Monday, January 15, 2007

Law of Symmetry: fact or fiction?

Basically the Law of Symmetry states that a singleton or void is usually not alone in a bridge hand. It claims that the frequencies of bad splits are increased. So, for example, when you have 9 trumps and a singleton somewhere, the law suggests you play for a 3-1 split since the probabilities have increased.

You probably wonder if this is true, or if it’s just rubbish and superstition. From now on, never forget this: it’s pure rubish! And there’s mathematical proof. Here it is:
- suppose you give North and South a 4333 distribution, give North 4 s, South 4 s.
- from the NS perspective, you have pretty normal chances of splits in all suits.
- now you put 2 s from the North hand to the South hand, creating a singleton. Put a and a from South to North.
- North now has a 4-1-5-3 and South has a 2-5-2-4.

Has anything changed to the existence of singletons? Yes.
Has anything changed to the probabilities from EW hands? No! They still own the exact same cards together, so the frequencies haven’t changed.
You can start with whatever hands you like, and the frequencies won’t change for opponent’s splits in the suits whenever you exchange some cards between North and South.

You might have the impression that when you have a freak distribution, you usually get bad splits. However, the reason why has nothing to do with the shortness you own. It’s about your long suits, the number of cards you control in another suit! For example, if you have a 10 card fit, it’s obvious someone will have a singleton. That however doesn’t change the frequency of 2-1 or 3-0 splits in the suit, no matter if you have a 5-5, 6-4, 7-3, 8-2, 9-1 or 10-0 fit.
To back this up: the more cards you have together with your partner in a suit, the more chance you get for opponents to have a singleton or void:
- 5 card fit: 7-1 or 8-0 split = 3,1%
- 6 card fit: 6-1 or 7-0 split = 7,3%
- 7 card fit: 5-1 or 6-0 split = 16,0%
- 8 card fit: 4-1 or 5-0 split = 32,17%
- 9 card fit: 3-1 or 4-0 split = 59,3%
- 10+ card fit: 100%
If you have monster fits, you have more chance to have some shortness yourself. And opponents have more chance to have some shortness as well. However, having a singleton or void doesn’t imply you have a monster fit with partner! So the law has no ground at all.

A final conclusion: a singleton or void in your or partner’s hand doesn’t increase the chance of a singleton or void in opponents’ hands. It’s your long suits that do the trick. But the absolute frequency of splits in a suit doesn’t change. So there’s no reason to choose a different line of play because of "the Law of Symmetry".


Ashok said...

This "law" was proposed in the early days of bridge. My guess is that at that time this was observable because of some weirdness owing to inadequate (hand) shuffling. If that's so, people should expect it to hold in the majority of clubs where people lazily riffle once or twice before dealing.

Barry Rogoff said...

Symmetry Revisited

By Barry Rogoff

It has long been theorized that in any given bridge deal, suit patterns tend to match hand patterns. In other words, if you’re looking at a hand that is divided 5-4-3-1, then it’s likely that one of the four suits is distributed 5-4-3-1 around the table. Conventional bridge wisdom, however, asserts that there is no relationship between suit and hand patterns.

The truth is that the theory has never been proved or disproved. According to probability theory, in any random deal each of the possible hand patterns has a specific probability of occurrence, balanced hands being the most common. Suit patterns, therefore, have the same probabilities of occurrence as hand patterns. Each type of pattern is simply a different way of looking at how 52 cards are distributed into four groups of 13.

The crux of the argument, then, is simply this:

The frequency distributions of hand patterns and suit patterns are not random. Therefore, the frequency distribution of hand patterns that match suit patterns is also not random and may be interesting.

It would seem that the theoretical result can be derived mathematically based on probability theory. Not being a mathematician, however, I have been forced to use a less sophisticated proof: a computer simulation: a simple Java program that generates pseudorandom deals and tallies the number of matching hand and suit patterns in each deal.

The result of the simulation is indeed interesting. The percentages of hand-suit matches in a run of one million deals are:

0 6.92%
1 22.91%
2 40.17%
4 30.01%

As you can see, more than 70 per cent of all bridge hands have at least two matching hand and suit patterns. Only about seven per cent have no matching patterns.

So why does this matter? Once in a while, you have to play for one of several possible layouts based on an inferential count. By counting the number of matching patterns in each layout, you can get a clue as to which is the most likely.

For example, suppose that you have to choose between two layouts, one with zero matches and one with two matches. Everything else being equal, the two-match layout is almost seven times more likely than the zero-match layout.

Anyone who would like to run a longer simulation or check the code for correctness is welcome to do so. Email me ( and I’ll send you the Java source code. And should the theory ever come up at your table, I’d be delighted to hear about it.