Vacant Places

In the card game bridge, the law or principle of vacant places is a simple method for estimating the probable location of any particular card in the four hands. It can be used both to aid in a decision at the table and to derive the entire suit division probability table.

At the beginning of a deal, each of four hands comprises thirteen cards and one may say there are thirteen vacant places in each hand. The probability that a particular card lies in a particular hand is one-quarter, or 13/52, the proportion of vacant places in that hand. From the perspective of a player who sees one hand, the probable lie of a missing card in a particular one of the other hands is one-third. The principle of vacant places is a rule for updating those uniform probabilities as one learns about the deal during the auction and the play. Essentially, as the lies of some cards become known – especially as the entire distributions of some suits become known – the odds on location of any other particular card remain proportional to the dwindling numbers of unidentified cards in all hands, i.e. to the numbers of so-called vacant places.

The principle of vacant places follows from Conditional Probability theory, which is based on Bayes Theorem. For a good background to bridge probabilities, and vacant places in particular, see Kelsey; see also the Official Encyclopedia of Bridge

We are the declarer in a heart contract with trump suit combination Kxxx in dummy and AJxxx in hand (see figure). There are four heart cards missing, the queen and three spot cards or ♥Qxxx. We play small to the king as both opponents follow low and lead another small heart, ♥2. The last of the three spot cards appears on our right leaving one outstanding heart, the queen. Because no one would play the queen while holding a spot card too, we have learned nothing about the location of the queen directly, only the distribution of the three spot cards, one at left and two at right. At the moment of decision we can perform a vacant places calculation.

First, suppose we know nothing about the other suits, probably because the opponents did not bid. Then we know only the one small heart observed at left and the two observed at right. That leaves twelve "vacant places" where ♥Q may reside at left and eleven vacant places at right. If the queen lies in 12 of the 23 vacant places, at left, we win by playing the ace; the queen drops. In 11 of the 23 vacant places, we win by playing the jack and then the ace, dropping the queen at right on the next heart trick. Thus the odds in favor of playing the ace are 12 to 11; the ace is a slight favorite to win an extra trick, i.e. to win five tricks in hearts. The proportion 12/23 = 52.174% is exactly the probability that appears in standard catalogs of suit combinations.

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