in its best suit. The "Law" says nothing about how the tricks will be divided between the two sides.
This article describes a corollary to the "Law" which is not well known - how to apply the "Law" to estimate the trick-taking potential of one side on its own.
n = t + (h–20)/2
where n is the number of tricks that any one side can take; and t is the total number of trumps they hold; and h is the total number of high cards they hold (using the standard Work count of 4321). Obviously if N is the "Total Tricks" (N=n1+n2) and T is the "Total Trumps" (T=t1+t2) then we have N = T (which is the "Law").
The important and simple features of the formula are:
The increment of two points per trick is surprisingly accurate and is borne out by the double-dummy analyses of Matthew L. Ginsberg. The expectation of tricks at no trump is as follows:
|
HCP |
20 |
21 |
22 |
23 |
24 |
25 |
26 |
27 |
28 |
29 |
30 |
|
Tricks |
6.1 |
6.7 |
7.2 |
7.6 |
8.2 |
8.7 |
9.1 |
9.7 |
10.1 |
10.6 |
11.1 |
This correspondence is essentially independent of the number of trumps. Beyond that range, the accuracy suffers, mostly I believe because the defenders have a big advantage being on lead. Declarer may have 12 easy tricks for a slam, but if the defenders get their two first, he's down. Further, as hands become more distributional, there is a greater likelihood of wastage (see Variations... below).
The formula above is indeed fairly accurate for non-distributional hands up to the game level. Beyond that, some adjustments need to be made to retain the accuracy. The following adjustments work well:
The formula now becomes:
n = t + (h'–19)/2
where h' is the adjusted HCP count.
Obviously, the formula has an elegance and simplicity that commends itself highly. But does it work? I have tested it on 300,000 deals – comparing the predicted results with the results of a double-dummy solver (many thanks to Matt Ginsberg for his double-dummy-solved dataset of deals) and the results are surprisingly good! In fact, the mean error is -0.02 tricks and the standard deviation is 1.22*, which compares reasonably with the "Law" accuracy (mean: 0.126, standard deviation 1.039). * Note that if we were dealing with a (continuous) normal distribution, a standard deviation of 1.22 would imply that 60% of all observations would be between -1 and +1.
The improved formula has a mean error of +0.14 (i.e. it is pessimistic) and a standard deviation of 1.17.
One of the reasons it is so useful in competitive situations (like "The Law") is that you don't always want to know what you can most likely make - but rather how many they can make and how many you will go down doubled if you outbid them. The HCP adjustment of the formula gives you a better handle on what you can expect to make (and therefore what they can). Of course, half of the time, you will end up with a fractional estimate of trick taking ability. You should round up or down as you see fit, depending on the scoring method, the level of the contract, whether competitive, the abilities of your opponents and so on.
How good is the formula at the table? It doesn't replace bridge judgment of course but it is easy to apply and very often you know how many cards your partner holds in a particular suit because of the auction so far. In competitive situations, it is very good, I believe. It is slightly pessimistic for part scores, which gives you a margin of error. How good is it for judging whether to bid slams and games? It is very accurate for game-level contracts and a little optimistic for slams (thereby urging a little extra caution when bidding slams). In a MP competition where the opponents are quiet and where you can more or less reasonably expect all other pairs to be faced with the same decision (to bid or not to bid) then what you care about most is what is most likely to make. Here the formula is excellent, although you have to remember that the formula gets less accurate (on both sides of the estimate) as the total trumps increase [this doesn't show up in the mean/std.dev figures but it is true to an extent – see below in law details].
However, at total points or IMPs, or when there is interference or a field of mixed abilities, then more care must be used. You are not so much concerned with what is likely to make, as you are with maximizing your score. For example, in a competitive IMPs situation you might bid on to 4S over 4H even though you expect both sides can likely only make 9 tricks – because it can be very expensive if you are wrong – as bad as a double-game swing. Or if you are thinking of bidding a grand slam and it appears that you can make "14" tricks by the formula, it still may make sense to bid only a small slam because if the formula is off by 3 (possible, though unlikely) it can be very expensive indeed, since the grand slam bonus is relatively small compared to the risk.
What about No-trump contracts? Consider that the total trumps held by your side is 61/2 and the formula will work out fine. For example, with 21 HCP, you can expect to make 1NT (actually, you should make about two-thirds of the time, otherwise you are down). With 25 expect to make 3NT (again you should make most of the time). Remember that the scoring table favors bidding games, especially when vulnerable. Of course, we know that long suits are helpful at NT contracts but generally most useful when the long suit is a minor, because of the scoring table. The formula doesn't take into account length of suits for NT play, so it is not particularly good for judging whether you have enough high cards to make 3NT with, say, an eight-bagger in clubs.
