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The Law of Total Tricks

Teymur Tahseen

The law of total tricks is much vaunted and has many fervent adherents. It is easy to state and understand and proves relevant to many bidding situations.

The Theory

We define the trick count to be the sum of the number of tricks North-South and East-West can make on a given hand.

The sum of the number of trumps for North-South and East-West is termed the trump count.

So, if both partnerships have eight-card fits and can make nine tricks each, the trump count is 16 and the trick count is 18.

The original Law of Total Tricks stated by Jean-Rene Vernes, which I shall call the weak law, states that the average trick count over all hands with the same trump count is approximately equal to the trump count.

The law most quoted, popularized by Larry Cohen, which I shall call the strong law, states that the trick count on any given hand is approximately equal to the trump count.

The Evidence

These statistics are lifted from I Fought the Law by Mike Lawrence.
The graph below shows average trick count for given trump counts plotted against trump count.
lol LOTT

For small trump counts, the Weak Law over-estimates the trick count. For large trump counts, the Weak Law under-estimates the trick count. The Weak Law is true.

The graph below shows the accuracy of the Strong Law.
lol LOTT

The accuracy of the Strong Law is decreasing with trick count. In other words, the Strong Law works better when the partnerships do not have big fits. The Strong Law is less true.

Applying the Strong Law

Suppose both sides have a fit. The opponents have shown an eight-card fit in hearts and neither pair is vulnerable. Partner has shown five diamonds and you have four-card support. In the pass-out seat, should you compete over two hearts?

The Strong Law says that you can expect there to be at least seventeen tricks. You can assume, since the opponents have stopped well short of game that the opponents will take at most nine tricks. In that case, three diamonds will go at most one off. If opponents are taking eight tricks in hearts, you expect to make three diamonds. If opponents are going one down in two hearts, you care expecting to make three diamonds. In each case, the diamond contract scores better, even if it is doubled when it goes off. The strong law thus implies that it is correct to bid three diamonds.

This is the typical line of reasoning when using the strong law. Make sure you understand it.

Adjustment Factors

Larry Cohen refined the Strong Law by identifying positive adjustments factor, which caused the Law to under-estimate the trick count and negative adjustment factors, which cause the Law to over-estimate the trick count. This table appears in To Bid or Not to Bid by Larry Cohen.

Positive FactorDescriptionNegative FactorDescription
Positive PurityNo minor honours in opponents' suits, good interiors in own suitsNegative PurityPoor interiors in own suits, minor honours in opponents' suits
Positive FitDouble/double fitNegative FitMisfits
Positive ShapeExtra length or voidsNegative ShapeFlats

These factors are used, possibly in only a qualitative manner, to determine the total number of tricks. Note that these factors are the same factors used to determine whether a hand is suitable for competing. This is logical. If there are more tricks available, bidding on makes more sense.