LAW OF SIMMETRY, does that exist?

When Declarer has 9 cards fit (AKJxx - 10xxx or AKJxxx - 10xx) missing only the Queen, if interference not happen, we play first the Ace and if Queen don't apears we go to the other side and play low to KJxx and if a low card apears we must decide: King (for Queen's drop) / Jack (finesse).
The technical way to decide between finesse or drop is using the math of probabilities that says: chance of finesse is 50% and chance for drop is 53,14%.
However following the Ely Culbertson's pseudo law of simmetry if in our hand or dummey there is a singleton we should finesse! 

Thus the question is: does the probability's calculus be influenced by the "Law of Simmetry" restrited by the distribution of the 52 cards?

A player that already played many hands of bridge have noticed the strange  frequency that some unbalanced division of the 4 suits for a player corresponds to another unbalanced division for other player. A void correspond other void, a singleton correspond other singleton.

Those situations are mostly often considered mere coincidences without being given due importance and without being considered valid elements of reference.

Despites mathematically we can't validate these inferences they occur with a meaning frequence that advance players prefer to use then for unbalenced hands.

So, the "Law of Simmetry" may be considered a guideline for infere the probable card distribution of opponents based in our card distribution.

Of course, the "Law of Simmetry" is just an additional tools with the math of probabilities that allows us to make a decision in the played of a hand when there are no bidding informations.

The Law of Simmetry can be summarized in two statements:

a) To a balanced hand usually corresponds another balanced hand and to a unbalanced hand usually corresponds another unbalanced hand with the same degree of balance.

b) A balanced hand usually corresponds to a suit divided with same balance between the four players, while an unbalanced hand corresponds a suit with unbalanced distribution between the players (this suit unbalanced usually belong to the player that had the more long suit)

According this pseudo law there is a correlation between types of hands distribution and the suit distribution among all players, so if a player has a distribution of cards 5422 it is a good chance that another player also have a identical hand, or one of the four suits have the same distribution 5422.

Thus this law statement that from the character of his own hand pattern a player can draw inferences concerning the pattern of other hands and of the distribution of the four suits.

To better illustrate this concept lets look for each hand in the diagram below where it is possible to set up a table and observe the perfect correspondence between the division of each player's cards and the distribution of the suits.  

Let's see the follow diagram in this hand:

                         KQx
                         AKJxx
                         xx
                         Vxx
 Ax              =======   xxx
Qxx           =     N      =   x
 Axxxxx      = W    E   =  Jx
 xx              =     S      =  AQxxxxx
                     =======
                       Jxxxx
                       xxxx
                       KQx
                        K

that can be represented in this table:

              types of hand
  NORTH SOUTH   WEST EAST
Distribution
of the
suits		 3 5 2 3
2 3 6 2
5 4 3 1
3 1 2 7


So in this table we clearly observe:
- the distribution 5332 in Spades is the same of the North cards;
- the distribution 5431 in Hearts is the same of the South cards; 
- the distribution 6322 in Diamonds is the same of the West cards;
- the distribution 7321 in Clubs is the same of the East cards.

It is also easy to observe that the success in a contract of 4 or 4 by N-S dependds on essentially in how the Hearts will be played.

If the general rule to play AK with 9 trumps (AKJxx-xxxx) be follow here based in the math of probabilities that presume, after the play of the Ace, that the drop of the Queen has chance of 52,12% and so the King should be played, this contract will not be done.

But who follows the pseudo Law of Simmetry, after noticed the singleton in Clubs, make the finesse supposing a correspondent singleton in Hearts.

Of course this argumentation has no math validation and Ely Culberson (1891-1955) knowing that had used his own experience in boards played  to became a defender of this psedo Law of Simmetry as also was Easley R. Blackwood (1903-1992) that used the rule: if our short suit of 4 cards is 2-2 then play for drop, but if it is 3-1 play finesse, or if our shorter suit of 5 cards is divided 3-2 play for drop but if it is divided 4-1 play finesse.

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