An upward adjustment should typically be made for double fits. Two long suits in the same hand (say a ten-bagger and a nine-bagger) are worth a half to a whole extra trick (for example 201/2 or 21 total tricks rather than the 20 predicted for each side having a 10 card fit). A downward adjustment of say a half of a trick might also be appropriate when your short suit(s) are doubletons.
Of course, as always, non-working honors tend to make your hand more defensively oriented, as they tend to reduce the total number of tricks available on the deal – you should already have accounted for these in the adjusted formula.
Here are some examples:
Non-vul vs vul at MPs, RHO deals and it is passed round to partner who opens 1H. You're looking at A952-KJT-43-KT62 and after RHO passes again, bid 1S. LHO now bids 2D and partner rebids 3C. Pass to you. Partner apparently has a nice hand probably 54xx in shape and let's guess 12-17 HCP. You've got 11 HCP and it looks like two eight card fits. The straight formula gives at least 8+11/2+1/2=10 and maybe 8+4+1/2=121/2 tricks. It looks like all our values are working well. On the other hand, your doubleton in the opponents suit is not a great holding (you'd prefer a singleton or even three) so perhaps you take that half of a trick off again. Slam looks possibly but by no means certain (even game may not be cold) and you have no really good way to show interest anyway. You jump to 4H and partner (who has 15 HCP, but with a fair amount of duplication with you) wraps up 10 tricks, just like everybody else.
All vulnerable, MPs, partner deals and opens 1S. RHO passes, and you pass with Q7-7653-9-KJ7632. LHO reopens with 2D and partner bids 2H. RHO passes and you raise to 3H. LHO bids 4D and it comes back to you. Now what? You have 6 HCP and partner hopefully has a good 14-17 HCP and some shape (at least 5-4, hopefully 5-5). I very much doubt if he has the 17+ HCP and the extra H or he might have bid 4 himself. In any case, your SQ looks good and but the CK – isn't worth full value – it looks like partner may be short in clubs (RHO hasn't ever raised the D bid). We have an eight card fit and that's it. So the available tricks should be 71/2 to 9. LHO is bidding quite strongly so it looks like a decent suit, probably seven cards. Let's guess that the total tricks on the deal is 17 (18 if partner has an extra heart). If we can make 8 then they're down one or making. If we can make 9, they're down 1 or 2. Should you bid 4? No way! Bidding four (you aren't doubled) gives you 37% for -100 (16% if you are doubled), passing gives you 45% or 85% (!) and doubling gives you 85% or 94%! It turns out that there are 17.5 total tricks (we always make 9, they make 8 or 9 about equally).
Nobody vulnerable, IMPs. LHO deals and passes. Partner opens 1S and RHO messes things up with 3C. You have KJ7-T6542-AKJ94-void so you're quite happy. Perhaps a 4C bid but you're haven't discussed splinters and anyway, would it be a splinter? Hell, bid 4S and make sure we don't miss game! LHO now joins the fun with 5C. Partner bids a fairly fast 5H (yes, I know you're not supposed to infer anything from his speed) so it looks like he's at least 5-4 in shape. Don't know if he's got more than a minimum or is just showing shape. Still he's unlikely to have only 10 or 11 HCP. So let's see we've got a double fit of 8/9, 9/9 or 9/10. That should be good for 91/2 to101/2 trick base. Let's put his HCP range at 12 to 15 (much more and he'd probably cue 6C or something) so that's 24 to 27 HCP total for the good guys. So our minimum tricks should be 111/2 and the max 14. However, we're counting a lot for a suit partner may be well be short in – let's say the range is 11 to 13. Well now, slam would be nice. You don't have very good trumps – partner better had! Hope he doesn't have too much wastage in clubs. Since this is a competitive auction and it looks like 19 or 20 total tricks, we should get some IMPs just for outbidding the opponents so 5H looks good. 5C doubled is no longer an option. 12 tricks might be there but it's not odds on and since this is a non-vulnerable slam, let's be conservative. Making 5 for a 10 IMP gain. 6H goes down at the other table.
A few details follow for the total tricks available for particular distributions (these don't involve HCP counts - but of course any implied adjustments to your thinking are the same):
This page was contributed by Robin Hillyard
(spider on OKBridge, Beowulf on BBO).
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