The principle of restricted choice is a guideline used in card games such as contract bridge to intuit hidden information.
The principle helps other players infer the locations of unobserved equivalent cards such as that spade ace after observing the king.
The increase or decrease in probability is an example of Bayesian updating as evidence accumulates and particular applications of restricted choice are similar to the Monty Hall problem.
Restricted choice is always introduced in terms of two touching cards – consecutive ranks in the same suit, such as ♥QJ or ♦KQ – where equivalence is manifest.
The probability calculations in coverage of restricted choice often take uniform randomization for granted but that is problematic.
There are four spade cards ♠8754 in the South (closed hand) and five ♠AJ1096 in the North (dummy, visible to all players).
Prior to play, 16 different West and East spade holdings or "lies" are possible from the perspective of South.
These are listed in Table 1, ordered first by "split" from equal to unequal numbers of cards, then by West's holding from strongest to weakest.
Later, after winning a side-suit trick, South leads another small spade and West follows low with the ♠3 (or ♠2).
South is at a decision point and knows that only two of the original 16 lies remain possible (bolded in Table 1), for West has played both low cards and East the king.
However, the principle of restricted choice tells us that while both lies of the cards are possible, the probabilities are 2:1 in favour of assuming West holds Q32 and to therefore play the ten.
During the course of many equivalent deals, East with ♠KQ should in theory win the first trick with the king or queen uniformly at random; that is, half each without any pattern.
The last column gives the a priori probability of any specific original holding such as 32 and KQ; that one is represented by row one covering the 2–2 split.
The other lie featured in our example play of the spade suit, Q32 and K, is represented by row two covering the 3–1 split.
Thus the table shows that the a priori odds on these two specific lies were not even but slightly in favor of the former, about 6.78 to 6.22 for ♠KQ against ♠K.
What are the odds a posteriori, at the moment of truth in our example play of the spade suit?
The principle of restricted choice is general but this specific probability calculation does suppose East would win with the king from ♠KQ precisely half the time (which is best).
The principle of restricted choice is an application of Bayes' theorem on conditional probability.
Further, based on the play to trick 1, only two of the original 16 (i.e., a priori) possible holdings shown in Table 1 above remain available for East, each equally possible.
Increases and decreases in the probabilities of original lies of the opposing cards, as the play of the hand proceeds, are examples of Bayesian updating as evidence accumulates.