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".