# Statistics

In preparing for our book, we did a lot of statistics by looking at thousands of hands in many different ways. But before us, Matthew Ginsberg had done a large study published in The Bridge World (May 1996). He had let his double dummy engine (the same that he has based his very strong computer program GIB on) analyze almost 450,000 deals in order to see how accurate the Law of Total Tricks really was.
The result of his study showed these things:
1. the average difference between total tricks and total trumps is 0.05
2. the average error per deal is 0.75
3. total tricks equals total trumps on 40% of the deals
4. on 46.9% of the deals there is a difference of one (up or down)
5. on 13.1% of the deals there is a difference of at least two (up or down)
From conclusion No. 1 we understand that Vernes was right in stating that "total trumps and total tricks are approximately equal", since the difference (0.05 tricks less than trumps on average) is so small.

From conclusion No. 2 we understand that on any given deal the average difference between trumps and tricks will be three quarter of a trick. Since on average there will an equality (almost), it follows that deals with extra tricks are balanced by deals with fewer tricks.

From conclusion No. 3 we see how often trumps and tricks really are equal. And the result, only 40%, is considerably less than one may expect given the incredible support the Law has got from players at all levels from all over the world.

From conclusion No. 4 we understand that deals where there is a difference of one (up or down) are more common than deals where there is an equality.

From conclusion No. 5 we learn that deals where there is a difference of at least two (up or down) will happen a little more than on every eighth deal. On a 32-board match, expect four such deals.

In deciding which trump suit was best and how many tricks the sides could take with each of the the four suits as trumps, Ginsberg specified that the contract was always played from the right side (for the declaring side).
Any bridge player knows the importance of this factor, but for some reason Vernes never mentioned it when he introduced the Law to the general public. It's easy to construct deals where North-South's best trump suit is, say, spades, but if North is declarer they take one, two or even three tricks more than if South is declarer. If that is the case: then, in counting the total tricks, how many should we count for North-South on such a deal? The correct answer is "It depends on who is the declarer", but Vernes and all his followers simply skipped the issue. As the Law is formulated, it is incomplete, and on many deals it can't give us an answer.
It was wise of Ginsberg not to repeat Vernes' omission, but when he also stated that "of two (or three) suits of the same length, the trump suit was picked randomly", he was no longer talking about total tricks. Why?
Because it is not unusual that if one side has two or more trumps suits of the same length, one of them will produce more tricks than the other. If we correct that – as we did ourselves in our statistics – the result will be that some of the deals which Ginsberg called -2 will instead be -1 or 0 (or even +1), some of his -1 deals will be 0 or more, some of his 0 deals will be +1 or more, etc. The effect of this correction is that Ginsberg's claim that there is a slight tendency towards fewer tricks than trumps is (a little bit) wrong. If we study total tricks (which assumes "best trump suit" for each side), there is instead a slight tendency towards more tricks than trumps. But only if the contract always is played from the right side.

In a way we like Ginsberg's approach, though. In real life it is not easy to realize when our eight diamonds are better than our eight spades, for instance, and since bridge is a game of errors, even at the very top, his "practical approach" appeal to us. But if he randomizes "the best suit", then why not randomize the declarer too. And if we do that, the result will be that everything move further to the left, i.e. that the tendency towards fewer tricks will be even stronger than in the study.
So, for practical purposes, to say that "The Law of Total Tricks" is right on 40% of the deals is a little too high. 35-37% will be closer to the mark. Or, put differently, slightly more often than every third deal.